From 194fd0dfd8ae7df0a4c2a8319eead24ef79be54b Mon Sep 17 00:00:00 2001
From: "David A. Ehrlich" <davidalexander.ehrlich@uni-goettingen.de>
Date: Thu, 18 Jan 2024 10:22:46 +0100
Subject: [PATCH 01/34] Fix: Corrected unsorting in fdr test

---
 idtxl/stats.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 5c2fc21..8174e70 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -297,7 +297,7 @@ def _perform_fdr_corretion(pval, constant, alpha):
     if np.invert(sign).any():
         first_false = np.where(np.invert(sign))[0][0]
         sign[first_false:] = False  # avoids false positives due to equal pvals
-    sign = sign[sort_idx]  # restore original ordering of significance values
+    sign[sort_idx] = sign.copy() # restore original ordering of significance values
     return sign, thresh
 
 
@@ -687,8 +687,8 @@ def max_statistic_sequential(analysis_setup, data):
             break
 
     # Get back original order and return results.
-    significance = significance[selected_vars_order]
-    pvalue = pvalue[selected_vars_order]
+    significance[selected_vars_order] = significance.copy()
+    pvalue[selected_vars_order] = pvalue.copy()
     return significance, pvalue, individual_stat
 
 

From a0e735e0d3b6020f1c46ef3e20da9883f3d42102 Mon Sep 17 00:00:00 2001
From: "David A. Ehrlich" <davidalexander.ehrlich@uni-goettingen.de>
Date: Thu, 18 Jan 2024 12:17:57 +0100
Subject: [PATCH 02/34] Fix: Removed filtering by omnibus significance in fdr
 correction

---
 idtxl/stats.py | 32 ++++++++++++++++++--------------
 1 file changed, 18 insertions(+), 14 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 8174e70..76e4c18 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -163,22 +163,26 @@ def network_fdr(settings=None, *results):
     cands = []
     if correct_by_target:  # whole target
         for target in results_comb.targets_analysed:
-            if results_comb._single_target[target].omnibus_sign:
-                pval = np.append(pval, results_comb._single_target[target].omnibus_pval)
-                target_idx = np.append(target_idx, target)
-                n_perm = np.append(n_perm, results_comb.settings.n_perm_omnibus)
+            next_pval = results_comb._single_target[target].omnibus_pval
+            pval = np.append(
+                pval, next_pval if next_pval is not None else 1)
+            target_idx = np.append(target_idx, target)
+            n_perm = np.append(
+                    n_perm, results_comb.settings.n_perm_omnibus)
     else:  # individual variables
         for target in results_comb.targets_analysed:
-            if results_comb._single_target[target].omnibus_sign:
-                n_sign = results_comb._single_target[target].selected_sources_pval.size
-                pval = np.append(
-                    pval, (results_comb._single_target[target].selected_sources_pval)
-                )
-                target_idx = np.append(target_idx, np.ones(n_sign) * target).astype(int)
-                cands = cands + (
-                    results_comb._single_target[target].selected_vars_sources
-                )
-                n_perm = np.append(n_perm, results_comb.settings.n_perm_max_seq)
+            n_sign = (results_comb._single_target[target].
+                        selected_sources_pval.size)
+            pval = np.append(
+                pval, (results_comb._single_target[target].
+                        selected_sources_pval))
+            target_idx = np.append(target_idx,
+                                    np.ones(n_sign) * target).astype(int)
+            cands = (cands +
+                        (results_comb._single_target[target].
+                        selected_vars_sources))
+            n_perm = np.append(
+                n_perm, results_comb.settings.n_perm_max_seq)
 
     if pval.size == 0:
         print("No links in final results ...")

From 3eb63049fa6d3013b6b8a05a5e9e36cacf5b0ed5 Mon Sep 17 00:00:00 2001
From: "David A. Ehrlich" <davidalexander.ehrlich@uni-goettingen.de>
Date: Thu, 25 Jan 2024 17:20:29 +0100
Subject: [PATCH 03/34] Fix: Removed filtering in fdr correction for AIS

---
 idtxl/stats.py | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 76e4c18..e54bff0 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -59,10 +59,10 @@ def ais_fdr(settings=None, *results):
     process_idx = np.arange(0).astype(int)
     n_perm = np.arange(0).astype(int)
     for process in results_comb.processes_analysed:
-        if results_comb._single_process[process].ais_sign:
-            pval = np.append(pval, results_comb._single_process[process].ais_pval)
-            process_idx = np.append(process_idx, process)
-            n_perm = np.append(n_perm, results_comb.settings.n_perm_mi)
+        next_pval = results_comb._single_process[process].ais_pval
+        pval = np.append(pval, next_pval if next_pval is not None else 1)
+        process_idx = np.append(process_idx, process)
+        n_perm = np.append(n_perm, results_comb.settings.n_perm_mi)
 
     if pval.size == 0:
         print("FDR correction: no links in final results ...\n")

From 1330787f30250d95ed13f734972025084b057ea5 Mon Sep 17 00:00:00 2001
From: "David A. Ehrlich" <davidalexander.ehrlich@uni-goettingen.de>
Date: Thu, 25 Jan 2024 17:25:50 +0100
Subject: [PATCH 04/34] Fix: Correctly compute the total number of tests for
 correct_by_target==False

---
 idtxl/stats.py | 35 +++++++++++++++++++++++------------
 1 file changed, 23 insertions(+), 12 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index e54bff0..d1467b0 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -69,7 +69,7 @@ def ais_fdr(settings=None, *results):
         results_comb._add_fdr(fdr=None, alpha=alpha, constant=constant)
         return results_comb
 
-    sign, thresh = _perform_fdr_corretion(pval, constant, alpha)
+    sign, thresh = _perform_fdr_corretion(pval, constant, alpha, len(results_comb.processes_analysed))
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
@@ -162,6 +162,9 @@ def network_fdr(settings=None, *results):
     n_perm = np.arange(0).astype(int)
     cands = []
     if correct_by_target:  # whole target
+        # The total number of tests is the number of targets
+        n_tests = len(results_comb.targets_analysed)
+
         for target in results_comb.targets_analysed:
             next_pval = results_comb._single_target[target].omnibus_pval
             pval = np.append(
@@ -170,12 +173,17 @@ def network_fdr(settings=None, *results):
             n_perm = np.append(
                     n_perm, results_comb.settings.n_perm_omnibus)
     else:  # individual variables
+        # The total number of tests is the number of targets times the number
+        # of source candidates (i.e. source processes * time lags) analyzed for each target
+        n_tests = np.sum(len(target.source_set) for target in results.targets_analysed) \
+                * (settings["max_lag_sources"] - settings["min_lag_sources"] + 1)
+
         for target in results_comb.targets_analysed:
             n_sign = (results_comb._single_target[target].
                         selected_sources_pval.size)
             pval = np.append(
-                pval, (results_comb._single_target[target].
-                        selected_sources_pval))
+                pval, [next_pval if next_pval is not None else 1 for next_pval in
+                       results_comb._single_target[target].selected_sources_pval])
             target_idx = np.append(target_idx,
                                     np.ones(n_sign) * target).astype(int)
             cands = (cands +
@@ -194,7 +202,7 @@ def network_fdr(settings=None, *results):
         )
         return results_comb
 
-    sign, thresh = _perform_fdr_corretion(pval, constant, alpha)
+    sign, thresh = _perform_fdr_corretion(pval, constant, alpha, n_tests)
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
@@ -247,7 +255,7 @@ def network_fdr(settings=None, *results):
     return results_comb
 
 
-def _perform_fdr_corretion(pval, constant, alpha):
+def _perform_fdr_corretion(pval, constant, alpha, n_tests):
     """Calculate sequential threshold for FDR-correction.
 
     Calculate sequential thresholds for FDR-correction of p-values. The
@@ -270,31 +278,34 @@ def _perform_fdr_corretion(pval, constant, alpha):
             according to Genovese (2002): 1 will divide alpha by 1, 2 will
             divide alpha by the sum_i(1/i); see the paper for details on the
             assumptions (default=2)
+        n_tests : int
+            total number of tests performed for calculating the FDR-thresholds
 
     Returns:
         array of bools
-            significance of p-values
+            significance of p-values in the order of the input array
         array of floats
-            FDR-thresholds for each p-value
+            FDR-thresholds for each p-value in increasing order
     """
     # Sort all p-values in ascending order.
     sort_idx = np.argsort(pval)
     pval.sort()
 
     # Calculate threshold
-    n = pval.size
     if constant == 2:  # pick the requested constant (see Genovese, p.872)
-        if n < 1000:
-            const = sum(1 / np.arange(1, n + 1))
+        if n_tests < 1000:
+            const = np.sum(1 / np.arange(1, n_tests + 1))
         else:
-            const = np.log(n) + np.e  # aprx. harmonic sum with Euler's number
+            const = np.log(n_tests) + np.e  # aprx. harmonic sum with Euler's number
     elif constant == 1:
         # This is less strict than the other one and corresponds to a
         # Bonoferroni-correction for the first p-value, however, it makes more
         # strict assumptions on the distribution of p-values, while constant 2
         # works for any joint distribution of the p-values.
         const = 1
-    thresh = (np.arange(1, n + 1) / n) * alpha / const
+
+    # Calculate threshold for each p-value.
+    thresh = (np.arange(1, len(pval) + 1) / n_tests) * alpha / const
 
     # Compare data to threshold.
     sign = pval <= thresh

From 84c88ff0ab0ed04b185f9bf702a6e84d7b0259d3 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Sat, 27 Jan 2024 16:54:33 +0100
Subject: [PATCH 05/34] Fix input checks in multivariate network inference

---
 idtxl/multivariate_mi.py | 18 ++++++++----------
 idtxl/multivariate_te.py | 26 ++++++++++----------------
 2 files changed, 18 insertions(+), 26 deletions(-)

diff --git a/idtxl/multivariate_mi.py b/idtxl/multivariate_mi.py
index 26d709b..0c4c244 100644
--- a/idtxl/multivariate_mi.py
+++ b/idtxl/multivariate_mi.py
@@ -7,9 +7,9 @@
 Note:
     Written for Python 3.4+
 """
-from .stats import network_fdr
 from .network_inference import NetworkInferenceMI, NetworkInferenceMultivariate
 from .results import ResultsNetworkInference
+from .stats import network_fdr
 
 
 class MultivariateMI(NetworkInferenceMI, NetworkInferenceMultivariate):
@@ -129,20 +129,18 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
 
         # Check which targets and sources are requested for analysis.
         if targets == "all":
-            targets = [t for t in range(data.n_processes)]
+            targets = list(range(data.n_processes))
         if sources == "all":
             sources = ["all" for t in targets]
-        if (type(sources) is list) and (type(sources[0]) is int):
+        elif isinstance(sources, list) and isinstance(sources[0], int):
             sources = [sources for t in targets]
-        if (type(sources) is list) and (type(sources[0]) is list):
+        elif isinstance(sources, list) and isinstance(sources[0], list):
             pass
         else:
-            ValueError("Sources was not specified correctly: {0}.".format(sources))
-        assert len(sources) == len(targets), (
-            "List of targets and list of "
-            "sources have to have the same "
-            "same length"
-        )
+            raise ValueError(f"Sources was not specified correctly: {sources}.")
+        assert len(sources) == len(
+            targets
+        ), "List of targets and list of sources have to have the same length"
 
         # Check and set defaults for checkpointing.
         settings = self._set_checkpointing_defaults(settings, data, sources, targets)
diff --git a/idtxl/multivariate_te.py b/idtxl/multivariate_te.py
index db8198a..b727b3f 100755
--- a/idtxl/multivariate_te.py
+++ b/idtxl/multivariate_te.py
@@ -7,9 +7,9 @@
 Note:
     Written for Python 3.4+
 """
-from .network_inference import NetworkInferenceTE, NetworkInferenceMultivariate
-from .stats import network_fdr
+from .network_inference import NetworkInferenceMultivariate, NetworkInferenceTE
 from .results import ResultsNetworkInference
+from .stats import network_fdr
 
 
 class MultivariateTE(NetworkInferenceTE, NetworkInferenceMultivariate):
@@ -132,20 +132,18 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
 
         # Check which targets and sources are requested for analysis.
         if targets == "all":
-            targets = [t for t in range(data.n_processes)]
+            targets = list(range(data.n_processes))
         if sources == "all":
             sources = ["all" for t in targets]
-        if (type(sources) is list) and (type(sources[0]) is int):
+        elif isinstance(sources is list) and isinstance(sources[0], int):
             sources = [sources for t in targets]
-        if (type(sources) is list) and (type(sources[0]) is list):
+        elif isinstance(sources, list) and isinstance(sources[0], list):
             pass
         else:
-            ValueError("Sources was not specified correctly: {0}.".format(sources))
-        assert len(sources) == len(targets), (
-            "List of targets and list of "
-            "sources have to have the same "
-            "same length"
-        )
+            raise ValueError(f"Sources was not specified correctly: {sources}.")
+        assert len(sources) == len(
+            targets
+        ), "List of targets and list of sources have to have the same length"
 
         # Check and set defaults for checkpointing.
         settings = self._set_checkpointing_defaults(settings, data, sources, targets)
@@ -158,11 +156,7 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
         )
         for t in range(len(targets)):
             if settings["verbose"]:
-                print(
-                    "\n####### analysing target with index {0} from list {1}".format(
-                        t, targets
-                    )
-                )
+                print(f"\n####### analysing target with index {t} from list {targets}")
             res_single = self.analyse_single_target(
                 settings, data, targets[t], sources[t]
             )

From e1ac96bab0b58357b984c9320b9ceac8913153e9 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Sat, 27 Jan 2024 17:00:05 +0100
Subject: [PATCH 06/34] Update version number

---
 docs/conf.py | 2 +-
 setup.py     | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/conf.py b/docs/conf.py
index 838f6e6..1611d4d 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -71,7 +71,7 @@
 # built documents.
 #
 # The short X.Y version.
-version = "1.5.1"
+version = "1.6.0"
 # The full version, including alpha/beta/rc tags.
 release = version
 
diff --git a/setup.py b/setup.py
index 301f05e..6197c07 100644
--- a/setup.py
+++ b/setup.py
@@ -8,7 +8,7 @@
     name="idtxl",
     packages=["idtxl"],
     include_package_data=True,
-    version="1.5.1",
+    version="1.6",
     description="Information Dynamics Toolkit xl",
     author="Patricia Wollstadt, Joseph T. Lizier, Raul Vicente, Conor Finn, Mario Martinez-Zarzuela, Pedro Mediano, Leonardo Novelli, Michael Wibral",
     author_email="p.wollstadt@gmail.com",

From cb959689878ae38c0ee407ba1480e37e1d0df4d6 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 08:06:45 +0100
Subject: [PATCH 07/34] Fix calculation of no. tests for candidate-wise FDR

Correctly calculate the number of tests when running FDR per candidate
test, i.e., with correct_by_target=False.

Fix formatting.
---
 idtxl/stats.py | 68 +++++++++++++++++++++++++-------------------------
 1 file changed, 34 insertions(+), 34 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index d1467b0..1169032 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -1,8 +1,11 @@
 """Provide statistics functions."""
+# pylint: disable=protected-access
 import copy as cp
+
 import numpy as np
-from . import idtxl_utils as utils
+
 from . import idtxl_exceptions as ex
+from . import idtxl_utils as utils
 
 
 def ais_fdr(settings=None, *results):
@@ -69,18 +72,16 @@ def ais_fdr(settings=None, *results):
         results_comb._add_fdr(fdr=None, alpha=alpha, constant=constant)
         return results_comb
 
-    sign, thresh = _perform_fdr_corretion(pval, constant, alpha, len(results_comb.processes_analysed))
+    sign, thresh = _perform_fdr_corretion(
+        pval, constant, alpha, len(results_comb.processes_analysed)
+    )
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
     if (1 / min(n_perm)) > thresh[0]:
         print(
-            "WARNING: Number of permutations ("
-            "n_perm_max_seq"
-            ") for at "
-            "least one target is too low to allow for FDR correction "
-            "(FDR-threshold: {0:.4f}, min. theoretically possible p-value: "
-            "{1}).".format(thresh[0], 1 / min(n_perm))
+            "WARNING: Number of permutations ('n_perm_max_seq') for at least one target is too low to allow for "
+            f"FDR correction (FDR-threshold: {thresh[0]:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
         )
         results_comb._add_fdr(fdr=None, alpha=alpha, constant=constant)
         return results_comb
@@ -167,30 +168,33 @@ def network_fdr(settings=None, *results):
 
         for target in results_comb.targets_analysed:
             next_pval = results_comb._single_target[target].omnibus_pval
-            pval = np.append(
-                pval, next_pval if next_pval is not None else 1)
+            pval = np.append(pval, next_pval if next_pval is not None else 1)
             target_idx = np.append(target_idx, target)
-            n_perm = np.append(
-                    n_perm, results_comb.settings.n_perm_omnibus)
+            n_perm = np.append(n_perm, results_comb.settings.n_perm_omnibus)
     else:  # individual variables
         # The total number of tests is the number of targets times the number
         # of source candidates (i.e. source processes * time lags) analyzed for each target
-        n_tests = np.sum(len(target.source_set) for target in results.targets_analysed) \
-                * (settings["max_lag_sources"] - settings["min_lag_sources"] + 1)
+        n_tests = np.sum(
+            len(results_comb._single_target[target].sources_tested)
+            for target in results_comb.targets_analysed
+        ) * (settings["max_lag_sources"] - settings["min_lag_sources"] + 1)
 
         for target in results_comb.targets_analysed:
-            n_sign = (results_comb._single_target[target].
-                        selected_sources_pval.size)
+            if results_comb._single_target[target].selected_sources_pval is None:
+                continue
+            n_sign = results_comb._single_target[target].selected_sources_pval.size
             pval = np.append(
-                pval, [next_pval if next_pval is not None else 1 for next_pval in
-                       results_comb._single_target[target].selected_sources_pval])
-            target_idx = np.append(target_idx,
-                                    np.ones(n_sign) * target).astype(int)
-            cands = (cands +
-                        (results_comb._single_target[target].
-                        selected_vars_sources))
-            n_perm = np.append(
-                n_perm, results_comb.settings.n_perm_max_seq)
+                pval,
+                [
+                    next_pval if next_pval is not None else 1
+                    for next_pval in results_comb._single_target[
+                        target
+                    ].selected_sources_pval
+                ],
+            )
+            target_idx = np.append(target_idx, np.ones(n_sign) * target).astype(int)
+            cands = cands + (results_comb._single_target[target].selected_vars_sources)
+            n_perm = np.append(n_perm, results_comb.settings.n_perm_max_seq)
 
     if pval.size == 0:
         print("No links in final results ...")
@@ -208,12 +212,8 @@ def network_fdr(settings=None, *results):
     # variables is too low, return without performing any correction.
     if (1 / min(n_perm)) > thresh[0]:
         print(
-            "WARNING: Number of permutations ("
-            "n_perm_max_seq"
-            ") for at "
-            "least one target is too low to allow for FDR correction "
-            "(FDR-threshold: {0:.4f}, min. theoretically possible p-value: "
-            "{1}).".format(thresh[0], 1 / min(n_perm))
+            "WARNING: Number of permutations ('n_perm_max_seq') for at least one target is too low to allow for "
+            f"FDR correction (FDR-threshold: {thresh[0]:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
         )
         results_comb._add_fdr(
             fdr=None,
@@ -243,7 +243,7 @@ def network_fdr(settings=None, *results):
                 t = target_idx[s]
                 cand = cands[s]
                 cand_ind = fdr[t].selected_vars_sources.index(cand)
-                fdr[t].selected_vars_sources.pop(cand_ind)
+                fdr[t].selected_vars_sources.remove(cand_ind)
                 fdr[t].selected_sources_pval = np.delete(
                     fdr[t].selected_sources_pval, cand_ind
                 )
@@ -312,7 +312,7 @@ def _perform_fdr_corretion(pval, constant, alpha, n_tests):
     if np.invert(sign).any():
         first_false = np.where(np.invert(sign))[0][0]
         sign[first_false:] = False  # avoids false positives due to equal pvals
-    sign[sort_idx] = sign.copy() # restore original ordering of significance values
+    sign[sort_idx] = sign.copy()  # restore original ordering of significance values
     return sign, thresh
 
 
@@ -347,7 +347,7 @@ def omnibus_test(analysis_setup, data):
         float
             the test's p-value
         float
-            the estimated test statisic, i.e., the information transfer from
+            the estimated test statistic, i.e., the information transfer from
             all sources into the target
 
     Raises:

From 490a3de85d08b18ac21236b6d0d598ebdd1eac62 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 08:52:57 +0100
Subject: [PATCH 08/34] Refactor removal of non-significant FDR results

---
 idtxl/stats.py | 13 +++++--------
 1 file changed, 5 insertions(+), 8 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 1169032..f7b7ec6 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -227,10 +227,9 @@ def network_fdr(settings=None, *results):
     # the results object. Create a copy of the results object to leave the
     # original intact.
     fdr = cp.deepcopy(results_comb._single_target)
-    for s in range(sign.shape[0]):
-        if not sign[s]:
+    for i, t in enumerate(target_idx):
+        if not sign[i]:
             if correct_by_target:
-                t = target_idx[s]
                 fdr[t].selected_vars_full = cp.deepcopy(
                     results_comb._single_target[t].selected_vars_target
                 )
@@ -240,17 +239,15 @@ def network_fdr(settings=None, *results):
                 fdr[t].omnibus_pval = 1
                 fdr[t].omnibus_sign = False
             else:
-                t = target_idx[s]
-                cand = cands[s]
-                cand_ind = fdr[t].selected_vars_sources.index(cand)
-                fdr[t].selected_vars_sources.remove(cand_ind)
+                cand_ind = fdr[t].selected_vars_sources.index(cands[i])
+                fdr[t].selected_vars_sources.remove(cands[i])
                 fdr[t].selected_sources_pval = np.delete(
                     fdr[t].selected_sources_pval, cand_ind
                 )
                 fdr[t].selected_sources_te = np.delete(
                     fdr[t].selected_sources_te, cand_ind
                 )
-                fdr[t].selected_vars_full.pop(fdr[t].selected_vars_full.index(cand))
+                fdr[t].selected_vars_full.remove(cands[i])
     results_comb._add_fdr(fdr, alpha, correct_by_target, constant)
     return results_comb
 

From b82945d6bd027006bfd83e1a054ee4478455dbc0 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 08:53:35 +0100
Subject: [PATCH 09/34] Fix future warning for using np.sum on generator

---
 idtxl/stats.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index f7b7ec6..133df2e 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -174,7 +174,7 @@ def network_fdr(settings=None, *results):
     else:  # individual variables
         # The total number of tests is the number of targets times the number
         # of source candidates (i.e. source processes * time lags) analyzed for each target
-        n_tests = np.sum(
+        n_tests = sum(
             len(results_comb._single_target[target].sources_tested)
             for target in results_comb.targets_analysed
         ) * (settings["max_lag_sources"] - settings["min_lag_sources"] + 1)

From 47c93d2d084073f0596690caccf0ada78fe95a5f Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 08:55:19 +0100
Subject: [PATCH 10/34] Fix tests for network FDR

---
 test/test_stats.py | 148 ++++++++++++++++++++++++++++++++-------------
 1 file changed, 106 insertions(+), 42 deletions(-)

diff --git a/test/test_stats.py b/test/test_stats.py
index 6eb51ed..e85d62f 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -1,13 +1,15 @@
 """Unit tests for stats module."""
-import pytest
+# pylint: disable=protected-access
 import numpy as np
+import pytest
+from test_estimators_jidt import _get_gauss_data
+
 from idtxl import stats
-from idtxl.multivariate_te import MultivariateTE
 from idtxl.active_information_storage import ActiveInformationStorage
-from idtxl.estimators_jidt import JidtDiscreteCMI
 from idtxl.data import Data
+from idtxl.estimators_jidt import JidtDiscreteCMI
+from idtxl.multivariate_te import MultivariateTE
 from idtxl.results import ResultsNetworkInference, ResultsSingleProcessAnalysis
-from test_estimators_jidt import _get_gauss_data
 
 
 def test_omnibus_test():
@@ -53,48 +55,52 @@ def test_max_statistic_sequential():
 
 
 def test_network_fdr():
-    settings = {"n_perm_max_seq": 1000, "n_perm_omnibus": 1000}
+    # Simulate results for a 3-node network, analyzed all-to-all. Set the
+    # omnibus p-value for targets 1 and 2 such that they does not survive
+    # FDR-correction.
+    settings_fdr = {"n_perm_max_seq": 2000, "n_perm_omnibus": 2000}
     target_0 = {
         "selected_vars_sources": [(1, 1), (1, 2), (1, 3), (2, 1), (2, 0)],
-        "selected_vars_full": [
-            (0, 1),
-            (0, 2),
-            (0, 3),
-            (1, 1),
-            (1, 2),
-            (1, 3),
-            (2, 1),
-            (2, 0),
-        ],
+        "selected_vars_target": [(0, 1)],
+        "sources_tested": [1, 2],
         "omnibus_pval": 0.0001,
         "omnibus_sign": True,
         "selected_sources_pval": np.array([0.001, 0.0014, 0.01, 0.045, 0.047]),
         "selected_sources_te": np.array([1.1, 1.0, 0.8, 0.7, 0.63]),
     }
+    target_0["selected_vars_full"] = (
+        target_0["selected_vars_sources"] + target_0["selected_vars_target"]
+    )
     target_1 = {
         "selected_vars_sources": [(1, 2), (2, 1), (2, 2)],
-        "selected_vars_full": [(1, 0), (1, 1), (1, 2), (2, 1), (2, 2)],
+        "selected_vars_target": [(1, 1)],
+        "sources_tested": [0, 2],
         "omnibus_pval": 0.031,
         "omnibus_sign": True,
         "selected_sources_pval": np.array([0.00001, 0.00014, 0.01]),
         "selected_sources_te": np.array([1.8, 1.75, 0.75]),
     }
+    target_1["selected_vars_full"] = (
+        target_1["selected_vars_sources"] + target_1["selected_vars_target"]
+    )
     target_2 = {
         "selected_vars_sources": [],
-        "selected_vars_full": [(2, 0), (2, 1)],
+        "selected_vars_target": [],
+        "selected_vars_full": [],
+        "sources_tested": [0, 1],
         "omnibus_pval": 0.41,
         "omnibus_sign": False,
         "selected_sources_pval": None,
         "selected_sources_te": np.array([]),
     }
     res_1 = ResultsNetworkInference(n_nodes=3, n_realisations=1000, normalised=True)
-    res_1._add_single_result(target=0, settings=settings, results=target_0)
-    res_1._add_single_result(target=1, settings=settings, results=target_1)
+    res_1._add_single_result(target=0, settings=settings_fdr, results=target_0)
+    res_1._add_single_result(target=1, settings=settings_fdr, results=target_1)
     res_2 = ResultsNetworkInference(n_nodes=3, n_realisations=1000, normalised=True)
-    res_2._add_single_result(target=2, settings=settings, results=target_2)
+    res_2._add_single_result(target=2, settings=settings_fdr, results=target_2)
 
     for correct_by_target in [True, False]:
-        settings = {
+        settings_fdr = {
             "cmi_estimator": "JidtKraskovCMI",
             "alpha_fdr": 0.05,
             "max_lag_sources": 3,
@@ -106,37 +112,94 @@ def test_network_fdr():
         data.generate_mute_data(n_samples=100, n_replications=3)
         analysis_setup = MultivariateTE()
         analysis_setup._initialise(
-            settings=settings, data=data, sources=[1, 2], target=0
+            settings=settings_fdr, data=data, sources=[1, 2], target=0
         )
-        res_pruned = stats.network_fdr(settings, res_1, res_2)
-        assert not res_pruned._single_target[
-            2
-        ].selected_vars_sources, "Target 2 has not been pruned from results."
-
-        for k in res_pruned.targets_analysed:
-            if res_pruned._single_target[k]["selected_sources_pval"] is None:
-                assert not res_pruned._single_target[k]["selected_vars_sources"]
-            else:
+        res_pruned = stats.network_fdr(settings_fdr, res_1, res_2)
+        if correct_by_target:
+            assert not res_pruned.get_single_target(1, fdr=True)[
+                "omnibus_sign"
+            ], "Target 1 has not been pruned from results."
+            assert not res_pruned.get_single_target(2, fdr=True)[
+                "omnibus_sign"
+            ], "Target 2 has not been pruned from results."
+            assert res_pruned.get_single_target(0, fdr=True)[
+                "omnibus_sign"
+            ], "Target 0 has been wrongly pruned from results."
+
+            # Ensure non-sign results were correctly removed after FDR correction.
+            for t in res_pruned.targets_analysed:
+                if not res_pruned.get_single_target(t, fdr=True)["omnibus_sign"]:
+                    assert not res_pruned._single_target_fdr[t]["selected_vars_sources"]
+                    assert (
+                        res_pruned._single_target_fdr[t]["selected_sources_te"] is None
+                    )
+                    assert (
+                        res_pruned._single_target_fdr[t]["selected_sources_pval"]
+                        is None
+                    )
+                    assert res_pruned._single_target_fdr[t]["omnibus_pval"] == 1
+                    assert not res_pruned._single_target_fdr[t]["omnibus_sign"]
+                else:
+                    assert len(
+                        res_pruned._single_target[t]["selected_vars_sources"]
+                    ) == len(
+                        res_pruned._single_target[t]["selected_sources_pval"]
+                    ), "Source list and list of p-values should have the same length."
+                    assert len(
+                        res_pruned._single_target[t]["selected_vars_sources"]
+                    ) == len(
+                        res_pruned._single_target_fdr[t]["selected_vars_sources"]
+                    ), "Corrected and uncorrected source list should have the same length."
+                    assert len(
+                        res_pruned._single_target[t]["selected_vars_target"]
+                    ) == len(
+                        res_pruned._single_target_fdr[t]["selected_vars_target"]
+                    ), "Corrected and uncorrected source list should have the same length."
+        else:
+            for t in [0, 1]:  # ensure correct output for FDR-corrected targets
                 assert len(
-                    res_pruned._single_target[k]["selected_vars_sources"]
-                ) == len(res_pruned._single_target[k]["selected_sources_pval"]), (
-                    "Source list and list of p-values should have " "the same length."
-                )
-
+                    res_pruned.get_single_target(t, fdr=True)["selected_vars_sources"]
+                ) < len(
+                    res_pruned.get_single_target(t, fdr=False)["selected_vars_sources"]
+                ), f"Non-sign sources for target {t} have not been pruned from results."
+                assert len(
+                    res_pruned._single_target_fdr[t]["selected_vars_sources"]
+                ) == len(res_pruned._single_target_fdr[t]["selected_sources_pval"])
+                assert len(
+                    res_pruned._single_target_fdr[t]["selected_vars_sources"]
+                ) == len(res_pruned._single_target_fdr[t]["selected_sources_te"])
     # Test function call for single result
-    res_pruned = stats.network_fdr(settings, res_1)
+    res_pruned = stats.network_fdr(settings_fdr, res_1)
     print("successful call on single result dict.")
 
-    # Test None result for insufficient no. permutations, no FDR-corrected
-    # results (the results class throws an error if no FDR-corrected results
-    # exist).
+    # Ensure that no FDR correction is performed if the no. permutations is
+    # insufficient (the results class throws an error if no FDR-corrected
+    # results exist).
+    settings_fdr["correct_by_target"] = True
+    res_1.settings["n_perm_omnibus"] = 2
+    res_2.settings["n_perm_omnibus"] = 2
+    res_pruned = stats.network_fdr(settings_fdr, res_1, res_2)
+    for target in [0, 1, 2]:
+        with pytest.raises(RuntimeError):
+            res_pruned.get_single_target(target, fdr=True)
+    with pytest.raises(RuntimeError):
+        res_pruned.get_adjacency_matrix("binary", fdr=True)
+    settings_fdr["correct_by_target"] = False
     res_1.settings["n_perm_max_seq"] = 2
     res_2.settings["n_perm_max_seq"] = 2
-    res_pruned = stats.network_fdr(settings, res_1, res_2)
+    res_pruned = stats.network_fdr(settings_fdr, res_1, res_2)
+    for target in [0, 1, 2]:
+        with pytest.raises(RuntimeError):
+            res_pruned.get_single_target(target, fdr=True)
     with pytest.raises(RuntimeError):
         res_pruned.get_adjacency_matrix("binary", fdr=True)
 
 
+def test_fdr_sorting():
+    # Test for correct ordering of p-vals and sign after _perform_fdr_correction
+    pass
+
+
 def test_ais_fdr():
     settings = {"n_perm_max_seq": 1000, "n_perm_mi": 1000}
     process_0 = {
@@ -354,10 +417,11 @@ def test_analytical_surrogates():
 
 
 if __name__ == "__main__":
-    test_ais_fdr()
+    # test_ais_fdr()
     # test_analytical_surrogates()
     # test_data_type()
     # test_network_fdr()
+    test_fdr_sorting()
     # test_find_pvalue()
     # test_find_table_max()
     # test_find_table_min()

From af7d90f2bc7db1bdd7124024be1c4b53b9f3312f Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 08:57:06 +0100
Subject: [PATCH 11/34] Add log output for seq. max. stats

---
 idtxl/network_inference.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/idtxl/network_inference.py b/idtxl/network_inference.py
index 80b829f..7f888b3 100755
--- a/idtxl/network_inference.py
+++ b/idtxl/network_inference.py
@@ -1,8 +1,9 @@
 """Parent class for all network inference."""
 import numpy as np
-from .network_analysis import NetworkAnalysis
-from . import stats
+
 from . import idtxl_exceptions as ex
+from . import stats
+from .network_analysis import NetworkAnalysis
 
 
 class NetworkInference(NetworkAnalysis):
@@ -235,6 +236,7 @@ def _force_conditionals(self, cond, data):
     def _remove_non_significant(self, s, p, stat):
         # Remove non-significant sources from the candidate set. Loop
         # backwards over the candidates to remove them iteratively.
+        print(f"removing {np.sum(np.invert(s))} variables after seq. max stats")
         for i in range(s.shape[0] - 1, -1, -1):
             if not s[i]:
                 self._remove_selected_var(self.selected_vars_sources[i])

From bba010087245004bc0655f460b066559a7604daa Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 09:22:52 +0100
Subject: [PATCH 12/34] Create a copy of p-values in FDR correction

By creating a copy, p-values are only sorted within the function and not
also in the calling function. In-place sorting does not create a copy,
while np.sort() does.
---
 idtxl/stats.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 133df2e..e7ad956 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -270,11 +270,11 @@ def _perform_fdr_corretion(pval, constant, alpha, n_tests):
             p-values to be corrected
         alpha : float
             critical alpha level
-        fdr_constant : int
+        constant : int
             one of two constants used for calculating the FDR-thresholds
             according to Genovese (2002): 1 will divide alpha by 1, 2 will
             divide alpha by the sum_i(1/i); see the paper for details on the
-            assumptions (default=2)
+            assumptions
         n_tests : int
             total number of tests performed for calculating the FDR-thresholds
 
@@ -286,7 +286,7 @@ def _perform_fdr_corretion(pval, constant, alpha, n_tests):
     """
     # Sort all p-values in ascending order.
     sort_idx = np.argsort(pval)
-    pval.sort()
+    pval_sorted = np.sort(pval)
 
     # Calculate threshold
     if constant == 2:  # pick the requested constant (see Genovese, p.872)
@@ -302,10 +302,10 @@ def _perform_fdr_corretion(pval, constant, alpha, n_tests):
         const = 1
 
     # Calculate threshold for each p-value.
-    thresh = (np.arange(1, len(pval) + 1) / n_tests) * alpha / const
+    thresh = (np.arange(1, len(pval_sorted) + 1) / n_tests) * alpha / const
 
     # Compare data to threshold.
-    sign = pval <= thresh
+    sign = pval_sorted <= thresh
     if np.invert(sign).any():
         first_false = np.where(np.invert(sign))[0][0]
         sign[first_false:] = False  # avoids false positives due to equal pvals

From 9253d7c86308d76ee07b25c72d14ad354f9f38e6 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 09:25:35 +0100
Subject: [PATCH 13/34] Add test for correct sorting in FDR stats

---
 test/test_stats.py | 17 +++++++++++++++--
 1 file changed, 15 insertions(+), 2 deletions(-)

diff --git a/test/test_stats.py b/test/test_stats.py
index e85d62f..ec70116 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -197,7 +197,20 @@ def test_network_fdr():
 
 def test_fdr_sorting():
     # Test for correct ordering of p-vals and sign after _perform_fdr_correction
-    pass
+    n_tests = 8
+    pval_sorted = np.arange(n_tests) / 300
+    constant = 2
+    alpha = 0.05
+    sign, _ = stats._perform_fdr_corretion(pval_sorted, constant, alpha, n_tests)
+    assert sign[:3].all()
+
+    permutation = np.random.permutation(n_tests)
+    pval_unsorted = pval_sorted[permutation].copy()
+    sign_unsorted, _ = stats._perform_fdr_corretion(
+        pval_unsorted, constant, alpha, n_tests
+    )
+    assert sign_unsorted.sum() == 3
+    assert np.array_equal(pval_sorted[sign], sorted(pval_unsorted[sign_unsorted]))
 
 
 def test_ais_fdr():
@@ -430,4 +443,4 @@ def test_analytical_surrogates():
     # test_omnibus_test()
     # test_max_statistic()
     # test_min_statistic()
-    test_max_statistic_sequential()
+    # test_max_statistic_sequential()

From 8abe8a000c166e29ea212c492490655d24ef6dc5 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:18:33 +0100
Subject: [PATCH 14/34] Fix formatting of error message

---
 idtxl/data_spiketime.py    | 12 +++++++-----
 idtxl/estimators_Rudelt.py | 18 ++++++++++--------
 test/test_fast_emb.py      |  7 +++++--
 3 files changed, 22 insertions(+), 15 deletions(-)

diff --git a/idtxl/data_spiketime.py b/idtxl/data_spiketime.py
index 2358685..691258c 100644
--- a/idtxl/data_spiketime.py
+++ b/idtxl/data_spiketime.py
@@ -1,10 +1,12 @@
 """Provide spiketime data structures for IDTxl analysis."""
 
-import numpy as np
+import os
 from sys import stderr
+
+import numpy as np
 from scipy.optimize import newton
+
 import idtxl.hde_utils as utl
-import os
 
 VERBOSE = False
 
@@ -15,9 +17,9 @@
     FAST_EMBEDDING_AVAILABLE = False
     print(
         """
-    Error importing Cython fast embedding module for HDE estimator.\n
-    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,\n
-    this may take a long time. Other estimators are not affected.\n
+    Error importing Cython fast embedding module for HDE estimator.
+    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,
+    this may take a long time. Other estimators are not affected.
     """,
         file=stderr,
         flush=True,
diff --git a/idtxl/estimators_Rudelt.py b/idtxl/estimators_Rudelt.py
index 69c22c6..f293578 100644
--- a/idtxl/estimators_Rudelt.py
+++ b/idtxl/estimators_Rudelt.py
@@ -1,14 +1,16 @@
 """Provide HDE estimators."""
 
 import logging
-import numpy as np
-from scipy.optimize import newton, minimize
 import sys
-from sys import stderr
-from idtxl.estimator import Estimator
-import idtxl.hde_utils as utl
 from collections import Counter
+from sys import stderr
+
 import mpmath as mp
+import numpy as np
+from scipy.optimize import minimize, newton
+
+import idtxl.hde_utils as utl
+from idtxl.estimator import Estimator
 
 FAST_EMBEDDING_AVAILABLE = True
 try:
@@ -17,9 +19,9 @@
     FAST_EMBEDDING_AVAILABLE = False
     print(
         """
-    Error importing Cython fast embedding module for HDE estimator.\n
-    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,\n
-    this may take a long time. Other estimators are not affected.\n
+    Error importing Cython fast embedding module for HDE estimator.
+    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,
+    this may take a long time. Other estimators are not affected.
     """,
         file=stderr,
         flush=True,
diff --git a/test/test_fast_emb.py b/test/test_fast_emb.py
index 1936273..9c81893 100644
--- a/test/test_fast_emb.py
+++ b/test/test_fast_emb.py
@@ -13,7 +13,10 @@
 
 try:
     import idtxl.hde_fast_embedding as fast_emb
+
     print("\nFast embedding tools are available.")
 except:
-    print("Error importing Cython fast embedding module. \n Please try the following command in a linux "
-          "terminal in the IDTxl/idtxl folder: \n>> python3 setup_hde_fast_embedding.py build_ext --inplace")
\ No newline at end of file
+    print(
+        "Error importing Cython fast embedding module. \n Please try the following command in a linux "
+        "terminal in the IDTxl/idtxl folder: \n>> python3 setup_hde_fast_embedding.py build_ext --inplace"
+    )

From fe4956c40da044618c6bd0dfcb7018ff481d16e2 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:18:57 +0100
Subject: [PATCH 15/34] Fix formatting and pylint errors

---
 idtxl/data.py | 37 +++++++++++++++++--------------------
 1 file changed, 17 insertions(+), 20 deletions(-)

diff --git a/idtxl/data.py b/idtxl/data.py
index 533f6fa..f658e01 100755
--- a/idtxl/data.py
+++ b/idtxl/data.py
@@ -1,5 +1,6 @@
 """Provide data structures for IDTxl analysis."""
 import numpy as np
+
 from . import idtxl_utils as utils
 
 VERBOSE = False
@@ -145,7 +146,7 @@ def set_data(self, data, dim_order):
                 must have the same length as number of dimensions in data
         """
         if len(dim_order) > 3:
-            raise RuntimeError("dim_order can not have more than three " "entries")
+            raise RuntimeError("dim_order can not have more than three entries")
         if len(dim_order) != data.ndim:
             raise RuntimeError(
                 f"Data array dimension ({data.ndim}) and length of "
@@ -157,8 +158,8 @@ def set_data(self, data, dim_order):
         data_ordered = self._reorder_data(data, dim_order)
         self._set_data_size(data_ordered)
         print(
-            "Adding data with properties: {0} processes, {1} samples, {2} "
-            "replications".format(self.n_processes, self.n_samples, self.n_replications)
+            f"Adding data with properties: {self.n_processes} processes, {self.n_samples} "
+            f"samples, {self.n_replications} replications"
         )
         try:
             delattr(self, "data")
@@ -256,7 +257,7 @@ def get_realisations(self, current_value, idx_list, shuffle=False):
                 samples * no.replications) x number of indices
         """
         if not hasattr(self, "data"):
-            raise AttributeError("No data has been added to this Data() " "instance.")
+            raise AttributeError("No data has been added to this Data() instance.")
         # Return None if index list is empty.
         if not idx_list:
             return None, None
@@ -296,12 +297,11 @@ def get_realisations(self, current_value, idx_list, shuffle=False):
                     realisations[r : r + n_real_time, i] = self.data[
                         idx[0], idx[1] : last_sample, replication
                     ]
-                except IndexError:
+                except IndexError as e:
                     raise IndexError(
-                        "You tried to access variable {0} in a "
-                        "data set with {1} processes and {2} "
-                        "samples.".format(idx, self.n_processes, self.n_samples)
-                    )
+                        f"You tried to access variable {idx} in a data set with {self.n_processes} "
+                        f"processes and {self.n_samples} samples."
+                    ) from e
                 r += n_real_time
 
             assert not np.isnan(
@@ -378,14 +378,11 @@ def _get_data_slice(self, process, offset_samples=0, shuffle=False):
 
         try:
             data_slice = self.data[process, offset_samples:, replication_index]
-        except IndexError:
+        except IndexError as e:
             raise IndexError(
-                "You tried to access process {0} with an offset "
-                "of {1} in a data set of {2} processes and {3} "
-                "samples.".format(
-                    process, offset_samples, self.n_processes, self.n_samples
-                )
-            )
+                "You tried to access process {process} with an offset of {offset_samples} in a data set "
+                "of {self.n_processes} processes and {self.n_samples} samples."
+            ) from e
         assert not np.isnan(
             data_slice
         ).any(), "There are nans in the retrieved data slice."
@@ -706,22 +703,22 @@ def _get_permutation_samples(self, n_samples, perm_settings):
         elif perm_type == "circular":
             max_shift = perm_settings["max_shift"]
             if not isinstance(max_shift, int) or max_shift < 1:
-                raise TypeError(" " "max_shift" " has to be an int > 0.")
+                raise TypeError("'max_shift' has to be an int > 0.")
             perm = self._circular_shift(n_samples, max_shift)[0]
 
         elif perm_type == "block":
             block_size = perm_settings["block_size"]
             perm_range = perm_settings["perm_range"]
             if not isinstance(block_size, int) or block_size < 1:
-                raise TypeError(" " "block_size" " has to be an int > 0.")
+                raise TypeError("'block_size' has to be an int > 0.")
             if not isinstance(perm_range, int) or perm_range < 1:
-                raise TypeError(" " "perm_range" " has to be an int > 0.")
+                raise TypeError("'perm_range' has to be an int > 0.")
             perm = self._swap_blocks(n_samples, block_size, perm_range)
 
         elif perm_type == "local":
             perm_range = perm_settings["perm_range"]
             if not isinstance(perm_range, int) or perm_range < 1:
-                raise TypeError(" " "perm_range" " has to be an int > 0.")
+                raise TypeError("'perm_range' has to be an int > 0.")
             perm = self._swap_local(n_samples, perm_range)
 
         else:

From 312f587ae83f7c5a60ec3ce22ae2041d8e2acf48 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:19:26 +0100
Subject: [PATCH 16/34] Fix unsorting and surrogate calculation in bivariate
 sequential max stats

Fix unsorting of values per source, and unsorting of variables across
source processes. Fix calculation of surrogate values per variable.

For bivariate sequential max stats, variables are tested per source
process. When returning TE values, p-values, and significance
information, take care to sort values in the same order as tested source
variables in the analysis object.

The surrogate statistics were not correctly calculated for every
candidate.

Fix formatting
---
 idtxl/stats.py | 221 ++++++++++++++++++++-----------------------------
 1 file changed, 89 insertions(+), 132 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index e7ad956..9d26a71 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -547,8 +547,7 @@ def max_statistic_sequential(analysis_setup, data):
 
     if analysis_setup.settings["verbose"]:
         print(
-            "sequential maximum statistic, n_perm: {0}, testing {1} selected"
-            " sources".format(n_permutations, len(analysis_setup.selected_vars_sources))
+            f"sequential maximum statistic, n_perm: {n_permutations}, testing {len(analysis_setup.selected_vars_sources)} selected sources"
         )
 
     assert analysis_setup.selected_vars_sources, "No sources to test."
@@ -637,11 +636,9 @@ def max_statistic_sequential(analysis_setup, data):
                 #  we'll terminate the max sequential stats test,
                 #  and declare all not significant
                 print(
-                    "AlgorithmExhaustedError encountered in estimations: {}.".format(
-                        aee.message
-                    )
+                    f"AlgorithmExhaustedError encountered in estimations: {aee.message}. "
+                    "Stopping sequential max stats at candidate with rank 0"
                 )
-                print("Stopping sequential max stats at candidate with rank 0")
                 return (
                     np.zeros(len(analysis_setup.selected_vars_sources)).astype(bool),
                     np.ones(len(analysis_setup.selected_vars_sources)),
@@ -662,11 +659,9 @@ def max_statistic_sequential(analysis_setup, data):
         #  we'll terminate the max sequential stats test,
         #  and declare all not significant
         print(
-            "AlgorithmExhaustedError encountered in estimations: {}.".format(
-                aee.message
-            )
+            f"AlgorithmExhaustedError encountered in estimations: {aee.message}. "
+            "Stopping sequential max stats at candidate with rank 0"
         )
-        print("Stopping sequential max stats at candidate with rank 0")
         # For now we don't need a stack trace:
         # traceback.print_tb(aee.__traceback__)
         # Return (signficance, pvalue, TEs):
@@ -692,10 +687,7 @@ def max_statistic_sequential(analysis_setup, data):
         pvalue[c] = p
         if not s:  # break as soon as a candidate is no longer significant
             if analysis_setup.settings["verbose"]:
-                print(
-                    "\nStopping sequential max stats at candidate with rank "
-                    "{0}.".format(c)
-                )
+                print(f"\nStopping sequential max stats at candidate with rank {c}.")
             break
 
     # Get back original order and return results.
@@ -765,8 +757,7 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
 
     if analysis_setup.settings["verbose"]:
         print(
-            "sequential maximum statistic, n_perm: {0}, testing {1} selected"
-            " sources".format(n_permutations, len(analysis_setup.selected_vars_sources))
+            f"sequential maximum statistic, n_perm: {n_permutations}, testing {len(analysis_setup.selected_vars_sources)} selected sources"
         )
 
     assert analysis_setup.selected_vars_sources, "No sources to test."
@@ -788,12 +779,17 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
     )
     significance = np.zeros(len(analysis_setup.selected_vars_sources)).astype(bool)
     pvalue = np.ones(len(analysis_setup.selected_vars_sources))
-    stat = np.zeros(len(analysis_setup.selected_vars_sources))
+    individual_stat = np.zeros(len(analysis_setup.selected_vars_sources))
     for source in significant_sources:
         # Find selected past variables for current source
         source_vars = [
             s for s in analysis_setup.selected_vars_sources if s[0] == source
         ]
+        source_vars_idx = [
+            i
+            for i, s in enumerate(analysis_setup.selected_vars_sources)
+            if s[0] == source
+        ]
 
         # Determine length of conditioning set and allocate memory.
         idx_conditional = source_vars.copy()
@@ -825,8 +821,8 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             temp_cand = data.get_realisations(
                 analysis_setup.current_value, [candidate]
             )[0]
-            # The following may happen if either the requested conditing is
-            # 'none' or if the conditiong set that is tested consists only of
+            # The following may happen if either the requested conditioning is
+            # 'none' or if the conditioning set that is tested consists only of
             # a single candidate.
             if temp_cond is None:
                 conditional_realisations = conditional_realisations_target
@@ -843,58 +839,60 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             i_1 = i_2
             i_2 += data.n_realisations(analysis_setup.current_value)
 
-        # Generate surrogates for the current candidate.
-        if (
-            analysis_setup._cmi_estimator.is_analytic_null_estimator()
-            and permute_in_time
-        ):
-            # Generate the surrogates analytically
-            surr_table[
-                idx_c, :
-            ] = analysis_setup._cmi_estimator.estimate_surrogates_analytic(
-                n_perm=n_permutations,
-                var1=data.get_realisations(analysis_setup.current_value, [candidate])[
-                    0
-                ],
-                var2=analysis_setup._current_value_realisations,
-                conditional=temp_cond,
-            )
-        else:
-            analysis_setup.settings["analytical_surrogates"] = False
-            surr_candidate_realisations = _get_surrogates(
-                data,
-                analysis_setup.current_value,
-                [candidate],
-                n_permutations,
-                analysis_setup.settings,
-            )
-            try:
-                surr_table[idx_c, :] = analysis_setup._cmi_estimator.estimate_parallel(
-                    n_chunks=n_permutations,
-                    re_use=["var2", "conditional"],
-                    var1=surr_candidate_realisations,
+            # Generate surrogates for the current candidate.
+            if (
+                analysis_setup._cmi_estimator.is_analytic_null_estimator()
+                and permute_in_time
+            ):
+                # Generate the surrogates analytically
+                surr_table[
+                    idx_c, :
+                ] = analysis_setup._cmi_estimator.estimate_surrogates_analytic(
+                    n_perm=n_permutations,
+                    var1=data.get_realisations(
+                        analysis_setup.current_value, [candidate]
+                    )[0],
                     var2=analysis_setup._current_value_realisations,
                     conditional=temp_cond,
                 )
-            except ex.AlgorithmExhaustedError as aee:
-                # The aglorithm cannot continue here, so
-                #  we'll terminate the max sequential stats test,
-                #  and declare all not significant
-                print(
-                    "AlgorithmExhaustedError encountered in estimations: {}.".format(
-                        aee.message
-                    )
-                )
-                print("Stopping sequential max stats at candidate with rank 0")
-                return (
-                    np.zeros(len(analysis_setup.selected_vars_sources)).astype(bool),
-                    np.ones(len(analysis_setup.selected_vars_sources)),
-                    np.zeros(len(analysis_setup.selected_vars_sources)),
+            else:
+                analysis_setup.settings["analytical_surrogates"] = False
+                surr_candidate_realisations = _get_surrogates(
+                    data,
+                    analysis_setup.current_value,
+                    [candidate],
+                    n_permutations,
+                    analysis_setup.settings,
                 )
+                try:
+                    surr_table[
+                        idx_c, :
+                    ] = analysis_setup._cmi_estimator.estimate_parallel(
+                        n_chunks=n_permutations,
+                        re_use=["var2", "conditional"],
+                        var1=surr_candidate_realisations,
+                        var2=analysis_setup._current_value_realisations,
+                        conditional=temp_cond,
+                    )
+                except ex.AlgorithmExhaustedError as aee:
+                    # The aglorithm cannot continue here, so
+                    #  we'll terminate the max sequential stats test,
+                    #  and declare all not significant
+                    print(
+                        f"AlgorithmExhaustedError encountered in estimations: {aee.message}. "
+                        "Stopping sequential max stats at candidate with rank 0"
+                    )
+                    return (
+                        np.zeros(len(analysis_setup.selected_vars_sources)).astype(
+                            bool
+                        ),
+                        np.ones(len(analysis_setup.selected_vars_sources)),
+                        np.zeros(len(analysis_setup.selected_vars_sources)),
+                    )
 
-        # Calculate original statistic (multivariate/bivariate TE/MI)
+        # Calculate original statistic (bivariate TE/MI)
         try:
-            individual_stat = analysis_setup._cmi_estimator.estimate_parallel(
+            individual_stat_source = analysis_setup._cmi_estimator.estimate_parallel(
                 n_chunks=len(source_vars),
                 re_use=re_use,
                 var1=candidate_realisations,
@@ -906,10 +904,9 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             #  we'll terminate the max sequential stats test,
             #  and declare all not significant
             print(
-                "AlgorithmExhaustedError encountered in "
-                "estimations: {}.".format(aee.message)
+                f"AlgorithmExhaustedError encountered in estimations: {aee.message}. "
+                "Stopping sequential max stats at candidate with rank 0"
             )
-            print("Stopping sequential max stats at candidate with rank 0")
             # For now we don't need a stack trace:
             # traceback.print_tb(aee.__traceback__)
             # Return (signficance, pvalue, TEs):
@@ -919,38 +916,40 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                 np.zeros(len(analysis_setup.selected_vars_sources)),
             )
 
-        selected_vars_order = utils.argsort_descending(individual_stat)
-        individual_stat_sorted = utils.sort_descending(individual_stat)
+        selected_vars_order = utils.argsort_descending(individual_stat_source)
+        individual_stat_source_sorted = utils.sort_descending(individual_stat_source)
         max_distribution = _sort_table_max(surr_table)
+        significance_source = np.zeros(individual_stat_source.shape[0], dtype=bool)
+        pvalue_source = np.ones(individual_stat_source.shape[0])
 
         # Compare each original value with the distribution of the same rank,
         # starting with the highest value.
-        for c in range(individual_stat.shape[0]):
-            [s, p] = _find_pvalue(
-                individual_stat_sorted[c],
+        for c in range(individual_stat_source.shape[0]):
+            s, p = _find_pvalue(
+                individual_stat_source_sorted[c],
                 max_distribution[c,],
                 alpha,
                 tail="one_bigger",
             )
-            # Write results into an array with the same order as the set of
-            # selected sources from all process. Find the currently tested
-            # variable and its index in the list of all selected variables.
-            current_var = source_vars[selected_vars_order[c]]
-            for ind, v in enumerate(analysis_setup.selected_vars_sources):
-                if v == current_var:
-                    break
-            significance[ind] = s
-            pvalue[ind] = p
-            stat[ind] = individual_stat_sorted[c]
+            significance_source[c] = s
+            pvalue_source[c] = p
+
             if not s:  # break as soon as a candidate is no longer significant
                 if analysis_setup.settings["verbose"]:
                     print(
-                        "\nStopping sequential max stats at candidate with "
-                        "rank {0}.".format(c)
+                        f"\nStopping sequential max stats at candidate with rank {c}."
                     )
                 break
 
-    return significance, pvalue, stat
+        # Get back original order of variables within the current source. Write
+        # re-ordered results into global results array at the respective
+        # variable locations.
+        significance_source[selected_vars_order] = significance_source.copy()
+        pvalue_source[selected_vars_order] = pvalue_source.copy()
+        significance[source_vars_idx] = significance_source
+        pvalue[source_vars_idx] = pvalue_source
+        individual_stat[source_vars_idx] = individual_stat_source
+    return significance, pvalue, individual_stat
 
 
 def min_statistic(
@@ -1069,38 +1068,9 @@ def mi_against_surrogates(analysis_setup, data):
     permute_in_time = _check_permute_in_time(analysis_setup, data, n_perm)
     if analysis_setup.settings["verbose"]:
         print(
-            "mi permutation test against surrogates, n_perm: {0}".format(
-                analysis_setup.settings["n_perm_mi"]
-            )
+            f"mi permutation test against surrogates, n_perm: {analysis_setup.settings['n_perm_mi']}"
         )
-    """
-    surr_realisations = np.empty(
-                        (data.n_realisations(analysis_setup.current_value) *
-                         (n_perm + 1), 1))
-    i_1 = 0
-    i_2 = data.n_realisations(analysis_setup.current_value)
-    # The first chunk holds the original data
-    surr_realisations[i_1:i_2, ] = analysis_setup._current_value_realisations
-    # Create surrogate data by shuffling the realisations of the current value.
-    for perm in range(n_perm):
-        i_1 = i_2
-        i_2 += data.n_realisations(analysis_setup.current_value)
-        # Check the permutation type for the current candidate.
-        if permute_over_replications:
-            surr_temp = data.permute_data(analysis_setup.current_value,
-                                          [analysis_setup.current_value])[0]
-        else:
-            [real, repl_idx] = data.get_realisations(
-                                            analysis_setup.current_value,
-                                            [analysis_setup.current_value])
-            surr_temp = _permute_realisations(real, repl_idx, perm_range)
-        # Add current shuffled realisation to the array of all realisations for
-        # parallel MI estimation.
-        # surr_realisations[i_1:i_2, ] = surr_temp
-        [real, repl_idx] = data.get_realisations(
-                                            analysis_setup.current_value,
-                                            [analysis_setup.current_value])
-        """
+
     if analysis_setup._cmi_estimator.is_analytic_null_estimator() and permute_in_time:
         # Generate the surrogates analytically
         analysis_setup.settings["analytical_surrogates"] = True
@@ -1524,7 +1494,7 @@ def _find_pvalue(statistic, distribution, alpha, tail):
     assert distribution.ndim == 1, "Test distribution must be 1D."
     check_n_perm(distribution.shape[0], alpha)
 
-    if tail == "one_bigger" or tail == "one":
+    if tail in ["one_bigger", "one"]:
         pvalue = sum(distribution >= statistic) / distribution.shape[0]
     elif tail == "one_smaller":
         pvalue = sum(distribution <= statistic) / distribution.shape[0]
@@ -1535,20 +1505,7 @@ def _find_pvalue(statistic, distribution, alpha, tail):
         alpha = alpha / 2
     else:
         raise ValueError(
-            (
-                "Unkown value for "
-                "tail"
-                ", should be "
-                "one"
-                ", "
-                "one_bigger"
-                ","
-                " "
-                "one_smaller"
-                ", or "
-                "two"
-                "): {0}.".format(tail)
-            )
+            f"Unkown value for tail: {tail}, should be one, one_bigger, one_smaller, or two"
         )
 
     # If the statistic is larger than all values in the test distribution, set

From b18403b7ae83a3a3414d739750340fb1e2194612 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:24:43 +0100
Subject: [PATCH 17/34] Add unit tests for sequential max. stats

---
 test/test_stats.py | 89 +++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 77 insertions(+), 12 deletions(-)

diff --git a/test/test_stats.py b/test/test_stats.py
index ec70116..872cd65 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -25,6 +25,7 @@ def test_min_statistic():
 
 
 def test_max_statistic_sequential():
+    np.random.seed(0)
     data = Data()
     data.generate_mute_data(104, 10)
     settings = {
@@ -38,20 +39,82 @@ def test_max_statistic_sequential():
         "max_lag_target": 5,
     }
     setup = MultivariateTE()
-    setup._initialise(settings, data, sources=[0, 1], target=2)
-    setup.current_value = (0, 4)
-    setup.selected_vars_sources = [(1, 1), (1, 2)]
-    setup.selected_vars_full = [(0, 1), (1, 1), (1, 2)]
-    setup._selected_vars_realisations = np.random.rand(
-        data.n_realisations(setup.current_value), len(setup.selected_vars_full)
-    )
-    setup._current_value_realisations = np.random.rand(
-        data.n_realisations(setup.current_value), 1
+    setup._initialise(settings, data, sources=[0, 2], target=1)
+    setup.current_value = (1, 4)
+    setup.selected_vars_target = [(1, 1)]
+    setup.selected_vars_sources = [(0, 1), (2, 1)]
+    setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
+
+    setup._current_value_realisations = data.get_realisations(
+        setup.current_value, [setup.current_value]
+    )[0]
+
+    setup._selected_vars_realisations = np.hstack(
+        (
+            data.get_realisations(  # use actual realisation for target var and first source var
+                setup.current_value,
+                [setup.selected_vars_target[0], setup.selected_vars_sources[0]],
+            )[
+                0
+            ],
+            np.random.rand(
+                data.n_realisations(setup.current_value), 1
+            ),  # use random data as realizations for second source var
+        )
     )
     [sign, p, te] = stats.max_statistic_sequential(analysis_setup=setup, data=data)
+    print(p)
+    print(sign)
+    print(te)
+    assert len(p) == len(te)
+    assert len(p) == len(sign)
+    assert len(p) == len(setup.selected_vars_sources)
+
+    setup = MultivariateTE()
+    setup._initialise(settings, data, sources=[0, 1, 4], target=3)
+    setup.current_value = (3, 4)
+    setup.selected_vars_target = [(1, 1)]
+    setup.selected_vars_sources = [(0, 1), (0, 2), (1, 1), (4, 1)]
+    setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
+    setup._current_value_realisations = data.get_realisations(
+        setup.current_value, [setup.current_value]
+    )[0]
+    setup._selected_vars_realisations = data.get_realisations(
+        setup.current_value, setup.selected_vars_full
+    )[0]
+
     [sign, p, te] = stats.max_statistic_sequential_bivariate(
         analysis_setup=setup, data=data
     )
+    assert len(p) == len(te)
+    assert len(p) == len(sign)
+    assert len(p) == len(setup.selected_vars_sources)
+    print(p)
+    print(setup.selected_vars_sources)
+    print(sign)
+    print(te, "\n\n")
+
+    for t in [0, 4]:
+        np.random.seed(0)
+        data.generate_mute_data(104, 10)
+        data._data[t, :, :] = np.random.rand(104, 10)
+        setup._current_value_realisations = data.get_realisations(
+            setup.current_value, [setup.current_value]
+        )[0]
+        setup._selected_vars_realisations = data.get_realisations(
+            setup.current_value, setup.selected_vars_full
+        )[0]
+        [sign, p, te] = stats.max_statistic_sequential_bivariate(
+            analysis_setup=setup, data=data
+        )
+        for i, var in enumerate(setup.selected_vars_sources):
+            if var[0] == t:
+                assert not sign[
+                    i
+                ], "Seq. max. stats returned sign result for random data"
+                assert (
+                    p[i] > setup.settings["alpha_max_seq"]
+                ), f"{p[i]} not smaller than critical alpha {setup.settings['alpha_max_seq']}"
 
 
 def test_network_fdr():
@@ -407,7 +470,9 @@ def test_analytical_surrogates():
     expected_mi, source1, source2, target = _get_gauss_data(covariance=0.4)
     settings = {"discretise_method": "equal", "n_discrete_bins": 5}
     est = JidtDiscreteCMI(settings)
-    source_dis, target_dis = est._discretise_vars(var1=source1, var2=target)
+    source_dis, target_dis = est._discretise_vars(  # pylint: disable=W0632
+        var1=source1, var2=target
+    )
     data = Data(np.hstack((source_dis, target_dis)), dim_order="sp", normalise=False)
     settings = {
         "cmi_estimator": "JidtDiscreteCMI",
@@ -434,7 +499,7 @@ def test_analytical_surrogates():
     # test_analytical_surrogates()
     # test_data_type()
     # test_network_fdr()
-    test_fdr_sorting()
+    # test_fdr_sorting()
     # test_find_pvalue()
     # test_find_table_max()
     # test_find_table_min()
@@ -443,4 +508,4 @@ def test_analytical_surrogates():
     # test_omnibus_test()
     # test_max_statistic()
     # test_min_statistic()
-    # test_max_statistic_sequential()
+    test_max_statistic_sequential()

From 935161bee4263ce48400f673d007906fbd14beb1 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:27:09 +0100
Subject: [PATCH 18/34] Fix error in input check

---
 idtxl/multivariate_te.py | 8 +++-----
 1 file changed, 3 insertions(+), 5 deletions(-)

diff --git a/idtxl/multivariate_te.py b/idtxl/multivariate_te.py
index b727b3f..8464344 100755
--- a/idtxl/multivariate_te.py
+++ b/idtxl/multivariate_te.py
@@ -135,7 +135,7 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
             targets = list(range(data.n_processes))
         if sources == "all":
             sources = ["all" for t in targets]
-        elif isinstance(sources is list) and isinstance(sources[0], int):
+        elif isinstance(sources, list) and isinstance(sources[0], int):
             sources = [sources for t in targets]
         elif isinstance(sources, list) and isinstance(sources[0], list):
             pass
@@ -154,12 +154,10 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
             n_realisations=data.n_realisations(),
             normalised=data.normalise,
         )
-        for t in range(len(targets)):
+        for t, target in enumerate(targets):
             if settings["verbose"]:
                 print(f"\n####### analysing target with index {t} from list {targets}")
-            res_single = self.analyse_single_target(
-                settings, data, targets[t], sources[t]
-            )
+            res_single = self.analyse_single_target(settings, data, target, sources[t])
             results.combine_results(res_single)
 
         # Get no. realisations actually used for estimation from single target

From b11c69c121e1d623eed1552a8fbb0371e538b296 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:41:26 +0100
Subject: [PATCH 19/34] Fix docstrings

---
 idtxl/estimators_multivariate_pid.py | 61 +++++++++++++++-------------
 idtxl/idtxl_utils.py                 | 23 +++++------
 idtxl/network_analysis.py            | 16 ++++----
 idtxl/results.py                     | 35 ++++++++--------
 idtxl/stats.py                       | 10 ++---
 5 files changed, 74 insertions(+), 71 deletions(-)

diff --git a/idtxl/estimators_multivariate_pid.py b/idtxl/estimators_multivariate_pid.py
index ad40463..46432f1 100755
--- a/idtxl/estimators_multivariate_pid.py
+++ b/idtxl/estimators_multivariate_pid.py
@@ -8,6 +8,7 @@
   http://arxiv.org/abs/2002.03356
 """
 import numpy as np
+
 from . import lattices as lt
 from . import pid_goettingen
 from .estimator import Estimator
@@ -47,7 +48,7 @@ class SxPID(Estimator):
     def __init__(self, settings):
         # get estimation parameters
         self.settings = settings.copy()
-        self.settings.setdefault('verbose', False)
+        self.settings.setdefault("verbose", False)
 
     def is_parallel():
         return False
@@ -56,7 +57,8 @@ def is_analytic_null_estimator(self):
         return False
 
     def estimate(self, s, t):
-        """
+        """Estimate SxPID from list of sources and a target
+
         Args:
             s : list of numpy arrays
                 1D arrays containing realizations of a discrete random variable
@@ -64,16 +66,12 @@ def estimate(self, s, t):
                 1D array containing realizations of a discrete random variable
 
         Returns:
-            dict of dict
-                {
-                 'ptw' -> { realization -> {alpha -> [float, float, float]} }
-
-                 'avg' -> {alpha -> [float, float, float]}
-                }
-            where the list of floats is ordered
-            [informative, misinformative, informative - misinformative]
-            ptw stands for pointwise decomposition
-            avg stands for average decomposition
+            dict
+                SxPID results, with entries
+                    - 'ptw' -> { realization -> {alpha -> [float, float, float]}}: pointwise decomposition
+                    - 'avg' -> {alpha -> [float, float, float]}: average decomposition
+
+                the list of floats is ordered [informative, misinformative, informative - misinformative]
         """
         s, t, self.settings = _check_input(s, t, self.settings)
         pdf = _get_pdf_dict(s, t)
@@ -90,19 +88,20 @@ def estimate(self, s, t):
             pdf_orig=pdf,
             chld=lattices[num_source_vars][0],
             achain=lattices[num_source_vars][1],
-            printing=self.settings['verbose'])
+            printing=self.settings["verbose"],
+        )
 
         # TODO AskM: Trivariate: does it make sense to name the alphas
         #    for example shared_syn_s1_s2__syn_s1_s3 ?
         results = {
-            'ptw': retval_ptw,
-            'avg': retval_avg,
+            "ptw": retval_ptw,
+            "avg": retval_avg,
         }
         return results
 
 
 def _get_pdf_dict(s, t):
-    """"Write probability mass function estimated via counting to a dict."""
+    """ "Write probability mass function estimated via counting to a dict."""
     # Create dictionary with probability mass function
     counts = dict()
     n_samples = s[0].shape[0]
@@ -126,10 +125,10 @@ def _check_input(s, t, settings):
     """Check input to PID estimators."""
     # Check if inputs are numpy arrays.
     if type(t) != np.ndarray:
-        raise TypeError('Input t must be a numpy array.')
+        raise TypeError("Input t must be a numpy array.")
     for i in range(len(s)):
         if type(s[i]) != np.ndarray:
-            raise TypeError('All inputs s{0} must be numpy arrays.'.format(i+1))
+            raise TypeError("All inputs s{0} must be numpy arrays.".format(i + 1))
 
     # In general, IDTxl expects 2D inputs because JIDT/JPYPE only accepts those
     # and we have a multivariate approach, i.e., a vector is a special case of
@@ -146,33 +145,37 @@ def _check_input(s, t, settings):
                 for col in range(1, s[i].shape[1]):
                     alph_col = len(np.unique(s[i][:, col]))
                     si_joint, alph_new = _join_variables(
-                        si_joint, s[i][:, col], alph_new, alph_col)
-                settings['alph_s'+str(i+1)] = alph_new
+                        si_joint, s[i][:, col], alph_new, alph_col
+                    )
+                settings["alph_s" + str(i + 1)] = alph_new
             else:
-                raise ValueError('Input source {0} s{0} has to be a 1D or 2D '
-                                 'numpy array.'.format(i+1))
+                raise ValueError(
+                    "Input source {0} s{0} has to be a 1D or 2D "
+                    "numpy array.".format(i + 1)
+                )
 
     if t.ndim != 1:
         if t.shape[1] == 1:
             t = np.squeeze(t)
         else:  # For now we only allow 1D-targets
-            raise ValueError('Input target t has to be a vector '
-                             '(t.shape[1]=1).')
+            raise ValueError("Input target t has to be a vector " "(t.shape[1]=1).")
 
     # Check types of remaining inputs.
     if type(settings) != dict:
-        raise TypeError('The settings argument should be a dictionary.')
+        raise TypeError("The settings argument should be a dictionary.")
     for i in range(len(s)):
         if not issubclass(s[i].dtype.type, np.integer):
-            raise TypeError('Input s{0} (source {0}) must be an integer numpy '
-                            'array.'.format(i+1))
+            raise TypeError(
+                "Input s{0} (source {0}) must be an integer numpy "
+                "array.".format(i + 1)
+            )
     # ^ for
     if not issubclass(t.dtype.type, np.integer):
-        raise TypeError('Input t (target) must be an integer numpy array.')
+        raise TypeError("Input t (target) must be an integer numpy array.")
 
     # Check if variables have equal length.
     for i in range(len(s)):
         if len(t) != len(s[i]):
-            raise ValueError('Number of samples s and t must be equal')
+            raise ValueError("Number of samples s and t must be equal")
 
     return s, t, settings
diff --git a/idtxl/idtxl_utils.py b/idtxl/idtxl_utils.py
index 50c85d9..f6ce0e4 100755
--- a/idtxl/idtxl_utils.py
+++ b/idtxl/idtxl_utils.py
@@ -1,9 +1,10 @@
 """Provide IDTxl utility functions."""
-import pprint
 import copy as cp
-import numpy as np
+import pprint
 import threading
 
+import numpy as np
+
 
 def swap_chars(s, i_1, i_2):
     """Swap to characters in a string.
@@ -316,14 +317,7 @@ def conflicting_entries(dict_1, dict_2):
     for k in intersect_keys:
         if np.array(dict_1[k] != dict_2[k]).any():
             print(
-                "Unequal entries for key "
-                "{0}"
-                ": dict_1: "
-                "{1}"
-                ", dict_2:"
-                " "
-                "{2}"
-                ".".format(k, dict_1[k], dict_2[k])
+                f"Unequal entries for key {k}: dict_1: {dict_1[k]}, dict_2: {dict_2[k]}"
             )
             return True
     return False
@@ -336,9 +330,12 @@ def calculate_mi(corr):
 
 class timeout(object):
     """Context manager for a timeout using threading module.
-    args:
-        timeout_duration: float, number of seconds to wait before timeout is triggered
-        exception_message: string, message to put in the exception
+
+    Args:
+        timeout_duration: float
+            number of seconds to wait before timeout is triggered
+        exception_message : string
+            message to put in the exception
     """
 
     def __init__(self, timeout_duration, exception_message="Timeout"):
diff --git a/idtxl/network_analysis.py b/idtxl/network_analysis.py
index d312674..7e9adde 100644
--- a/idtxl/network_analysis.py
+++ b/idtxl/network_analysis.py
@@ -1,16 +1,18 @@
 """Parent class for network inference and network comparison.
 """
-import os.path
-from datetime import datetime
-from shutil import copyfile
-from pprint import pprint
 import ast
 import copy as cp
 import itertools as it
+import os.path
+from datetime import datetime
+from pprint import pprint
+from shutil import copyfile
+
 import numpy as np
-from .estimator import get_estimator
-from . import idtxl_utils as utils
+
 from . import idtxl_io as io
+from . import idtxl_utils as utils
+from .estimator import get_estimator
 
 
 class NetworkAnalysis:
@@ -662,7 +664,7 @@ def resume_checkpoint(self, file_path):
 
         Args:
             file_path : str
-                path to checkpoint file (excluding extension: *.ckp)
+                path to checkpoint file (excluding extension: .ckp)
         """
 
         # Read checkpoint
diff --git a/idtxl/results.py b/idtxl/results.py
index 84f3702..ce33470 100644
--- a/idtxl/results.py
+++ b/idtxl/results.py
@@ -1,8 +1,10 @@
 """Provide results class for IDTxl network analysis."""
+import copy as cp
 import sys
 import warnings
-import copy as cp
+
 import numpy as np
+
 from . import idtxl_utils as utils
 
 warnings.simplefilter(action="ignore", category=FutureWarning)
@@ -1158,38 +1160,39 @@ def get_single_process(self, process):
                 (result['selected_vars']) or via dot-notation
                 (result.selected_vars). Contains keys
 
-                Process : int
+                - Process : int
                     Process that was optimized
-                estimation_method : String
+                - estimation_method : String
                     Estimation method that was used for optimization
-                T_D : float
+                - T_D : float
                     Estimated optimal value for the temporal depth TD
-                tau_R :
+                - tau_R :
                     Information timescale tau_R, a characteristic timescale of history
                     dependence similar to an autocorrelation time.
-                R_tot : float
+                - R_tot : float
                     Estimated value for the total history dependence Rtot,
-                AIS_tot : float
+                - AIS_tot : float
                     Estimated value for the total active information storage
-                opt_number_of_bins_d : int
+                - opt_number_of_bins_d : int
                     Number of bins d for the embedding that yields (R̂tot ,T̂D)
-                opt_scaling_k : int
+                - opt_scaling_k : int
                     Scaling exponent κ for the embedding that yields (R̂tot , T̂D)
-                opt_first_bin_size : int
+                - opt_first_bin_size : int
                     Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),
-                history_dependence : array with floating-point values
+                - history_dependence : array with floating-point values
                     Estimated history dependence for each embedding
-                firing_rate : float
+                - firing_rate : float
                     Firing rate of the neuron/ spike train
-                recording_length : float
+                - recording_length : float
                     Length of the recording (in seconds)
-                H_spiking : float
+                - H_spiking : float
                     Entropy of the spike times
 
                 if analyse_auto_MI was set to True additionally:
-                auto_MI : dict
+
+                - auto_MI : dict
                     numpy array of MI values for each delay
-                auto_MI_delays : list of int
+                - auto_MI_delays : list of int
                     list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay
 
         """
diff --git a/idtxl/stats.py b/idtxl/stats.py
index 9d26a71..7ee3852 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -513,13 +513,12 @@ def max_statistic_sequential(analysis_setup, data):
     calculation of surrogates for this statistic.
 
     Args:
-
         analysis_setup : MultivariateTE instance
             information on the current analysis, can have an optional attribute
-            'settings', a dictionary with parameters for statistical testing:
+            settings, a dictionary with parameters for statistical testing:
 
             - n_perm_max_seq : int [optional] - number of permutations
-              (default='n_perm_min_stat'|500)
+              (default=n_perm_min_stat|500)
             - alpha_max_seq : float [optional] - critical alpha level
               (default=0.05)
             - permute_in_time : bool [optional] - generate surrogates by
@@ -723,13 +722,12 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
     calculation of surrogates for this statistic.
 
     Args:
-
         analysis_setup : MultivariateTE instance
             information on the current analysis, can have an optional attribute
-            'settings', a dictionary with parameters for statistical testing:
+            settings, a dictionary with parameters for statistical testing:
 
             - n_perm_max_seq : int [optional] - number of permutations
-              (default='n_perm_min_stat'|500)
+              (default=n_perm_min_stat|500)
             - alpha_max_seq : float [optional] - critical alpha level
               (default=0.05)
             - permute_in_time : bool [optional] - generate surrogates by

From 4c6228ba48829eef6b84c429446ca8c8bdcb5889 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 14:41:41 +0100
Subject: [PATCH 20/34] Update sphinx documentation

---
 docs/doctrees/environment.pickle              | Bin 921680 -> 920870 bytes
 docs/doctrees/idtxl.doctree                   | Bin 1843274 -> 1845713 bytes
 docs/doctrees/idtxl_helper.doctree            | Bin 163080 -> 164078 bytes
 docs/doctrees/idtxl_results_class.doctree     | Bin 149071 -> 151963 bytes
 docs/html/.buildinfo                          |   2 +-
 .../idtxl/active_information_storage.html     |   6 +-
 docs/html/_modules/idtxl/bivariate_mi.html    |   6 +-
 docs/html/_modules/idtxl/bivariate_pid.html   |   6 +-
 docs/html/_modules/idtxl/bivariate_te.html    |   6 +-
 docs/html/_modules/idtxl/data.html            |  43 +--
 .../embedding_optimization_ais_Rudelt.html    |   6 +-
 .../_modules/idtxl/estimators_Rudelt.html     |  24 +-
 docs/html/_modules/idtxl/estimators_jidt.html |   6 +-
 docs/html/_modules/idtxl/estimators_mpi.html  |   6 +-
 .../idtxl/estimators_multivariate_pid.html    |  67 ++--
 .../_modules/idtxl/estimators_opencl.html     |   6 +-
 docs/html/_modules/idtxl/estimators_pid.html  |   6 +-
 .../_modules/idtxl/estimators_python.html     |   6 +-
 .../html/_modules/idtxl/idtxl_exceptions.html |   6 +-
 docs/html/_modules/idtxl/idtxl_io.html        |   6 +-
 docs/html/_modules/idtxl/idtxl_utils.html     |  29 +-
 docs/html/_modules/idtxl/multivariate_mi.html |  24 +-
 .../html/_modules/idtxl/multivariate_pid.html |   6 +-
 docs/html/_modules/idtxl/multivariate_te.html |  38 +-
 .../html/_modules/idtxl/network_analysis.html |  22 +-
 .../_modules/idtxl/network_comparison.html    |   6 +-
 .../_modules/idtxl/network_inference.html     |  12 +-
 docs/html/_modules/idtxl/postprocessing.html  |   6 +-
 docs/html/_modules/idtxl/results.html         |  41 +-
 .../idtxl/single_process_analysis.html        |   6 +-
 docs/html/_modules/idtxl/stats.html           | 363 ++++++++----------
 docs/html/_modules/idtxl/visualise_graph.html |   6 +-
 docs/html/_modules/index.html                 |   6 +-
 docs/html/_static/documentation_options.js    |   2 +-
 docs/html/genindex.html                       |   6 +-
 docs/html/idtxl.html                          | 124 ++++--
 docs/html/idtxl_data_class.html               |   6 +-
 docs/html/idtxl_estimators.html               |   6 +-
 docs/html/idtxl_helper.html                   |  48 +--
 docs/html/idtxl_network_comparison.html       |   6 +-
 docs/html/idtxl_network_inference.html        |   6 +-
 docs/html/idtxl_postprocessing.html           |   6 +-
 docs/html/idtxl_process_analysis.html         |   6 +-
 docs/html/idtxl_results_class.html            |  63 ++-
 docs/html/index.html                          |   6 +-
 docs/html/modules.html                        |   6 +-
 docs/html/objects.inv                         | Bin 2761 -> 2761 bytes
 docs/html/py-modindex.html                    |   6 +-
 docs/html/search.html                         |   6 +-
 docs/html/searchindex.js                      |   2 +-
 50 files changed, 563 insertions(+), 515 deletions(-)

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index 9863270cf0dd7068c9d32998da6e6b2f726e3f64..e4299c3eb2af5a3ee48a3dba0db1c6eced1edd1a 100644
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diff --git a/docs/doctrees/idtxl.doctree b/docs/doctrees/idtxl.doctree
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diff --git a/docs/doctrees/idtxl_helper.doctree b/docs/doctrees/idtxl_helper.doctree
index 6c68fcd5948baa734a4a7d1db163998cded013f6..ab1fc9a0536f4e76147bfd41b2de085f4c257b65 100644
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diff --git a/docs/html/.buildinfo b/docs/html/.buildinfo
index 85f9be7..579a468 100644
--- a/docs/html/.buildinfo
+++ b/docs/html/.buildinfo
@@ -1,4 +1,4 @@
 # Sphinx build info version 1
 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
-config: b08cb61c48070e6f8470e0fc861bc3eb
+config: 67ee60b2474765b8bfee49628ba7a89b
 tags: 645f666f9bcd5a90fca523b33c5a78b7
diff --git a/docs/html/_modules/idtxl/active_information_storage.html b/docs/html/_modules/idtxl/active_information_storage.html
index 27dc08d..9a7f274 100644
--- a/docs/html/_modules/idtxl/active_information_storage.html
+++ b/docs/html/_modules/idtxl/active_information_storage.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.active_information_storage &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.active_information_storage &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.active_information_storage</a></li> 
       </ul>
@@ -741,7 +741,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.active_information_storage</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/bivariate_mi.html b/docs/html/_modules/idtxl/bivariate_mi.html
index 53a8d8e..a8ec368 100644
--- a/docs/html/_modules/idtxl/bivariate_mi.html
+++ b/docs/html/_modules/idtxl/bivariate_mi.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.bivariate_mi &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.bivariate_mi &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_mi</a></li> 
       </ul>
@@ -397,7 +397,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_mi</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/bivariate_pid.html b/docs/html/_modules/idtxl/bivariate_pid.html
index 7ae3f85..92392a4 100644
--- a/docs/html/_modules/idtxl/bivariate_pid.html
+++ b/docs/html/_modules/idtxl/bivariate_pid.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.bivariate_pid &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.bivariate_pid &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_pid</a></li> 
       </ul>
@@ -387,7 +387,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_pid</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/bivariate_te.html b/docs/html/_modules/idtxl/bivariate_te.html
index fb12e49..e2cd536 100644
--- a/docs/html/_modules/idtxl/bivariate_te.html
+++ b/docs/html/_modules/idtxl/bivariate_te.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.bivariate_te &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.bivariate_te &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_te</a></li> 
       </ul>
@@ -400,7 +400,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.bivariate_te</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/data.html b/docs/html/_modules/idtxl/data.html
index ecb1981..8b2cd38 100644
--- a/docs/html/_modules/idtxl/data.html
+++ b/docs/html/_modules/idtxl/data.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.data &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.data &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.data</a></li> 
       </ul>
@@ -45,6 +45,7 @@ <h3>Navigation</h3>
   <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Provide data structures for IDTxl analysis.&quot;&quot;&quot;</span>
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
 
 <span class="n">VERBOSE</span> <span class="o">=</span> <span class="kc">False</span>
@@ -190,7 +191,7 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
 <span class="sd">                must have the same length as number of dimensions in data</span>
 <span class="sd">        &quot;&quot;&quot;</span>
         <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dim_order</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">&quot;dim_order can not have more than three &quot;</span> <span class="s2">&quot;entries&quot;</span><span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">&quot;dim_order can not have more than three entries&quot;</span><span class="p">)</span>
         <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dim_order</span><span class="p">)</span> <span class="o">!=</span> <span class="n">data</span><span class="o">.</span><span class="n">ndim</span><span class="p">:</span>
             <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span>
                 <span class="sa">f</span><span class="s2">&quot;Data array dimension (</span><span class="si">{</span><span class="n">data</span><span class="o">.</span><span class="n">ndim</span><span class="si">}</span><span class="s2">) and length of &quot;</span>
@@ -202,8 +203,8 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
         <span class="n">data_ordered</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_reorder_data</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">dim_order</span><span class="p">)</span>
         <span class="bp">self</span><span class="o">.</span><span class="n">_set_data_size</span><span class="p">(</span><span class="n">data_ordered</span><span class="p">)</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;Adding data with properties: </span><span class="si">{0}</span><span class="s2"> processes, </span><span class="si">{1}</span><span class="s2"> samples, </span><span class="si">{2}</span><span class="s2"> &quot;</span>
-            <span class="s2">&quot;replications&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_processes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_samples</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_replications</span><span class="p">)</span>
+            <span class="sa">f</span><span class="s2">&quot;Adding data with properties: </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">n_processes</span><span class="si">}</span><span class="s2"> processes, </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">n_samples</span><span class="si">}</span><span class="s2"> &quot;</span>
+            <span class="sa">f</span><span class="s2">&quot;samples, </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">n_replications</span><span class="si">}</span><span class="s2"> replications&quot;</span>
         <span class="p">)</span>
         <span class="k">try</span><span class="p">:</span>
             <span class="nb">delattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s2">&quot;data&quot;</span><span class="p">)</span>
@@ -301,7 +302,7 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
 <span class="sd">                samples * no.replications) x number of indices</span>
 <span class="sd">        &quot;&quot;&quot;</span>
         <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s2">&quot;data&quot;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;No data has been added to this Data() &quot;</span> <span class="s2">&quot;instance.&quot;</span><span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;No data has been added to this Data() instance.&quot;</span><span class="p">)</span>
         <span class="c1"># Return None if index list is empty.</span>
         <span class="k">if</span> <span class="ow">not</span> <span class="n">idx_list</span><span class="p">:</span>
             <span class="k">return</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span>
@@ -341,12 +342,11 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
                     <span class="n">realisations</span><span class="p">[</span><span class="n">r</span> <span class="p">:</span> <span class="n">r</span> <span class="o">+</span> <span class="n">n_real_time</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span>
                         <span class="n">idx</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">idx</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="p">:</span> <span class="n">last_sample</span><span class="p">,</span> <span class="n">replication</span>
                     <span class="p">]</span>
-                <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
+                <span class="k">except</span> <span class="ne">IndexError</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
                     <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span>
-                        <span class="s2">&quot;You tried to access variable </span><span class="si">{0}</span><span class="s2"> in a &quot;</span>
-                        <span class="s2">&quot;data set with </span><span class="si">{1}</span><span class="s2"> processes and </span><span class="si">{2}</span><span class="s2"> &quot;</span>
-                        <span class="s2">&quot;samples.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_processes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_samples</span><span class="p">)</span>
-                    <span class="p">)</span>
+                        <span class="sa">f</span><span class="s2">&quot;You tried to access variable </span><span class="si">{</span><span class="n">idx</span><span class="si">}</span><span class="s2"> in a data set with </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">n_processes</span><span class="si">}</span><span class="s2"> &quot;</span>
+                        <span class="sa">f</span><span class="s2">&quot;processes and </span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">n_samples</span><span class="si">}</span><span class="s2"> samples.&quot;</span>
+                    <span class="p">)</span> <span class="kn">from</span> <span class="nn">e</span>
                 <span class="n">r</span> <span class="o">+=</span> <span class="n">n_real_time</span>
 
             <span class="k">assert</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span>
@@ -423,14 +423,11 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
 
         <span class="k">try</span><span class="p">:</span>
             <span class="n">data_slice</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">process</span><span class="p">,</span> <span class="n">offset_samples</span><span class="p">:,</span> <span class="n">replication_index</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
+        <span class="k">except</span> <span class="ne">IndexError</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
             <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span>
-                <span class="s2">&quot;You tried to access process </span><span class="si">{0}</span><span class="s2"> with an offset &quot;</span>
-                <span class="s2">&quot;of </span><span class="si">{1}</span><span class="s2"> in a data set of </span><span class="si">{2}</span><span class="s2"> processes and </span><span class="si">{3}</span><span class="s2"> &quot;</span>
-                <span class="s2">&quot;samples.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                    <span class="n">process</span><span class="p">,</span> <span class="n">offset_samples</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_processes</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_samples</span>
-                <span class="p">)</span>
-            <span class="p">)</span>
+                <span class="s2">&quot;You tried to access process </span><span class="si">{process}</span><span class="s2"> with an offset of </span><span class="si">{offset_samples}</span><span class="s2"> in a data set &quot;</span>
+                <span class="s2">&quot;of </span><span class="si">{self.n_processes}</span><span class="s2"> processes and </span><span class="si">{self.n_samples}</span><span class="s2"> samples.&quot;</span>
+            <span class="p">)</span> <span class="kn">from</span> <span class="nn">e</span>
         <span class="k">assert</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span>
             <span class="n">data_slice</span>
         <span class="p">)</span><span class="o">.</span><span class="n">any</span><span class="p">(),</span> <span class="s2">&quot;There are nans in the retrieved data slice.&quot;</span>
@@ -751,22 +748,22 @@ <h1>Source code for idtxl.data</h1><div class="highlight"><pre>
         <span class="k">elif</span> <span class="n">perm_type</span> <span class="o">==</span> <span class="s2">&quot;circular&quot;</span><span class="p">:</span>
             <span class="n">max_shift</span> <span class="o">=</span> <span class="n">perm_settings</span><span class="p">[</span><span class="s2">&quot;max_shift&quot;</span><span class="p">]</span>
             <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">max_shift</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">or</span> <span class="n">max_shift</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot; &quot;</span> <span class="s2">&quot;max_shift&quot;</span> <span class="s2">&quot; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
+                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;&#39;max_shift&#39; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
             <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_circular_shift</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">max_shift</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
 
         <span class="k">elif</span> <span class="n">perm_type</span> <span class="o">==</span> <span class="s2">&quot;block&quot;</span><span class="p">:</span>
             <span class="n">block_size</span> <span class="o">=</span> <span class="n">perm_settings</span><span class="p">[</span><span class="s2">&quot;block_size&quot;</span><span class="p">]</span>
             <span class="n">perm_range</span> <span class="o">=</span> <span class="n">perm_settings</span><span class="p">[</span><span class="s2">&quot;perm_range&quot;</span><span class="p">]</span>
             <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">block_size</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">or</span> <span class="n">block_size</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot; &quot;</span> <span class="s2">&quot;block_size&quot;</span> <span class="s2">&quot; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
+                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;&#39;block_size&#39; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
             <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">perm_range</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">or</span> <span class="n">perm_range</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot; &quot;</span> <span class="s2">&quot;perm_range&quot;</span> <span class="s2">&quot; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
+                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;&#39;perm_range&#39; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
             <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_swap_blocks</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">block_size</span><span class="p">,</span> <span class="n">perm_range</span><span class="p">)</span>
 
         <span class="k">elif</span> <span class="n">perm_type</span> <span class="o">==</span> <span class="s2">&quot;local&quot;</span><span class="p">:</span>
             <span class="n">perm_range</span> <span class="o">=</span> <span class="n">perm_settings</span><span class="p">[</span><span class="s2">&quot;perm_range&quot;</span><span class="p">]</span>
             <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">perm_range</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">or</span> <span class="n">perm_range</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot; &quot;</span> <span class="s2">&quot;perm_range&quot;</span> <span class="s2">&quot; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
+                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;&#39;perm_range&#39; has to be an int &gt; 0.&quot;</span><span class="p">)</span>
             <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_swap_local</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">perm_range</span><span class="p">)</span>
 
         <span class="k">else</span><span class="p">:</span>
@@ -1133,7 +1130,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.data</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/embedding_optimization_ais_Rudelt.html b/docs/html/_modules/idtxl/embedding_optimization_ais_Rudelt.html
index 7888684..4d38b07 100644
--- a/docs/html/_modules/idtxl/embedding_optimization_ais_Rudelt.html
+++ b/docs/html/_modules/idtxl/embedding_optimization_ais_Rudelt.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.embedding_optimization_ais_Rudelt &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.embedding_optimization_ais_Rudelt &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.embedding_optimization_ais_Rudelt</a></li> 
       </ul>
@@ -1311,7 +1311,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.embedding_optimization_ais_Rudelt</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_Rudelt.html b/docs/html/_modules/idtxl/estimators_Rudelt.html
index b352b7b..e3e2a92 100644
--- a/docs/html/_modules/idtxl/estimators_Rudelt.html
+++ b/docs/html/_modules/idtxl/estimators_Rudelt.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_Rudelt &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_Rudelt &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_Rudelt</a></li> 
       </ul>
@@ -46,14 +46,16 @@ <h1>Source code for idtxl.estimators_Rudelt</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Provide HDE estimators.&quot;&quot;&quot;</span>
 
 <span class="kn">import</span> <span class="nn">logging</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">newton</span><span class="p">,</span> <span class="n">minimize</span>
 <span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">from</span> <span class="nn">sys</span> <span class="kn">import</span> <span class="n">stderr</span>
-<span class="kn">from</span> <span class="nn">idtxl.estimator</span> <span class="kn">import</span> <span class="n">Estimator</span>
-<span class="kn">import</span> <span class="nn">idtxl.hde_utils</span> <span class="k">as</span> <span class="nn">utl</span>
 <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">Counter</span>
+<span class="kn">from</span> <span class="nn">sys</span> <span class="kn">import</span> <span class="n">stderr</span>
+
 <span class="kn">import</span> <span class="nn">mpmath</span> <span class="k">as</span> <span class="nn">mp</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">minimize</span><span class="p">,</span> <span class="n">newton</span>
+
+<span class="kn">import</span> <span class="nn">idtxl.hde_utils</span> <span class="k">as</span> <span class="nn">utl</span>
+<span class="kn">from</span> <span class="nn">idtxl.estimator</span> <span class="kn">import</span> <span class="n">Estimator</span>
 
 <span class="n">FAST_EMBEDDING_AVAILABLE</span> <span class="o">=</span> <span class="kc">True</span>
 <span class="k">try</span><span class="p">:</span>
@@ -62,9 +64,9 @@ <h1>Source code for idtxl.estimators_Rudelt</h1><div class="highlight"><pre>
     <span class="n">FAST_EMBEDDING_AVAILABLE</span> <span class="o">=</span> <span class="kc">False</span>
     <span class="nb">print</span><span class="p">(</span>
         <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Error importing Cython fast embedding module for HDE estimator.\n</span>
-<span class="sd">    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,\n</span>
-<span class="sd">    this may take a long time. Other estimators are not affected.\n</span>
+<span class="sd">    Error importing Cython fast embedding module for HDE estimator.</span>
+<span class="sd">    When running the HDE estimator, the slow Python implementation for optimizing the HDE embedding will be used,</span>
+<span class="sd">    this may take a long time. Other estimators are not affected.</span>
 <span class="sd">    &quot;&quot;&quot;</span><span class="p">,</span>
         <span class="n">file</span><span class="o">=</span><span class="n">stderr</span><span class="p">,</span>
         <span class="n">flush</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
@@ -1350,7 +1352,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_Rudelt</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_jidt.html b/docs/html/_modules/idtxl/estimators_jidt.html
index 907b2e5..2f67f8f 100644
--- a/docs/html/_modules/idtxl/estimators_jidt.html
+++ b/docs/html/_modules/idtxl/estimators_jidt.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_jidt &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_jidt &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_jidt</a></li> 
       </ul>
@@ -1874,7 +1874,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_jidt</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_mpi.html b/docs/html/_modules/idtxl/estimators_mpi.html
index e6752b2..b400224 100644
--- a/docs/html/_modules/idtxl/estimators_mpi.html
+++ b/docs/html/_modules/idtxl/estimators_mpi.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_mpi &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_mpi &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_mpi</a></li> 
       </ul>
@@ -267,7 +267,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_mpi</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_multivariate_pid.html b/docs/html/_modules/idtxl/estimators_multivariate_pid.html
index 66407fc..61906c0 100644
--- a/docs/html/_modules/idtxl/estimators_multivariate_pid.html
+++ b/docs/html/_modules/idtxl/estimators_multivariate_pid.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_multivariate_pid &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_multivariate_pid &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_multivariate_pid</a></li> 
       </ul>
@@ -53,6 +53,7 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
 <span class="sd">  http://arxiv.org/abs/2002.03356</span>
 <span class="sd">&quot;&quot;&quot;</span>
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">lattices</span> <span class="k">as</span> <span class="n">lt</span>
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">pid_goettingen</span>
 <span class="kn">from</span> <span class="nn">.estimator</span> <span class="kn">import</span> <span class="n">Estimator</span>
@@ -92,7 +93,7 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
     <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">):</span>
         <span class="c1"># get estimation parameters</span>
         <span class="bp">self</span><span class="o">.</span><span class="n">settings</span> <span class="o">=</span> <span class="n">settings</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="s1">&#39;verbose&#39;</span><span class="p">,</span> <span class="kc">False</span><span class="p">)</span>
+        <span class="bp">self</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="s2">&quot;verbose&quot;</span><span class="p">,</span> <span class="kc">False</span><span class="p">)</span>
 
 <div class="viewcode-block" id="SxPID.is_parallel"><a class="viewcode-back" href="../../idtxl.html#idtxl.estimators_multivariate_pid.SxPID.is_parallel">[docs]</a>    <span class="k">def</span> <span class="nf">is_parallel</span><span class="p">():</span>
         <span class="k">return</span> <span class="kc">False</span></div>
@@ -101,7 +102,8 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
         <span class="k">return</span> <span class="kc">False</span></div>
 
 <div class="viewcode-block" id="SxPID.estimate"><a class="viewcode-back" href="../../idtxl.html#idtxl.estimators_multivariate_pid.SxPID.estimate">[docs]</a>    <span class="k">def</span> <span class="nf">estimate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
+        <span class="sd">&quot;&quot;&quot;Estimate SxPID from list of sources and a target</span>
+
 <span class="sd">        Args:</span>
 <span class="sd">            s : list of numpy arrays</span>
 <span class="sd">                1D arrays containing realizations of a discrete random variable</span>
@@ -109,16 +111,12 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
 <span class="sd">                1D array containing realizations of a discrete random variable</span>
 
 <span class="sd">        Returns:</span>
-<span class="sd">            dict of dict</span>
-<span class="sd">                {</span>
-<span class="sd">                 &#39;ptw&#39; -&gt; { realization -&gt; {alpha -&gt; [float, float, float]} }</span>
-
-<span class="sd">                 &#39;avg&#39; -&gt; {alpha -&gt; [float, float, float]}</span>
-<span class="sd">                }</span>
-<span class="sd">            where the list of floats is ordered</span>
-<span class="sd">            [informative, misinformative, informative - misinformative]</span>
-<span class="sd">            ptw stands for pointwise decomposition</span>
-<span class="sd">            avg stands for average decomposition</span>
+<span class="sd">            dict</span>
+<span class="sd">                SxPID results, with entries</span>
+<span class="sd">                    - &#39;ptw&#39; -&gt; { realization -&gt; {alpha -&gt; [float, float, float]}}: pointwise decomposition</span>
+<span class="sd">                    - &#39;avg&#39; -&gt; {alpha -&gt; [float, float, float]}: average decomposition</span>
+
+<span class="sd">                the list of floats is ordered [informative, misinformative, informative - misinformative]</span>
 <span class="sd">        &quot;&quot;&quot;</span>
         <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">settings</span> <span class="o">=</span> <span class="n">_check_input</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">settings</span><span class="p">)</span>
         <span class="n">pdf</span> <span class="o">=</span> <span class="n">_get_pdf_dict</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
@@ -135,19 +133,20 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
             <span class="n">pdf_orig</span><span class="o">=</span><span class="n">pdf</span><span class="p">,</span>
             <span class="n">chld</span><span class="o">=</span><span class="n">lattices</span><span class="p">[</span><span class="n">num_source_vars</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span>
             <span class="n">achain</span><span class="o">=</span><span class="n">lattices</span><span class="p">[</span><span class="n">num_source_vars</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span>
-            <span class="n">printing</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s1">&#39;verbose&#39;</span><span class="p">])</span>
+            <span class="n">printing</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">],</span>
+        <span class="p">)</span>
 
         <span class="c1"># TODO AskM: Trivariate: does it make sense to name the alphas</span>
         <span class="c1">#    for example shared_syn_s1_s2__syn_s1_s3 ?</span>
         <span class="n">results</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s1">&#39;ptw&#39;</span><span class="p">:</span> <span class="n">retval_ptw</span><span class="p">,</span>
-            <span class="s1">&#39;avg&#39;</span><span class="p">:</span> <span class="n">retval_avg</span><span class="p">,</span>
+            <span class="s2">&quot;ptw&quot;</span><span class="p">:</span> <span class="n">retval_ptw</span><span class="p">,</span>
+            <span class="s2">&quot;avg&quot;</span><span class="p">:</span> <span class="n">retval_avg</span><span class="p">,</span>
         <span class="p">}</span>
         <span class="k">return</span> <span class="n">results</span></div></div>
 
 
 <span class="k">def</span> <span class="nf">_get_pdf_dict</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;&quot;Write probability mass function estimated via counting to a dict.&quot;&quot;&quot;</span>
+    <span class="sd">&quot;&quot;&quot; &quot;Write probability mass function estimated via counting to a dict.&quot;&quot;&quot;</span>
     <span class="c1"># Create dictionary with probability mass function</span>
     <span class="n">counts</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
     <span class="n">n_samples</span> <span class="o">=</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
@@ -171,10 +170,10 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
     <span class="sd">&quot;&quot;&quot;Check input to PID estimators.&quot;&quot;&quot;</span>
     <span class="c1"># Check if inputs are numpy arrays.</span>
     <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">!=</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;Input t must be a numpy array.&#39;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;Input t must be a numpy array.&quot;</span><span class="p">)</span>
     <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)):</span>
         <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">!=</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;All inputs s</span><span class="si">{0}</span><span class="s1"> must be numpy arrays.&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
+            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;All inputs s</span><span class="si">{0}</span><span class="s2"> must be numpy arrays.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
 
     <span class="c1"># In general, IDTxl expects 2D inputs because JIDT/JPYPE only accepts those</span>
     <span class="c1"># and we have a multivariate approach, i.e., a vector is a special case of</span>
@@ -191,34 +190,38 @@ <h1>Source code for idtxl.estimators_multivariate_pid</h1><div class="highlight"
                 <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
                     <span class="n">alph_col</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">][:,</span> <span class="n">col</span><span class="p">]))</span>
                     <span class="n">si_joint</span><span class="p">,</span> <span class="n">alph_new</span> <span class="o">=</span> <span class="n">_join_variables</span><span class="p">(</span>
-                        <span class="n">si_joint</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">][:,</span> <span class="n">col</span><span class="p">],</span> <span class="n">alph_new</span><span class="p">,</span> <span class="n">alph_col</span><span class="p">)</span>
-                <span class="n">settings</span><span class="p">[</span><span class="s1">&#39;alph_s&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">alph_new</span>
+                        <span class="n">si_joint</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">][:,</span> <span class="n">col</span><span class="p">],</span> <span class="n">alph_new</span><span class="p">,</span> <span class="n">alph_col</span>
+                    <span class="p">)</span>
+                <span class="n">settings</span><span class="p">[</span><span class="s2">&quot;alph_s&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">alph_new</span>
             <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;Input source </span><span class="si">{0}</span><span class="s1"> s</span><span class="si">{0}</span><span class="s1"> has to be a 1D or 2D &#39;</span>
-                                 <span class="s1">&#39;numpy array.&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
+                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
+                    <span class="s2">&quot;Input source </span><span class="si">{0}</span><span class="s2"> s</span><span class="si">{0}</span><span class="s2"> has to be a 1D or 2D &quot;</span>
+                    <span class="s2">&quot;numpy array.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+                <span class="p">)</span>
 
     <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">ndim</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
         <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
             <span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
         <span class="k">else</span><span class="p">:</span>  <span class="c1"># For now we only allow 1D-targets</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;Input target t has to be a vector &#39;</span>
-                             <span class="s1">&#39;(t.shape[1]=1).&#39;</span><span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Input target t has to be a vector &quot;</span> <span class="s2">&quot;(t.shape[1]=1).&quot;</span><span class="p">)</span>
 
     <span class="c1"># Check types of remaining inputs.</span>
     <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">dict</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;The settings argument should be a dictionary.&#39;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;The settings argument should be a dictionary.&quot;</span><span class="p">)</span>
     <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)):</span>
         <span class="k">if</span> <span class="ow">not</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">dtype</span><span class="o">.</span><span class="n">type</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">integer</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;Input s</span><span class="si">{0}</span><span class="s1"> (source </span><span class="si">{0}</span><span class="s1">) must be an integer numpy &#39;</span>
-                            <span class="s1">&#39;array.&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
+            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
+                <span class="s2">&quot;Input s</span><span class="si">{0}</span><span class="s2"> (source </span><span class="si">{0}</span><span class="s2">) must be an integer numpy &quot;</span>
+                <span class="s2">&quot;array.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+            <span class="p">)</span>
     <span class="c1"># ^ for</span>
     <span class="k">if</span> <span class="ow">not</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">dtype</span><span class="o">.</span><span class="n">type</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">integer</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;Input t (target) must be an integer numpy array.&#39;</span><span class="p">)</span>
+        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">&quot;Input t (target) must be an integer numpy array.&quot;</span><span class="p">)</span>
 
     <span class="c1"># Check if variables have equal length.</span>
     <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)):</span>
         <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;Number of samples s and t must be equal&#39;</span><span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Number of samples s and t must be equal&quot;</span><span class="p">)</span>
 
     <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">settings</span>
 </pre></div>
@@ -252,7 +255,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_multivariate_pid</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_opencl.html b/docs/html/_modules/idtxl/estimators_opencl.html
index e1d13da..0d1397d 100644
--- a/docs/html/_modules/idtxl/estimators_opencl.html
+++ b/docs/html/_modules/idtxl/estimators_opencl.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_opencl &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_opencl &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_opencl</a></li> 
       </ul>
@@ -889,7 +889,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_opencl</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_pid.html b/docs/html/_modules/idtxl/estimators_pid.html
index 45b51bb..73189db 100644
--- a/docs/html/_modules/idtxl/estimators_pid.html
+++ b/docs/html/_modules/idtxl/estimators_pid.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_pid &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_pid &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_pid</a></li> 
       </ul>
@@ -712,7 +712,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_pid</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/estimators_python.html b/docs/html/_modules/idtxl/estimators_python.html
index ef7d804..af04df4 100644
--- a/docs/html/_modules/idtxl/estimators_python.html
+++ b/docs/html/_modules/idtxl/estimators_python.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.estimators_python &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.estimators_python &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_python</a></li> 
       </ul>
@@ -235,7 +235,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.estimators_python</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/idtxl_exceptions.html b/docs/html/_modules/idtxl/idtxl_exceptions.html
index fab5651..2859fad 100644
--- a/docs/html/_modules/idtxl/idtxl_exceptions.html
+++ b/docs/html/_modules/idtxl/idtxl_exceptions.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.idtxl_exceptions &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.idtxl_exceptions &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_exceptions</a></li> 
       </ul>
@@ -114,7 +114,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_exceptions</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/idtxl_io.html b/docs/html/_modules/idtxl/idtxl_io.html
index 6883e51..7efa6bf 100644
--- a/docs/html/_modules/idtxl/idtxl_io.html
+++ b/docs/html/_modules/idtxl/idtxl_io.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.idtxl_io &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.idtxl_io &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_io</a></li> 
       </ul>
@@ -643,7 +643,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_io</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/idtxl_utils.html b/docs/html/_modules/idtxl/idtxl_utils.html
index 8e8fc0c..3706152 100644
--- a/docs/html/_modules/idtxl/idtxl_utils.html
+++ b/docs/html/_modules/idtxl/idtxl_utils.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.idtxl_utils &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.idtxl_utils &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_utils</a></li> 
       </ul>
@@ -44,11 +44,12 @@ <h3>Navigation</h3>
             
   <h1>Source code for idtxl.idtxl_utils</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Provide IDTxl utility functions.&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">pprint</span>
 <span class="kn">import</span> <span class="nn">copy</span> <span class="k">as</span> <span class="nn">cp</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pprint</span>
 <span class="kn">import</span> <span class="nn">threading</span>
 
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
 
 <div class="viewcode-block" id="swap_chars"><a class="viewcode-back" href="../../idtxl_helper.html#idtxl.idtxl_utils.swap_chars">[docs]</a><span class="k">def</span> <span class="nf">swap_chars</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">i_1</span><span class="p">,</span> <span class="n">i_2</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;Swap to characters in a string.</span>
@@ -361,14 +362,7 @@ <h1>Source code for idtxl.idtxl_utils</h1><div class="highlight"><pre>
     <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">intersect_keys</span><span class="p">:</span>
         <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">dict_1</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">!=</span> <span class="n">dict_2</span><span class="p">[</span><span class="n">k</span><span class="p">])</span><span class="o">.</span><span class="n">any</span><span class="p">():</span>
             <span class="nb">print</span><span class="p">(</span>
-                <span class="s2">&quot;Unequal entries for key &quot;</span>
-                <span class="s2">&quot;</span><span class="si">{0}</span><span class="s2">&quot;</span>
-                <span class="s2">&quot;: dict_1: &quot;</span>
-                <span class="s2">&quot;</span><span class="si">{1}</span><span class="s2">&quot;</span>
-                <span class="s2">&quot;, dict_2:&quot;</span>
-                <span class="s2">&quot; &quot;</span>
-                <span class="s2">&quot;</span><span class="si">{2}</span><span class="s2">&quot;</span>
-                <span class="s2">&quot;.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">dict_1</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">dict_2</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
+                <span class="sa">f</span><span class="s2">&quot;Unequal entries for key </span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">: dict_1: </span><span class="si">{</span><span class="n">dict_1</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="si">}</span><span class="s2">, dict_2: </span><span class="si">{</span><span class="n">dict_2</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span>
             <span class="p">)</span>
             <span class="k">return</span> <span class="kc">True</span>
     <span class="k">return</span> <span class="kc">False</span></div>
@@ -381,9 +375,12 @@ <h1>Source code for idtxl.idtxl_utils</h1><div class="highlight"><pre>
 
 <div class="viewcode-block" id="timeout"><a class="viewcode-back" href="../../idtxl_helper.html#idtxl.idtxl_utils.timeout">[docs]</a><span class="k">class</span> <span class="nc">timeout</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;Context manager for a timeout using threading module.</span>
-<span class="sd">    args:</span>
-<span class="sd">        timeout_duration: float, number of seconds to wait before timeout is triggered</span>
-<span class="sd">        exception_message: string, message to put in the exception</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">        timeout_duration: float</span>
+<span class="sd">            number of seconds to wait before timeout is triggered</span>
+<span class="sd">        exception_message : string</span>
+<span class="sd">            message to put in the exception</span>
 <span class="sd">    &quot;&quot;&quot;</span>
 
     <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">timeout_duration</span><span class="p">,</span> <span class="n">exception_message</span><span class="o">=</span><span class="s2">&quot;Timeout&quot;</span><span class="p">):</span>
@@ -431,7 +428,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.idtxl_utils</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/multivariate_mi.html b/docs/html/_modules/idtxl/multivariate_mi.html
index 8eb897e..d2f713d 100644
--- a/docs/html/_modules/idtxl/multivariate_mi.html
+++ b/docs/html/_modules/idtxl/multivariate_mi.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.multivariate_mi &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.multivariate_mi &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_mi</a></li> 
       </ul>
@@ -52,9 +52,9 @@ <h1>Source code for idtxl.multivariate_mi</h1><div class="highlight"><pre>
 <span class="sd">Note:</span>
 <span class="sd">    Written for Python 3.4+</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">.stats</span> <span class="kn">import</span> <span class="n">network_fdr</span>
 <span class="kn">from</span> <span class="nn">.network_inference</span> <span class="kn">import</span> <span class="n">NetworkInferenceMI</span><span class="p">,</span> <span class="n">NetworkInferenceMultivariate</span>
 <span class="kn">from</span> <span class="nn">.results</span> <span class="kn">import</span> <span class="n">ResultsNetworkInference</span>
+<span class="kn">from</span> <span class="nn">.stats</span> <span class="kn">import</span> <span class="n">network_fdr</span>
 
 
 <div class="viewcode-block" id="MultivariateMI"><a class="viewcode-back" href="../../idtxl_network_inference.html#idtxl.multivariate_mi.MultivariateMI">[docs]</a><span class="k">class</span> <span class="nc">MultivariateMI</span><span class="p">(</span><span class="n">NetworkInferenceMI</span><span class="p">,</span> <span class="n">NetworkInferenceMultivariate</span><span class="p">):</span>
@@ -174,20 +174,18 @@ <h1>Source code for idtxl.multivariate_mi</h1><div class="highlight"><pre>
 
         <span class="c1"># Check which targets and sources are requested for analysis.</span>
         <span class="k">if</span> <span class="n">targets</span> <span class="o">==</span> <span class="s2">&quot;all&quot;</span><span class="p">:</span>
-            <span class="n">targets</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">n_processes</span><span class="p">)]</span>
+            <span class="n">targets</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">n_processes</span><span class="p">))</span>
         <span class="k">if</span> <span class="n">sources</span> <span class="o">==</span> <span class="s2">&quot;all&quot;</span><span class="p">:</span>
             <span class="n">sources</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;all&quot;</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">]</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">):</span>
+        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
             <span class="n">sources</span> <span class="o">=</span> <span class="p">[</span><span class="n">sources</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">]</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">):</span>
+        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">list</span><span class="p">):</span>
             <span class="k">pass</span>
         <span class="k">else</span><span class="p">:</span>
-            <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Sources was not specified correctly: </span><span class="si">{0}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">sources</span><span class="p">))</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">),</span> <span class="p">(</span>
-            <span class="s2">&quot;List of targets and list of &quot;</span>
-            <span class="s2">&quot;sources have to have the same &quot;</span>
-            <span class="s2">&quot;same length&quot;</span>
-        <span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sources was not specified correctly: </span><span class="si">{</span><span class="n">sources</span><span class="si">}</span><span class="s2">.&quot;</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span>
+            <span class="n">targets</span>
+        <span class="p">),</span> <span class="s2">&quot;List of targets and list of sources have to have the same length&quot;</span>
 
         <span class="c1"># Check and set defaults for checkpointing.</span>
         <span class="n">settings</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_checkpointing_defaults</span><span class="p">(</span><span class="n">settings</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">targets</span><span class="p">)</span>
@@ -399,7 +397,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_mi</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/multivariate_pid.html b/docs/html/_modules/idtxl/multivariate_pid.html
index 309215d..9ea31ee 100644
--- a/docs/html/_modules/idtxl/multivariate_pid.html
+++ b/docs/html/_modules/idtxl/multivariate_pid.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.multivariate_pid &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.multivariate_pid &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_pid</a></li> 
       </ul>
@@ -394,7 +394,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_pid</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/multivariate_te.html b/docs/html/_modules/idtxl/multivariate_te.html
index 229e31e..cce9c69 100644
--- a/docs/html/_modules/idtxl/multivariate_te.html
+++ b/docs/html/_modules/idtxl/multivariate_te.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.multivariate_te &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.multivariate_te &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_te</a></li> 
       </ul>
@@ -52,9 +52,9 @@ <h1>Source code for idtxl.multivariate_te</h1><div class="highlight"><pre>
 <span class="sd">Note:</span>
 <span class="sd">    Written for Python 3.4+</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">.network_inference</span> <span class="kn">import</span> <span class="n">NetworkInferenceTE</span><span class="p">,</span> <span class="n">NetworkInferenceMultivariate</span>
-<span class="kn">from</span> <span class="nn">.stats</span> <span class="kn">import</span> <span class="n">network_fdr</span>
+<span class="kn">from</span> <span class="nn">.network_inference</span> <span class="kn">import</span> <span class="n">NetworkInferenceMultivariate</span><span class="p">,</span> <span class="n">NetworkInferenceTE</span>
 <span class="kn">from</span> <span class="nn">.results</span> <span class="kn">import</span> <span class="n">ResultsNetworkInference</span>
+<span class="kn">from</span> <span class="nn">.stats</span> <span class="kn">import</span> <span class="n">network_fdr</span>
 
 
 <div class="viewcode-block" id="MultivariateTE"><a class="viewcode-back" href="../../idtxl_network_inference.html#idtxl.multivariate_te.MultivariateTE">[docs]</a><span class="k">class</span> <span class="nc">MultivariateTE</span><span class="p">(</span><span class="n">NetworkInferenceTE</span><span class="p">,</span> <span class="n">NetworkInferenceMultivariate</span><span class="p">):</span>
@@ -177,20 +177,18 @@ <h1>Source code for idtxl.multivariate_te</h1><div class="highlight"><pre>
 
         <span class="c1"># Check which targets and sources are requested for analysis.</span>
         <span class="k">if</span> <span class="n">targets</span> <span class="o">==</span> <span class="s2">&quot;all&quot;</span><span class="p">:</span>
-            <span class="n">targets</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">n_processes</span><span class="p">)]</span>
+            <span class="n">targets</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">n_processes</span><span class="p">))</span>
         <span class="k">if</span> <span class="n">sources</span> <span class="o">==</span> <span class="s2">&quot;all&quot;</span><span class="p">:</span>
             <span class="n">sources</span> <span class="o">=</span> <span class="p">[</span><span class="s2">&quot;all&quot;</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">]</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">):</span>
+        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
             <span class="n">sources</span> <span class="o">=</span> <span class="p">[</span><span class="n">sources</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">]</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">):</span>
+        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sources</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">list</span><span class="p">):</span>
             <span class="k">pass</span>
         <span class="k">else</span><span class="p">:</span>
-            <span class="ne">ValueError</span><span class="p">(</span><span class="s2">&quot;Sources was not specified correctly: </span><span class="si">{0}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">sources</span><span class="p">))</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">),</span> <span class="p">(</span>
-            <span class="s2">&quot;List of targets and list of &quot;</span>
-            <span class="s2">&quot;sources have to have the same &quot;</span>
-            <span class="s2">&quot;same length&quot;</span>
-        <span class="p">)</span>
+            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Sources was not specified correctly: </span><span class="si">{</span><span class="n">sources</span><span class="si">}</span><span class="s2">.&quot;</span><span class="p">)</span>
+        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span>
+            <span class="n">targets</span>
+        <span class="p">),</span> <span class="s2">&quot;List of targets and list of sources have to have the same length&quot;</span>
 
         <span class="c1"># Check and set defaults for checkpointing.</span>
         <span class="n">settings</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_checkpointing_defaults</span><span class="p">(</span><span class="n">settings</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">targets</span><span class="p">)</span>
@@ -201,16 +199,10 @@ <h1>Source code for idtxl.multivariate_te</h1><div class="highlight"><pre>
             <span class="n">n_realisations</span><span class="o">=</span><span class="n">data</span><span class="o">.</span><span class="n">n_realisations</span><span class="p">(),</span>
             <span class="n">normalised</span><span class="o">=</span><span class="n">data</span><span class="o">.</span><span class="n">normalise</span><span class="p">,</span>
         <span class="p">)</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">)):</span>
+        <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">target</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">targets</span><span class="p">):</span>
             <span class="k">if</span> <span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
-                <span class="nb">print</span><span class="p">(</span>
-                    <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">####### analysing target with index </span><span class="si">{0}</span><span class="s2"> from list </span><span class="si">{1}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                        <span class="n">t</span><span class="p">,</span> <span class="n">targets</span>
-                    <span class="p">)</span>
-                <span class="p">)</span>
-            <span class="n">res_single</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">analyse_single_target</span><span class="p">(</span>
-                <span class="n">settings</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">targets</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">sources</span><span class="p">[</span><span class="n">t</span><span class="p">]</span>
-            <span class="p">)</span>
+                <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">####### analysing target with index </span><span class="si">{</span><span class="n">t</span><span class="si">}</span><span class="s2"> from list </span><span class="si">{</span><span class="n">targets</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+            <span class="n">res_single</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">analyse_single_target</span><span class="p">(</span><span class="n">settings</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">sources</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
             <span class="n">results</span><span class="o">.</span><span class="n">combine_results</span><span class="p">(</span><span class="n">res_single</span><span class="p">)</span>
 
         <span class="c1"># Get no. realisations actually used for estimation from single target</span>
@@ -415,7 +407,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.multivariate_te</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/network_analysis.html b/docs/html/_modules/idtxl/network_analysis.html
index e0c61a4..77eb9a2 100644
--- a/docs/html/_modules/idtxl/network_analysis.html
+++ b/docs/html/_modules/idtxl/network_analysis.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.network_analysis &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.network_analysis &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_analysis</a></li> 
       </ul>
@@ -45,17 +45,19 @@ <h3>Navigation</h3>
   <h1>Source code for idtxl.network_analysis</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Parent class for network inference and network comparison.</span>
 <span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">os.path</span>
-<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
-<span class="kn">from</span> <span class="nn">shutil</span> <span class="kn">import</span> <span class="n">copyfile</span>
-<span class="kn">from</span> <span class="nn">pprint</span> <span class="kn">import</span> <span class="n">pprint</span>
 <span class="kn">import</span> <span class="nn">ast</span>
 <span class="kn">import</span> <span class="nn">copy</span> <span class="k">as</span> <span class="nn">cp</span>
 <span class="kn">import</span> <span class="nn">itertools</span> <span class="k">as</span> <span class="nn">it</span>
+<span class="kn">import</span> <span class="nn">os.path</span>
+<span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">datetime</span>
+<span class="kn">from</span> <span class="nn">pprint</span> <span class="kn">import</span> <span class="n">pprint</span>
+<span class="kn">from</span> <span class="nn">shutil</span> <span class="kn">import</span> <span class="n">copyfile</span>
+
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">.estimator</span> <span class="kn">import</span> <span class="n">get_estimator</span>
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_io</span> <span class="k">as</span> <span class="n">io</span>
+<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
+<span class="kn">from</span> <span class="nn">.estimator</span> <span class="kn">import</span> <span class="n">get_estimator</span>
 
 
 <div class="viewcode-block" id="NetworkAnalysis"><a class="viewcode-back" href="../../idtxl.html#idtxl.network_analysis.NetworkAnalysis">[docs]</a><span class="k">class</span> <span class="nc">NetworkAnalysis</span><span class="p">:</span>
@@ -707,7 +709,7 @@ <h1>Source code for idtxl.network_analysis</h1><div class="highlight"><pre>
 
 <span class="sd">        Args:</span>
 <span class="sd">            file_path : str</span>
-<span class="sd">                path to checkpoint file (excluding extension: *.ckp)</span>
+<span class="sd">                path to checkpoint file (excluding extension: .ckp)</span>
 <span class="sd">        &quot;&quot;&quot;</span>
 
         <span class="c1"># Read checkpoint</span>
@@ -783,7 +785,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_analysis</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/network_comparison.html b/docs/html/_modules/idtxl/network_comparison.html
index dbd6d57..cf6801d 100644
--- a/docs/html/_modules/idtxl/network_comparison.html
+++ b/docs/html/_modules/idtxl/network_comparison.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.network_comparison &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.network_comparison &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_comparison</a></li> 
       </ul>
@@ -1115,7 +1115,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_comparison</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/network_inference.html b/docs/html/_modules/idtxl/network_inference.html
index dcf79de..2db0a45 100644
--- a/docs/html/_modules/idtxl/network_inference.html
+++ b/docs/html/_modules/idtxl/network_inference.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.network_inference &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.network_inference &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_inference</a></li> 
       </ul>
@@ -45,9 +45,10 @@ <h3>Navigation</h3>
   <h1>Source code for idtxl.network_inference</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Parent class for all network inference.&quot;&quot;&quot;</span>
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">.network_analysis</span> <span class="kn">import</span> <span class="n">NetworkAnalysis</span>
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">stats</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_exceptions</span> <span class="k">as</span> <span class="n">ex</span>
+<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">stats</span>
+<span class="kn">from</span> <span class="nn">.network_analysis</span> <span class="kn">import</span> <span class="n">NetworkAnalysis</span>
 
 
 <div class="viewcode-block" id="NetworkInference"><a class="viewcode-back" href="../../idtxl.html#idtxl.network_inference.NetworkInference">[docs]</a><span class="k">class</span> <span class="nc">NetworkInference</span><span class="p">(</span><span class="n">NetworkAnalysis</span><span class="p">):</span>
@@ -280,6 +281,7 @@ <h1>Source code for idtxl.network_inference</h1><div class="highlight"><pre>
     <span class="k">def</span> <span class="nf">_remove_non_significant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">stat</span><span class="p">):</span>
         <span class="c1"># Remove non-significant sources from the candidate set. Loop</span>
         <span class="c1"># backwards over the candidates to remove them iteratively.</span>
+        <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;removing </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">s</span><span class="p">))</span><span class="si">}</span><span class="s2"> variables after seq. max stats&quot;</span><span class="p">)</span>
         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
             <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
                 <span class="bp">self</span><span class="o">.</span><span class="n">_remove_selected_var</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
@@ -1269,7 +1271,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.network_inference</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/postprocessing.html b/docs/html/_modules/idtxl/postprocessing.html
index 8aaa192..c31edb8 100644
--- a/docs/html/_modules/idtxl/postprocessing.html
+++ b/docs/html/_modules/idtxl/postprocessing.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.postprocessing &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.postprocessing &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.postprocessing</a></li> 
       </ul>
@@ -1726,7 +1726,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.postprocessing</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/results.html b/docs/html/_modules/idtxl/results.html
index 9369472..eaf93d9 100644
--- a/docs/html/_modules/idtxl/results.html
+++ b/docs/html/_modules/idtxl/results.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.results &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.results &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.results</a></li> 
       </ul>
@@ -44,10 +44,12 @@ <h3>Navigation</h3>
             
   <h1>Source code for idtxl.results</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Provide results class for IDTxl network analysis.&quot;&quot;&quot;</span>
+<span class="kn">import</span> <span class="nn">copy</span> <span class="k">as</span> <span class="nn">cp</span>
 <span class="kn">import</span> <span class="nn">sys</span>
 <span class="kn">import</span> <span class="nn">warnings</span>
-<span class="kn">import</span> <span class="nn">copy</span> <span class="k">as</span> <span class="nn">cp</span>
+
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
 
 <span class="n">warnings</span><span class="o">.</span><span class="n">simplefilter</span><span class="p">(</span><span class="n">action</span><span class="o">=</span><span class="s2">&quot;ignore&quot;</span><span class="p">,</span> <span class="n">category</span><span class="o">=</span><span class="ne">FutureWarning</span><span class="p">)</span>
@@ -1203,38 +1205,39 @@ <h1>Source code for idtxl.results</h1><div class="highlight"><pre>
 <span class="sd">                (result[&#39;selected_vars&#39;]) or via dot-notation</span>
 <span class="sd">                (result.selected_vars). Contains keys</span>
 
-<span class="sd">                Process : int</span>
+<span class="sd">                - Process : int</span>
 <span class="sd">                    Process that was optimized</span>
-<span class="sd">                estimation_method : String</span>
+<span class="sd">                - estimation_method : String</span>
 <span class="sd">                    Estimation method that was used for optimization</span>
-<span class="sd">                T_D : float</span>
+<span class="sd">                - T_D : float</span>
 <span class="sd">                    Estimated optimal value for the temporal depth TD</span>
-<span class="sd">                tau_R :</span>
+<span class="sd">                - tau_R :</span>
 <span class="sd">                    Information timescale tau_R, a characteristic timescale of history</span>
 <span class="sd">                    dependence similar to an autocorrelation time.</span>
-<span class="sd">                R_tot : float</span>
+<span class="sd">                - R_tot : float</span>
 <span class="sd">                    Estimated value for the total history dependence Rtot,</span>
-<span class="sd">                AIS_tot : float</span>
+<span class="sd">                - AIS_tot : float</span>
 <span class="sd">                    Estimated value for the total active information storage</span>
-<span class="sd">                opt_number_of_bins_d : int</span>
+<span class="sd">                - opt_number_of_bins_d : int</span>
 <span class="sd">                    Number of bins d for the embedding that yields (R̂tot ,T̂D)</span>
-<span class="sd">                opt_scaling_k : int</span>
+<span class="sd">                - opt_scaling_k : int</span>
 <span class="sd">                    Scaling exponent κ for the embedding that yields (R̂tot , T̂D)</span>
-<span class="sd">                opt_first_bin_size : int</span>
+<span class="sd">                - opt_first_bin_size : int</span>
 <span class="sd">                    Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),</span>
-<span class="sd">                history_dependence : array with floating-point values</span>
+<span class="sd">                - history_dependence : array with floating-point values</span>
 <span class="sd">                    Estimated history dependence for each embedding</span>
-<span class="sd">                firing_rate : float</span>
+<span class="sd">                - firing_rate : float</span>
 <span class="sd">                    Firing rate of the neuron/ spike train</span>
-<span class="sd">                recording_length : float</span>
+<span class="sd">                - recording_length : float</span>
 <span class="sd">                    Length of the recording (in seconds)</span>
-<span class="sd">                H_spiking : float</span>
+<span class="sd">                - H_spiking : float</span>
 <span class="sd">                    Entropy of the spike times</span>
 
 <span class="sd">                if analyse_auto_MI was set to True additionally:</span>
-<span class="sd">                auto_MI : dict</span>
+
+<span class="sd">                - auto_MI : dict</span>
 <span class="sd">                    numpy array of MI values for each delay</span>
-<span class="sd">                auto_MI_delays : list of int</span>
+<span class="sd">                - auto_MI_delays : list of int</span>
 <span class="sd">                    list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay</span>
 
 <span class="sd">        &quot;&quot;&quot;</span>
@@ -1279,7 +1282,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.results</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/single_process_analysis.html b/docs/html/_modules/idtxl/single_process_analysis.html
index d696b68..436434d 100644
--- a/docs/html/_modules/idtxl/single_process_analysis.html
+++ b/docs/html/_modules/idtxl/single_process_analysis.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.single_process_analysis &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.single_process_analysis &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.single_process_analysis</a></li> 
       </ul>
@@ -82,7 +82,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.single_process_analysis</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/stats.html b/docs/html/_modules/idtxl/stats.html
index 145f8fb..0b8cf67 100644
--- a/docs/html/_modules/idtxl/stats.html
+++ b/docs/html/_modules/idtxl/stats.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.stats &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.stats &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.stats</a></li> 
       </ul>
@@ -44,10 +44,13 @@ <h3>Navigation</h3>
             
   <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 <span></span><span class="sd">&quot;&quot;&quot;Provide statistics functions.&quot;&quot;&quot;</span>
+<span class="c1"># pylint: disable=protected-access</span>
 <span class="kn">import</span> <span class="nn">copy</span> <span class="k">as</span> <span class="nn">cp</span>
+
 <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
+
 <span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_exceptions</span> <span class="k">as</span> <span class="n">ex</span>
+<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">idtxl_utils</span> <span class="k">as</span> <span class="n">utils</span>
 
 
 <div class="viewcode-block" id="ais_fdr"><a class="viewcode-back" href="../../idtxl_helper.html#idtxl.stats.ais_fdr">[docs]</a><span class="k">def</span> <span class="nf">ais_fdr</span><span class="p">(</span><span class="n">settings</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="o">*</span><span class="n">results</span><span class="p">):</span>
@@ -104,28 +107,26 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="n">process_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
     <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
     <span class="k">for</span> <span class="n">process</span> <span class="ow">in</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">processes_analysed</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_process</span><span class="p">[</span><span class="n">process</span><span class="p">]</span><span class="o">.</span><span class="n">ais_sign</span><span class="p">:</span>
-            <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_process</span><span class="p">[</span><span class="n">process</span><span class="p">]</span><span class="o">.</span><span class="n">ais_pval</span><span class="p">)</span>
-            <span class="n">process_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">process_idx</span><span class="p">,</span> <span class="n">process</span><span class="p">)</span>
-            <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_mi</span><span class="p">)</span>
+        <span class="n">next_pval</span> <span class="o">=</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_process</span><span class="p">[</span><span class="n">process</span><span class="p">]</span><span class="o">.</span><span class="n">ais_pval</span>
+        <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">next_pval</span> <span class="k">if</span> <span class="n">next_pval</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span>
+        <span class="n">process_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">process_idx</span><span class="p">,</span> <span class="n">process</span><span class="p">)</span>
+        <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_mi</span><span class="p">)</span>
 
     <span class="k">if</span> <span class="n">pval</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
         <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;FDR correction: no links in final results ...</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
         <span class="n">results_comb</span><span class="o">.</span><span class="n">_add_fdr</span><span class="p">(</span><span class="n">fdr</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">,</span> <span class="n">constant</span><span class="o">=</span><span class="n">constant</span><span class="p">)</span>
         <span class="k">return</span> <span class="n">results_comb</span>
 
-    <span class="n">sign</span><span class="p">,</span> <span class="n">thresh</span> <span class="o">=</span> <span class="n">_perform_fdr_corretion</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span>
+    <span class="n">sign</span><span class="p">,</span> <span class="n">thresh</span> <span class="o">=</span> <span class="n">_perform_fdr_corretion</span><span class="p">(</span>
+        <span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">processes_analysed</span><span class="p">)</span>
+    <span class="p">)</span>
 
     <span class="c1"># If the number of permutations for calculating p-values for individual</span>
     <span class="c1"># variables is too low, return without performing any correction.</span>
     <span class="k">if</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">))</span> <span class="o">&gt;</span> <span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;WARNING: Number of permutations (&quot;</span>
-            <span class="s2">&quot;n_perm_max_seq&quot;</span>
-            <span class="s2">&quot;) for at &quot;</span>
-            <span class="s2">&quot;least one target is too low to allow for FDR correction &quot;</span>
-            <span class="s2">&quot;(FDR-threshold: </span><span class="si">{0:.4f}</span><span class="s2">, min. theoretically possible p-value: &quot;</span>
-            <span class="s2">&quot;</span><span class="si">{1}</span><span class="s2">).&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">))</span>
+            <span class="s2">&quot;WARNING: Number of permutations (&#39;n_perm_max_seq&#39;) for at least one target is too low to allow for &quot;</span>
+            <span class="sa">f</span><span class="s2">&quot;FDR correction (FDR-threshold: </span><span class="si">{</span><span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, min. theoretically possible p-value: </span><span class="si">{</span><span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">)</span><span class="si">}</span><span class="s2">).&quot;</span>
         <span class="p">)</span>
         <span class="n">results_comb</span><span class="o">.</span><span class="n">_add_fdr</span><span class="p">(</span><span class="n">fdr</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">,</span> <span class="n">constant</span><span class="o">=</span><span class="n">constant</span><span class="p">)</span>
         <span class="k">return</span> <span class="n">results_comb</span>
@@ -207,23 +208,38 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
     <span class="n">cands</span> <span class="o">=</span> <span class="p">[]</span>
     <span class="k">if</span> <span class="n">correct_by_target</span><span class="p">:</span>  <span class="c1"># whole target</span>
+        <span class="c1"># The total number of tests is the number of targets</span>
+        <span class="n">n_tests</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">targets_analysed</span><span class="p">)</span>
+
         <span class="k">for</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">targets_analysed</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_sign</span><span class="p">:</span>
-                <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_pval</span><span class="p">)</span>
-                <span class="n">target_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">target_idx</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
-                <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_omnibus</span><span class="p">)</span>
+            <span class="n">next_pval</span> <span class="o">=</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_pval</span>
+            <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">next_pval</span> <span class="k">if</span> <span class="n">next_pval</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span>
+            <span class="n">target_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">target_idx</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
+            <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_omnibus</span><span class="p">)</span>
     <span class="k">else</span><span class="p">:</span>  <span class="c1"># individual variables</span>
+        <span class="c1"># The total number of tests is the number of targets times the number</span>
+        <span class="c1"># of source candidates (i.e. source processes * time lags) analyzed for each target</span>
+        <span class="n">n_tests</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span>
+            <span class="nb">len</span><span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">sources_tested</span><span class="p">)</span>
+            <span class="k">for</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">targets_analysed</span>
+        <span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;max_lag_sources&quot;</span><span class="p">]</span> <span class="o">-</span> <span class="n">settings</span><span class="p">[</span><span class="s2">&quot;min_lag_sources&quot;</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+
         <span class="k">for</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">targets_analysed</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_sign</span><span class="p">:</span>
-                <span class="n">n_sign</span> <span class="o">=</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span><span class="o">.</span><span class="n">size</span>
-                <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                    <span class="n">pval</span><span class="p">,</span> <span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span><span class="p">)</span>
-                <span class="p">)</span>
-                <span class="n">target_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">target_idx</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">n_sign</span><span class="p">)</span> <span class="o">*</span> <span class="n">target</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-                <span class="n">cands</span> <span class="o">=</span> <span class="n">cands</span> <span class="o">+</span> <span class="p">(</span>
-                    <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span>
-                <span class="p">)</span>
-                <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_max_seq</span><span class="p">)</span>
+            <span class="k">if</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+                <span class="k">continue</span>
+            <span class="n">n_sign</span> <span class="o">=</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span><span class="o">.</span><span class="n">size</span>
+            <span class="n">pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+                <span class="n">pval</span><span class="p">,</span>
+                <span class="p">[</span>
+                    <span class="n">next_pval</span> <span class="k">if</span> <span class="n">next_pval</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="mi">1</span>
+                    <span class="k">for</span> <span class="n">next_pval</span> <span class="ow">in</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span>
+                        <span class="n">target</span>
+                    <span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span>
+                <span class="p">],</span>
+            <span class="p">)</span>
+            <span class="n">target_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">target_idx</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">n_sign</span><span class="p">)</span> <span class="o">*</span> <span class="n">target</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
+            <span class="n">cands</span> <span class="o">=</span> <span class="n">cands</span> <span class="o">+</span> <span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">target</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)</span>
+            <span class="n">n_perm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n_perm</span><span class="p">,</span> <span class="n">results_comb</span><span class="o">.</span><span class="n">settings</span><span class="o">.</span><span class="n">n_perm_max_seq</span><span class="p">)</span>
 
     <span class="k">if</span> <span class="n">pval</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
         <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;No links in final results ...&quot;</span><span class="p">)</span>
@@ -235,18 +251,14 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
         <span class="p">)</span>
         <span class="k">return</span> <span class="n">results_comb</span>
 
-    <span class="n">sign</span><span class="p">,</span> <span class="n">thresh</span> <span class="o">=</span> <span class="n">_perform_fdr_corretion</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span>
+    <span class="n">sign</span><span class="p">,</span> <span class="n">thresh</span> <span class="o">=</span> <span class="n">_perform_fdr_corretion</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">n_tests</span><span class="p">)</span>
 
     <span class="c1"># If the number of permutations for calculating p-values for individual</span>
     <span class="c1"># variables is too low, return without performing any correction.</span>
     <span class="k">if</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">))</span> <span class="o">&gt;</span> <span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;WARNING: Number of permutations (&quot;</span>
-            <span class="s2">&quot;n_perm_max_seq&quot;</span>
-            <span class="s2">&quot;) for at &quot;</span>
-            <span class="s2">&quot;least one target is too low to allow for FDR correction &quot;</span>
-            <span class="s2">&quot;(FDR-threshold: </span><span class="si">{0:.4f}</span><span class="s2">, min. theoretically possible p-value: &quot;</span>
-            <span class="s2">&quot;</span><span class="si">{1}</span><span class="s2">).&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">))</span>
+            <span class="s2">&quot;WARNING: Number of permutations (&#39;n_perm_max_seq&#39;) for at least one target is too low to allow for &quot;</span>
+            <span class="sa">f</span><span class="s2">&quot;FDR correction (FDR-threshold: </span><span class="si">{</span><span class="n">thresh</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, min. theoretically possible p-value: </span><span class="si">{</span><span class="mi">1</span> <span class="o">/</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_perm</span><span class="p">)</span><span class="si">}</span><span class="s2">).&quot;</span>
         <span class="p">)</span>
         <span class="n">results_comb</span><span class="o">.</span><span class="n">_add_fdr</span><span class="p">(</span>
             <span class="n">fdr</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
@@ -260,10 +272,9 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="c1"># the results object. Create a copy of the results object to leave the</span>
     <span class="c1"># original intact.</span>
     <span class="n">fdr</span> <span class="o">=</span> <span class="n">cp</span><span class="o">.</span><span class="n">deepcopy</span><span class="p">(</span><span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">sign</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sign</span><span class="p">[</span><span class="n">s</span><span class="p">]:</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">target_idx</span><span class="p">):</span>
+        <span class="k">if</span> <span class="ow">not</span> <span class="n">sign</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
             <span class="k">if</span> <span class="n">correct_by_target</span><span class="p">:</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">target_idx</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
                 <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_full</span> <span class="o">=</span> <span class="n">cp</span><span class="o">.</span><span class="n">deepcopy</span><span class="p">(</span>
                     <span class="n">results_comb</span><span class="o">.</span><span class="n">_single_target</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_target</span>
                 <span class="p">)</span>
@@ -273,22 +284,20 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
                 <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_pval</span> <span class="o">=</span> <span class="mi">1</span>
                 <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">omnibus_sign</span> <span class="o">=</span> <span class="kc">False</span>
             <span class="k">else</span><span class="p">:</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">target_idx</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
-                <span class="n">cand</span> <span class="o">=</span> <span class="n">cands</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
-                <span class="n">cand_ind</span> <span class="o">=</span> <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">cand</span><span class="p">)</span>
-                <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">cand_ind</span><span class="p">)</span>
+                <span class="n">cand_ind</span> <span class="o">=</span> <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">cands</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+                <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">cands</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
                 <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span>
                     <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_pval</span><span class="p">,</span> <span class="n">cand_ind</span>
                 <span class="p">)</span>
                 <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_te</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span>
                     <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_sources_te</span><span class="p">,</span> <span class="n">cand_ind</span>
                 <span class="p">)</span>
-                <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_full</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_full</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">cand</span><span class="p">))</span>
+                <span class="n">fdr</span><span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">.</span><span class="n">selected_vars_full</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">cands</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
     <span class="n">results_comb</span><span class="o">.</span><span class="n">_add_fdr</span><span class="p">(</span><span class="n">fdr</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">correct_by_target</span><span class="p">,</span> <span class="n">constant</span><span class="p">)</span>
     <span class="k">return</span> <span class="n">results_comb</span></div>
 
 
-<span class="k">def</span> <span class="nf">_perform_fdr_corretion</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
+<span class="k">def</span> <span class="nf">_perform_fdr_corretion</span><span class="p">(</span><span class="n">pval</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">n_tests</span><span class="p">):</span>
     <span class="sd">&quot;&quot;&quot;Calculate sequential threshold for FDR-correction.</span>
 
 <span class="sd">    Calculate sequential thresholds for FDR-correction of p-values. The</span>
@@ -306,43 +315,46 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 <span class="sd">            p-values to be corrected</span>
 <span class="sd">        alpha : float</span>
 <span class="sd">            critical alpha level</span>
-<span class="sd">        fdr_constant : int</span>
+<span class="sd">        constant : int</span>
 <span class="sd">            one of two constants used for calculating the FDR-thresholds</span>
 <span class="sd">            according to Genovese (2002): 1 will divide alpha by 1, 2 will</span>
 <span class="sd">            divide alpha by the sum_i(1/i); see the paper for details on the</span>
-<span class="sd">            assumptions (default=2)</span>
+<span class="sd">            assumptions</span>
+<span class="sd">        n_tests : int</span>
+<span class="sd">            total number of tests performed for calculating the FDR-thresholds</span>
 
 <span class="sd">    Returns:</span>
 <span class="sd">        array of bools</span>
-<span class="sd">            significance of p-values</span>
+<span class="sd">            significance of p-values in the order of the input array</span>
 <span class="sd">        array of floats</span>
-<span class="sd">            FDR-thresholds for each p-value</span>
+<span class="sd">            FDR-thresholds for each p-value in increasing order</span>
 <span class="sd">    &quot;&quot;&quot;</span>
     <span class="c1"># Sort all p-values in ascending order.</span>
     <span class="n">sort_idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argsort</span><span class="p">(</span><span class="n">pval</span><span class="p">)</span>
-    <span class="n">pval</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
+    <span class="n">pval_sorted</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">pval</span><span class="p">)</span>
 
     <span class="c1"># Calculate threshold</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">pval</span><span class="o">.</span><span class="n">size</span>
     <span class="k">if</span> <span class="n">constant</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>  <span class="c1"># pick the requested constant (see Genovese, p.872)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1000</span><span class="p">:</span>
-            <span class="n">const</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
+        <span class="k">if</span> <span class="n">n_tests</span> <span class="o">&lt;</span> <span class="mi">1000</span><span class="p">:</span>
+            <span class="n">const</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_tests</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
         <span class="k">else</span><span class="p">:</span>
-            <span class="n">const</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">e</span>  <span class="c1"># aprx. harmonic sum with Euler&#39;s number</span>
+            <span class="n">const</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">n_tests</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">e</span>  <span class="c1"># aprx. harmonic sum with Euler&#39;s number</span>
     <span class="k">elif</span> <span class="n">constant</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
         <span class="c1"># This is less strict than the other one and corresponds to a</span>
         <span class="c1"># Bonoferroni-correction for the first p-value, however, it makes more</span>
         <span class="c1"># strict assumptions on the distribution of p-values, while constant 2</span>
         <span class="c1"># works for any joint distribution of the p-values.</span>
         <span class="n">const</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">thresh</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">alpha</span> <span class="o">/</span> <span class="n">const</span>
+
+    <span class="c1"># Calculate threshold for each p-value.</span>
+    <span class="n">thresh</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">pval_sorted</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_tests</span><span class="p">)</span> <span class="o">*</span> <span class="n">alpha</span> <span class="o">/</span> <span class="n">const</span>
 
     <span class="c1"># Compare data to threshold.</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="n">pval</span> <span class="o">&lt;=</span> <span class="n">thresh</span>
+    <span class="n">sign</span> <span class="o">=</span> <span class="n">pval_sorted</span> <span class="o">&lt;=</span> <span class="n">thresh</span>
     <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">sign</span><span class="p">)</span><span class="o">.</span><span class="n">any</span><span class="p">():</span>
         <span class="n">first_false</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">sign</span><span class="p">))[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
         <span class="n">sign</span><span class="p">[</span><span class="n">first_false</span><span class="p">:]</span> <span class="o">=</span> <span class="kc">False</span>  <span class="c1"># avoids false positives due to equal pvals</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="n">sign</span><span class="p">[</span><span class="n">sort_idx</span><span class="p">]</span>  <span class="c1"># restore original ordering of significance values</span>
+    <span class="n">sign</span><span class="p">[</span><span class="n">sort_idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">sign</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>  <span class="c1"># restore original ordering of significance values</span>
     <span class="k">return</span> <span class="n">sign</span><span class="p">,</span> <span class="n">thresh</span>
 
 
@@ -377,7 +389,7 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 <span class="sd">        float</span>
 <span class="sd">            the test&#39;s p-value</span>
 <span class="sd">        float</span>
-<span class="sd">            the estimated test statisic, i.e., the information transfer from</span>
+<span class="sd">            the estimated test statistic, i.e., the information transfer from</span>
 <span class="sd">            all sources into the target</span>
 
 <span class="sd">    Raises:</span>
@@ -546,13 +558,12 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 <span class="sd">    calculation of surrogates for this statistic.</span>
 
 <span class="sd">    Args:</span>
-
 <span class="sd">        analysis_setup : MultivariateTE instance</span>
 <span class="sd">            information on the current analysis, can have an optional attribute</span>
-<span class="sd">            &#39;settings&#39;, a dictionary with parameters for statistical testing:</span>
+<span class="sd">            settings, a dictionary with parameters for statistical testing:</span>
 
 <span class="sd">            - n_perm_max_seq : int [optional] - number of permutations</span>
-<span class="sd">              (default=&#39;n_perm_min_stat&#39;|500)</span>
+<span class="sd">              (default=n_perm_min_stat|500)</span>
 <span class="sd">            - alpha_max_seq : float [optional] - critical alpha level</span>
 <span class="sd">              (default=0.05)</span>
 <span class="sd">            - permute_in_time : bool [optional] - generate surrogates by</span>
@@ -580,8 +591,7 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 
     <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;sequential maximum statistic, n_perm: </span><span class="si">{0}</span><span class="s2">, testing </span><span class="si">{1}</span><span class="s2"> selected&quot;</span>
-            <span class="s2">&quot; sources&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n_permutations</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span>
+            <span class="sa">f</span><span class="s2">&quot;sequential maximum statistic, n_perm: </span><span class="si">{</span><span class="n">n_permutations</span><span class="si">}</span><span class="s2">, testing </span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)</span><span class="si">}</span><span class="s2"> selected sources&quot;</span>
         <span class="p">)</span>
 
     <span class="k">assert</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">,</span> <span class="s2">&quot;No sources to test.&quot;</span>
@@ -670,11 +680,9 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
                 <span class="c1">#  we&#39;ll terminate the max sequential stats test,</span>
                 <span class="c1">#  and declare all not significant</span>
                 <span class="nb">print</span><span class="p">(</span>
-                    <span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                        <span class="n">aee</span><span class="o">.</span><span class="n">message</span>
-                    <span class="p">)</span>
+                    <span class="sa">f</span><span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{</span><span class="n">aee</span><span class="o">.</span><span class="n">message</span><span class="si">}</span><span class="s2">. &quot;</span>
+                    <span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span>
                 <span class="p">)</span>
-                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span><span class="p">)</span>
                 <span class="k">return</span> <span class="p">(</span>
                     <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">),</span>
                     <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
@@ -695,11 +703,9 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
         <span class="c1">#  we&#39;ll terminate the max sequential stats test,</span>
         <span class="c1">#  and declare all not significant</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                <span class="n">aee</span><span class="o">.</span><span class="n">message</span>
-            <span class="p">)</span>
+            <span class="sa">f</span><span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{</span><span class="n">aee</span><span class="o">.</span><span class="n">message</span><span class="si">}</span><span class="s2">. &quot;</span>
+            <span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span>
         <span class="p">)</span>
-        <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span><span class="p">)</span>
         <span class="c1"># For now we don&#39;t need a stack trace:</span>
         <span class="c1"># traceback.print_tb(aee.__traceback__)</span>
         <span class="c1"># Return (signficance, pvalue, TEs):</span>
@@ -725,15 +731,12 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
         <span class="n">pvalue</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span>
         <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">:</span>  <span class="c1"># break as soon as a candidate is no longer significant</span>
             <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
-                <span class="nb">print</span><span class="p">(</span>
-                    <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Stopping sequential max stats at candidate with rank &quot;</span>
-                    <span class="s2">&quot;</span><span class="si">{0}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-                <span class="p">)</span>
+                <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Stopping sequential max stats at candidate with rank </span><span class="si">{</span><span class="n">c</span><span class="si">}</span><span class="s2">.&quot;</span><span class="p">)</span>
             <span class="k">break</span>
 
     <span class="c1"># Get back original order and return results.</span>
-    <span class="n">significance</span> <span class="o">=</span> <span class="n">significance</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span>
-    <span class="n">pvalue</span> <span class="o">=</span> <span class="n">pvalue</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span>
+    <span class="n">significance</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span> <span class="o">=</span> <span class="n">significance</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+    <span class="n">pvalue</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span> <span class="o">=</span> <span class="n">pvalue</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
     <span class="k">return</span> <span class="n">significance</span><span class="p">,</span> <span class="n">pvalue</span><span class="p">,</span> <span class="n">individual_stat</span></div>
 
 
@@ -764,13 +767,12 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 <span class="sd">    calculation of surrogates for this statistic.</span>
 
 <span class="sd">    Args:</span>
-
 <span class="sd">        analysis_setup : MultivariateTE instance</span>
 <span class="sd">            information on the current analysis, can have an optional attribute</span>
-<span class="sd">            &#39;settings&#39;, a dictionary with parameters for statistical testing:</span>
+<span class="sd">            settings, a dictionary with parameters for statistical testing:</span>
 
 <span class="sd">            - n_perm_max_seq : int [optional] - number of permutations</span>
-<span class="sd">              (default=&#39;n_perm_min_stat&#39;|500)</span>
+<span class="sd">              (default=n_perm_min_stat|500)</span>
 <span class="sd">            - alpha_max_seq : float [optional] - critical alpha level</span>
 <span class="sd">              (default=0.05)</span>
 <span class="sd">            - permute_in_time : bool [optional] - generate surrogates by</span>
@@ -798,8 +800,7 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
 
     <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;sequential maximum statistic, n_perm: </span><span class="si">{0}</span><span class="s2">, testing </span><span class="si">{1}</span><span class="s2"> selected&quot;</span>
-            <span class="s2">&quot; sources&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n_permutations</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span>
+            <span class="sa">f</span><span class="s2">&quot;sequential maximum statistic, n_perm: </span><span class="si">{</span><span class="n">n_permutations</span><span class="si">}</span><span class="s2">, testing </span><span class="si">{</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)</span><span class="si">}</span><span class="s2"> selected sources&quot;</span>
         <span class="p">)</span>
 
     <span class="k">assert</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">,</span> <span class="s2">&quot;No sources to test.&quot;</span>
@@ -821,12 +822,17 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="p">)</span>
     <span class="n">significance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>
     <span class="n">pvalue</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span>
-    <span class="n">stat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span>
+    <span class="n">individual_stat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span>
     <span class="k">for</span> <span class="n">source</span> <span class="ow">in</span> <span class="n">significant_sources</span><span class="p">:</span>
         <span class="c1"># Find selected past variables for current source</span>
         <span class="n">source_vars</span> <span class="o">=</span> <span class="p">[</span>
             <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span> <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">source</span>
         <span class="p">]</span>
+        <span class="n">source_vars_idx</span> <span class="o">=</span> <span class="p">[</span>
+            <span class="n">i</span>
+            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)</span>
+            <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">source</span>
+        <span class="p">]</span>
 
         <span class="c1"># Determine length of conditioning set and allocate memory.</span>
         <span class="n">idx_conditional</span> <span class="o">=</span> <span class="n">source_vars</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
@@ -858,8 +864,8 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
             <span class="n">temp_cand</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">get_realisations</span><span class="p">(</span>
                 <span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">,</span> <span class="p">[</span><span class="n">candidate</span><span class="p">]</span>
             <span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="c1"># The following may happen if either the requested conditing is</span>
-            <span class="c1"># &#39;none&#39; or if the conditiong set that is tested consists only of</span>
+            <span class="c1"># The following may happen if either the requested conditioning is</span>
+            <span class="c1"># &#39;none&#39; or if the conditioning set that is tested consists only of</span>
             <span class="c1"># a single candidate.</span>
             <span class="k">if</span> <span class="n">temp_cond</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
                 <span class="n">conditional_realisations</span> <span class="o">=</span> <span class="n">conditional_realisations_target</span>
@@ -876,58 +882,60 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
             <span class="n">i_1</span> <span class="o">=</span> <span class="n">i_2</span>
             <span class="n">i_2</span> <span class="o">+=</span> <span class="n">data</span><span class="o">.</span><span class="n">n_realisations</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">)</span>
 
-        <span class="c1"># Generate surrogates for the current candidate.</span>
-        <span class="k">if</span> <span class="p">(</span>
-            <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">is_analytic_null_estimator</span><span class="p">()</span>
-            <span class="ow">and</span> <span class="n">permute_in_time</span>
-        <span class="p">):</span>
-            <span class="c1"># Generate the surrogates analytically</span>
-            <span class="n">surr_table</span><span class="p">[</span>
-                <span class="n">idx_c</span><span class="p">,</span> <span class="p">:</span>
-            <span class="p">]</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_surrogates_analytic</span><span class="p">(</span>
-                <span class="n">n_perm</span><span class="o">=</span><span class="n">n_permutations</span><span class="p">,</span>
-                <span class="n">var1</span><span class="o">=</span><span class="n">data</span><span class="o">.</span><span class="n">get_realisations</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">,</span> <span class="p">[</span><span class="n">candidate</span><span class="p">])[</span>
-                    <span class="mi">0</span>
-                <span class="p">],</span>
-                <span class="n">var2</span><span class="o">=</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">_current_value_realisations</span><span class="p">,</span>
-                <span class="n">conditional</span><span class="o">=</span><span class="n">temp_cond</span><span class="p">,</span>
-            <span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;analytical_surrogates&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
-            <span class="n">surr_candidate_realisations</span> <span class="o">=</span> <span class="n">_get_surrogates</span><span class="p">(</span>
-                <span class="n">data</span><span class="p">,</span>
-                <span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">,</span>
-                <span class="p">[</span><span class="n">candidate</span><span class="p">],</span>
-                <span class="n">n_permutations</span><span class="p">,</span>
-                <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">,</span>
-            <span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">surr_table</span><span class="p">[</span><span class="n">idx_c</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_parallel</span><span class="p">(</span>
-                    <span class="n">n_chunks</span><span class="o">=</span><span class="n">n_permutations</span><span class="p">,</span>
-                    <span class="n">re_use</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;var2&quot;</span><span class="p">,</span> <span class="s2">&quot;conditional&quot;</span><span class="p">],</span>
-                    <span class="n">var1</span><span class="o">=</span><span class="n">surr_candidate_realisations</span><span class="p">,</span>
+            <span class="c1"># Generate surrogates for the current candidate.</span>
+            <span class="k">if</span> <span class="p">(</span>
+                <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">is_analytic_null_estimator</span><span class="p">()</span>
+                <span class="ow">and</span> <span class="n">permute_in_time</span>
+            <span class="p">):</span>
+                <span class="c1"># Generate the surrogates analytically</span>
+                <span class="n">surr_table</span><span class="p">[</span>
+                    <span class="n">idx_c</span><span class="p">,</span> <span class="p">:</span>
+                <span class="p">]</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_surrogates_analytic</span><span class="p">(</span>
+                    <span class="n">n_perm</span><span class="o">=</span><span class="n">n_permutations</span><span class="p">,</span>
+                    <span class="n">var1</span><span class="o">=</span><span class="n">data</span><span class="o">.</span><span class="n">get_realisations</span><span class="p">(</span>
+                        <span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">,</span> <span class="p">[</span><span class="n">candidate</span><span class="p">]</span>
+                    <span class="p">)[</span><span class="mi">0</span><span class="p">],</span>
                     <span class="n">var2</span><span class="o">=</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">_current_value_realisations</span><span class="p">,</span>
                     <span class="n">conditional</span><span class="o">=</span><span class="n">temp_cond</span><span class="p">,</span>
                 <span class="p">)</span>
-            <span class="k">except</span> <span class="n">ex</span><span class="o">.</span><span class="n">AlgorithmExhaustedError</span> <span class="k">as</span> <span class="n">aee</span><span class="p">:</span>
-                <span class="c1"># The aglorithm cannot continue here, so</span>
-                <span class="c1">#  we&#39;ll terminate the max sequential stats test,</span>
-                <span class="c1">#  and declare all not significant</span>
-                <span class="nb">print</span><span class="p">(</span>
-                    <span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                        <span class="n">aee</span><span class="o">.</span><span class="n">message</span>
-                    <span class="p">)</span>
-                <span class="p">)</span>
-                <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">(</span>
-                    <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">),</span>
-                    <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
-                    <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;analytical_surrogates&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
+                <span class="n">surr_candidate_realisations</span> <span class="o">=</span> <span class="n">_get_surrogates</span><span class="p">(</span>
+                    <span class="n">data</span><span class="p">,</span>
+                    <span class="n">analysis_setup</span><span class="o">.</span><span class="n">current_value</span><span class="p">,</span>
+                    <span class="p">[</span><span class="n">candidate</span><span class="p">],</span>
+                    <span class="n">n_permutations</span><span class="p">,</span>
+                    <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">,</span>
                 <span class="p">)</span>
+                <span class="k">try</span><span class="p">:</span>
+                    <span class="n">surr_table</span><span class="p">[</span>
+                        <span class="n">idx_c</span><span class="p">,</span> <span class="p">:</span>
+                    <span class="p">]</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_parallel</span><span class="p">(</span>
+                        <span class="n">n_chunks</span><span class="o">=</span><span class="n">n_permutations</span><span class="p">,</span>
+                        <span class="n">re_use</span><span class="o">=</span><span class="p">[</span><span class="s2">&quot;var2&quot;</span><span class="p">,</span> <span class="s2">&quot;conditional&quot;</span><span class="p">],</span>
+                        <span class="n">var1</span><span class="o">=</span><span class="n">surr_candidate_realisations</span><span class="p">,</span>
+                        <span class="n">var2</span><span class="o">=</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">_current_value_realisations</span><span class="p">,</span>
+                        <span class="n">conditional</span><span class="o">=</span><span class="n">temp_cond</span><span class="p">,</span>
+                    <span class="p">)</span>
+                <span class="k">except</span> <span class="n">ex</span><span class="o">.</span><span class="n">AlgorithmExhaustedError</span> <span class="k">as</span> <span class="n">aee</span><span class="p">:</span>
+                    <span class="c1"># The aglorithm cannot continue here, so</span>
+                    <span class="c1">#  we&#39;ll terminate the max sequential stats test,</span>
+                    <span class="c1">#  and declare all not significant</span>
+                    <span class="nb">print</span><span class="p">(</span>
+                        <span class="sa">f</span><span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{</span><span class="n">aee</span><span class="o">.</span><span class="n">message</span><span class="si">}</span><span class="s2">. &quot;</span>
+                        <span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span>
+                    <span class="p">)</span>
+                    <span class="k">return</span> <span class="p">(</span>
+                        <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span>
+                            <span class="nb">bool</span>
+                        <span class="p">),</span>
+                        <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
+                        <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
+                    <span class="p">)</span>
 
-        <span class="c1"># Calculate original statistic (multivariate/bivariate TE/MI)</span>
+        <span class="c1"># Calculate original statistic (bivariate TE/MI)</span>
         <span class="k">try</span><span class="p">:</span>
-            <span class="n">individual_stat</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_parallel</span><span class="p">(</span>
+            <span class="n">individual_stat_source</span> <span class="o">=</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">estimate_parallel</span><span class="p">(</span>
                 <span class="n">n_chunks</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">source_vars</span><span class="p">),</span>
                 <span class="n">re_use</span><span class="o">=</span><span class="n">re_use</span><span class="p">,</span>
                 <span class="n">var1</span><span class="o">=</span><span class="n">candidate_realisations</span><span class="p">,</span>
@@ -939,10 +947,9 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
             <span class="c1">#  we&#39;ll terminate the max sequential stats test,</span>
             <span class="c1">#  and declare all not significant</span>
             <span class="nb">print</span><span class="p">(</span>
-                <span class="s2">&quot;AlgorithmExhaustedError encountered in &quot;</span>
-                <span class="s2">&quot;estimations: </span><span class="si">{}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">aee</span><span class="o">.</span><span class="n">message</span><span class="p">)</span>
+                <span class="sa">f</span><span class="s2">&quot;AlgorithmExhaustedError encountered in estimations: </span><span class="si">{</span><span class="n">aee</span><span class="o">.</span><span class="n">message</span><span class="si">}</span><span class="s2">. &quot;</span>
+                <span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span>
             <span class="p">)</span>
-            <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Stopping sequential max stats at candidate with rank 0&quot;</span><span class="p">)</span>
             <span class="c1"># For now we don&#39;t need a stack trace:</span>
             <span class="c1"># traceback.print_tb(aee.__traceback__)</span>
             <span class="c1"># Return (signficance, pvalue, TEs):</span>
@@ -952,38 +959,40 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
                 <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">)),</span>
             <span class="p">)</span>
 
-        <span class="n">selected_vars_order</span> <span class="o">=</span> <span class="n">utils</span><span class="o">.</span><span class="n">argsort_descending</span><span class="p">(</span><span class="n">individual_stat</span><span class="p">)</span>
-        <span class="n">individual_stat_sorted</span> <span class="o">=</span> <span class="n">utils</span><span class="o">.</span><span class="n">sort_descending</span><span class="p">(</span><span class="n">individual_stat</span><span class="p">)</span>
+        <span class="n">selected_vars_order</span> <span class="o">=</span> <span class="n">utils</span><span class="o">.</span><span class="n">argsort_descending</span><span class="p">(</span><span class="n">individual_stat_source</span><span class="p">)</span>
+        <span class="n">individual_stat_source_sorted</span> <span class="o">=</span> <span class="n">utils</span><span class="o">.</span><span class="n">sort_descending</span><span class="p">(</span><span class="n">individual_stat_source</span><span class="p">)</span>
         <span class="n">max_distribution</span> <span class="o">=</span> <span class="n">_sort_table_max</span><span class="p">(</span><span class="n">surr_table</span><span class="p">)</span>
+        <span class="n">significance_source</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">individual_stat_source</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">bool</span><span class="p">)</span>
+        <span class="n">pvalue_source</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">individual_stat_source</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
 
         <span class="c1"># Compare each original value with the distribution of the same rank,</span>
         <span class="c1"># starting with the highest value.</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">individual_stat</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="p">[</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">_find_pvalue</span><span class="p">(</span>
-                <span class="n">individual_stat_sorted</span><span class="p">[</span><span class="n">c</span><span class="p">],</span>
+        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">individual_stat_source</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
+            <span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">_find_pvalue</span><span class="p">(</span>
+                <span class="n">individual_stat_source_sorted</span><span class="p">[</span><span class="n">c</span><span class="p">],</span>
                 <span class="n">max_distribution</span><span class="p">[</span><span class="n">c</span><span class="p">,],</span>
                 <span class="n">alpha</span><span class="p">,</span>
                 <span class="n">tail</span><span class="o">=</span><span class="s2">&quot;one_bigger&quot;</span><span class="p">,</span>
             <span class="p">)</span>
-            <span class="c1"># Write results into an array with the same order as the set of</span>
-            <span class="c1"># selected sources from all process. Find the currently tested</span>
-            <span class="c1"># variable and its index in the list of all selected variables.</span>
-            <span class="n">current_var</span> <span class="o">=</span> <span class="n">source_vars</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">[</span><span class="n">c</span><span class="p">]]</span>
-            <span class="k">for</span> <span class="n">ind</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">selected_vars_sources</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">v</span> <span class="o">==</span> <span class="n">current_var</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="n">significance</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span>
-            <span class="n">pvalue</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span>
-            <span class="n">stat</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">=</span> <span class="n">individual_stat_sorted</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
+            <span class="n">significance_source</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span>
+            <span class="n">pvalue_source</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span>
+
             <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">:</span>  <span class="c1"># break as soon as a candidate is no longer significant</span>
                 <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
                     <span class="nb">print</span><span class="p">(</span>
-                        <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Stopping sequential max stats at candidate with &quot;</span>
-                        <span class="s2">&quot;rank </span><span class="si">{0}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
+                        <span class="sa">f</span><span class="s2">&quot;</span><span class="se">\n</span><span class="s2">Stopping sequential max stats at candidate with rank </span><span class="si">{</span><span class="n">c</span><span class="si">}</span><span class="s2">.&quot;</span>
                     <span class="p">)</span>
                 <span class="k">break</span>
 
-    <span class="k">return</span> <span class="n">significance</span><span class="p">,</span> <span class="n">pvalue</span><span class="p">,</span> <span class="n">stat</span></div>
+        <span class="c1"># Get back original order of variables within the current source. Write</span>
+        <span class="c1"># re-ordered results into global results array at the respective</span>
+        <span class="c1"># variable locations.</span>
+        <span class="n">significance_source</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span> <span class="o">=</span> <span class="n">significance_source</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="n">pvalue_source</span><span class="p">[</span><span class="n">selected_vars_order</span><span class="p">]</span> <span class="o">=</span> <span class="n">pvalue_source</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+        <span class="n">significance</span><span class="p">[</span><span class="n">source_vars_idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">significance_source</span>
+        <span class="n">pvalue</span><span class="p">[</span><span class="n">source_vars_idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">pvalue_source</span>
+        <span class="n">individual_stat</span><span class="p">[</span><span class="n">source_vars_idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">individual_stat_source</span>
+    <span class="k">return</span> <span class="n">significance</span><span class="p">,</span> <span class="n">pvalue</span><span class="p">,</span> <span class="n">individual_stat</span></div>
 
 
 <div class="viewcode-block" id="min_statistic"><a class="viewcode-back" href="../../idtxl_helper.html#idtxl.stats.min_statistic">[docs]</a><span class="k">def</span> <span class="nf">min_statistic</span><span class="p">(</span>
@@ -1102,38 +1111,9 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="n">permute_in_time</span> <span class="o">=</span> <span class="n">_check_permute_in_time</span><span class="p">(</span><span class="n">analysis_setup</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">n_perm</span><span class="p">)</span>
     <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;verbose&quot;</span><span class="p">]:</span>
         <span class="nb">print</span><span class="p">(</span>
-            <span class="s2">&quot;mi permutation test against surrogates, n_perm: </span><span class="si">{0}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
-                <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;n_perm_mi&quot;</span><span class="p">]</span>
-            <span class="p">)</span>
+            <span class="sa">f</span><span class="s2">&quot;mi permutation test against surrogates, n_perm: </span><span class="si">{</span><span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s1">&#39;n_perm_mi&#39;</span><span class="p">]</span><span class="si">}</span><span class="s2">&quot;</span>
         <span class="p">)</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    surr_realisations = np.empty(</span>
-<span class="sd">                        (data.n_realisations(analysis_setup.current_value) *</span>
-<span class="sd">                         (n_perm + 1), 1))</span>
-<span class="sd">    i_1 = 0</span>
-<span class="sd">    i_2 = data.n_realisations(analysis_setup.current_value)</span>
-<span class="sd">    # The first chunk holds the original data</span>
-<span class="sd">    surr_realisations[i_1:i_2, ] = analysis_setup._current_value_realisations</span>
-<span class="sd">    # Create surrogate data by shuffling the realisations of the current value.</span>
-<span class="sd">    for perm in range(n_perm):</span>
-<span class="sd">        i_1 = i_2</span>
-<span class="sd">        i_2 += data.n_realisations(analysis_setup.current_value)</span>
-<span class="sd">        # Check the permutation type for the current candidate.</span>
-<span class="sd">        if permute_over_replications:</span>
-<span class="sd">            surr_temp = data.permute_data(analysis_setup.current_value,</span>
-<span class="sd">                                          [analysis_setup.current_value])[0]</span>
-<span class="sd">        else:</span>
-<span class="sd">            [real, repl_idx] = data.get_realisations(</span>
-<span class="sd">                                            analysis_setup.current_value,</span>
-<span class="sd">                                            [analysis_setup.current_value])</span>
-<span class="sd">            surr_temp = _permute_realisations(real, repl_idx, perm_range)</span>
-<span class="sd">        # Add current shuffled realisation to the array of all realisations for</span>
-<span class="sd">        # parallel MI estimation.</span>
-<span class="sd">        # surr_realisations[i_1:i_2, ] = surr_temp</span>
-<span class="sd">        [real, repl_idx] = data.get_realisations(</span>
-<span class="sd">                                            analysis_setup.current_value,</span>
-<span class="sd">                                            [analysis_setup.current_value])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
+
     <span class="k">if</span> <span class="n">analysis_setup</span><span class="o">.</span><span class="n">_cmi_estimator</span><span class="o">.</span><span class="n">is_analytic_null_estimator</span><span class="p">()</span> <span class="ow">and</span> <span class="n">permute_in_time</span><span class="p">:</span>
         <span class="c1"># Generate the surrogates analytically</span>
         <span class="n">analysis_setup</span><span class="o">.</span><span class="n">settings</span><span class="p">[</span><span class="s2">&quot;analytical_surrogates&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
@@ -1557,7 +1537,7 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
     <span class="k">assert</span> <span class="n">distribution</span><span class="o">.</span><span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">,</span> <span class="s2">&quot;Test distribution must be 1D.&quot;</span>
     <span class="n">check_n_perm</span><span class="p">(</span><span class="n">distribution</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">alpha</span><span class="p">)</span>
 
-    <span class="k">if</span> <span class="n">tail</span> <span class="o">==</span> <span class="s2">&quot;one_bigger&quot;</span> <span class="ow">or</span> <span class="n">tail</span> <span class="o">==</span> <span class="s2">&quot;one&quot;</span><span class="p">:</span>
+    <span class="k">if</span> <span class="n">tail</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">&quot;one_bigger&quot;</span><span class="p">,</span> <span class="s2">&quot;one&quot;</span><span class="p">]:</span>
         <span class="n">pvalue</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">distribution</span> <span class="o">&gt;=</span> <span class="n">statistic</span><span class="p">)</span> <span class="o">/</span> <span class="n">distribution</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
     <span class="k">elif</span> <span class="n">tail</span> <span class="o">==</span> <span class="s2">&quot;one_smaller&quot;</span><span class="p">:</span>
         <span class="n">pvalue</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">distribution</span> <span class="o">&lt;=</span> <span class="n">statistic</span><span class="p">)</span> <span class="o">/</span> <span class="n">distribution</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
@@ -1568,20 +1548,7 @@ <h1>Source code for idtxl.stats</h1><div class="highlight"><pre>
         <span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">/</span> <span class="mi">2</span>
     <span class="k">else</span><span class="p">:</span>
         <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="p">(</span>
-                <span class="s2">&quot;Unkown value for &quot;</span>
-                <span class="s2">&quot;tail&quot;</span>
-                <span class="s2">&quot;, should be &quot;</span>
-                <span class="s2">&quot;one&quot;</span>
-                <span class="s2">&quot;, &quot;</span>
-                <span class="s2">&quot;one_bigger&quot;</span>
-                <span class="s2">&quot;,&quot;</span>
-                <span class="s2">&quot; &quot;</span>
-                <span class="s2">&quot;one_smaller&quot;</span>
-                <span class="s2">&quot;, or &quot;</span>
-                <span class="s2">&quot;two&quot;</span>
-                <span class="s2">&quot;): </span><span class="si">{0}</span><span class="s2">.&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tail</span><span class="p">)</span>
-            <span class="p">)</span>
+            <span class="sa">f</span><span class="s2">&quot;Unkown value for tail: </span><span class="si">{</span><span class="n">tail</span><span class="si">}</span><span class="s2">, should be one, one_bigger, one_smaller, or two&quot;</span>
         <span class="p">)</span>
 
     <span class="c1"># If the statistic is larger than all values in the test distribution, set</span>
@@ -1777,7 +1744,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.stats</a></li> 
       </ul>
diff --git a/docs/html/_modules/idtxl/visualise_graph.html b/docs/html/_modules/idtxl/visualise_graph.html
index 118204c..f046873 100644
--- a/docs/html/_modules/idtxl/visualise_graph.html
+++ b/docs/html/_modules/idtxl/visualise_graph.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>idtxl.visualise_graph &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl.visualise_graph &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../../_static/pyramid.css" />
     <script data-url_root="../../" id="documentation_options" src="../../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" accesskey="U">Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.visualise_graph</a></li> 
       </ul>
@@ -401,7 +401,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
           <li class="nav-item nav-item-1"><a href="../index.html" >Module code</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl.visualise_graph</a></li> 
       </ul>
diff --git a/docs/html/_modules/index.html b/docs/html/_modules/index.html
index c7c6937..d735a55 100644
--- a/docs/html/_modules/index.html
+++ b/docs/html/_modules/index.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Overview: module code &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Overview: module code &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="../_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="../_static/pyramid.css" />
     <script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Overview: module code</a></li> 
       </ul>
     </div>  
@@ -100,7 +100,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="../py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="../index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="../index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Overview: module code</a></li> 
       </ul>
     </div>
diff --git a/docs/html/_static/documentation_options.js b/docs/html/_static/documentation_options.js
index eb42507..50f8de0 100644
--- a/docs/html/_static/documentation_options.js
+++ b/docs/html/_static/documentation_options.js
@@ -1,6 +1,6 @@
 var DOCUMENTATION_OPTIONS = {
     URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'),
-    VERSION: '1.5.1',
+    VERSION: '1.6.0',
     LANGUAGE: 'None',
     COLLAPSE_INDEX: false,
     BUILDER: 'html',
diff --git a/docs/html/genindex.html b/docs/html/genindex.html
index 5f30632..72024e7 100644
--- a/docs/html/genindex.html
+++ b/docs/html/genindex.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Index &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Index &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -31,7 +31,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Index</a></li> 
       </ul>
     </div>  
@@ -1058,7 +1058,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Index</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl.html b/docs/html/idtxl.html
index c3a7a26..22f704a 100644
--- a/docs/html/idtxl.html
+++ b/docs/html/idtxl.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>idtxl package &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl package &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -36,7 +36,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_helper.html" title="Helper functions"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl package</a></li> 
       </ul>
     </div>  
@@ -5212,7 +5212,8 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dl class="py method">
 <dt class="sig sig-object py" id="idtxl.estimators_multivariate_pid.SxPID.estimate">
 <span class="sig-name descname"><span class="pre">estimate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">s</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">t</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/idtxl/estimators_multivariate_pid.html#SxPID.estimate"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#idtxl.estimators_multivariate_pid.SxPID.estimate" title="Permalink to this definition">¶</a></dt>
-<dd><dl>
+<dd><p>Estimate SxPID from list of sources and a target</p>
+<dl>
 <dt>Args:</dt><dd><dl class="simple">
 <dt>s<span class="classifier">list of numpy arrays</span></dt><dd><p>1D arrays containing realizations of a discrete random variable</p>
 </dd>
@@ -5221,18 +5222,16 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 </dl>
 </dd>
 <dt>Returns:</dt><dd><dl>
-<dt>dict of dict</dt><dd><dl>
-<dt>{</dt><dd><p>‘ptw’ -&gt; { realization -&gt; {alpha -&gt; [float, float, float]} }</p>
-<p>‘avg’ -&gt; {alpha -&gt; [float, float, float]}</p>
+<dt>dict</dt><dd><dl class="simple">
+<dt>SxPID results, with entries</dt><dd><ul class="simple">
+<li><p>‘ptw’ -&gt; { realization -&gt; {alpha -&gt; [float, float, float]}}: pointwise decomposition</p></li>
+<li><p>‘avg’ -&gt; {alpha -&gt; [float, float, float]}: average decomposition</p></li>
+</ul>
 </dd>
 </dl>
-<p>}</p>
+<p>the list of floats is ordered [informative, misinformative, informative - misinformative]</p>
 </dd>
 </dl>
-<p>where the list of floats is ordered
-[informative, misinformative, informative - misinformative]
-ptw stands for pointwise decomposition
-avg stands for average decomposition</p>
 </dd>
 </dl>
 </dd></dl>
@@ -5971,12 +5970,16 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dt class="sig sig-object py" id="idtxl.idtxl_utils.timeout">
 <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">idtxl.idtxl_utils.</span></span><span class="sig-name descname"><span class="pre">timeout</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">timeout_duration</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">exception_message</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'Timeout'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/idtxl/idtxl_utils.html#timeout"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#idtxl.idtxl_utils.timeout" title="Permalink to this definition">¶</a></dt>
 <dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
-<p>Context manager for a timeout using threading module.
-args:</p>
-<blockquote>
-<div><p>timeout_duration: float, number of seconds to wait before timeout is triggered
-exception_message: string, message to put in the exception</p>
-</div></blockquote>
+<p>Context manager for a timeout using threading module.</p>
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
+<dt>timeout_duration: float</dt><dd><p>number of seconds to wait before timeout is triggered</p>
+</dd>
+<dt>exception_message<span class="classifier">string</span></dt><dd><p>message to put in the exception</p>
+</dd>
+</dl>
+</dd>
+</dl>
 <dl class="py method">
 <dt class="sig sig-object py" id="idtxl.idtxl_utils.timeout.timeout_handler">
 <span class="sig-name descname"><span class="pre">timeout_handler</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/idtxl/idtxl_utils.html#timeout.timeout_handler"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#idtxl.idtxl_utils.timeout.timeout_handler" title="Permalink to this definition">¶</a></dt>
@@ -6006,7 +6009,7 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dd><p>Resume analysis from a checkpoint saved to disk.</p>
 <dl class="simple">
 <dt>Args:</dt><dd><dl class="simple">
-<dt>file_path<span class="classifier">str</span></dt><dd><p>path to checkpoint file (excluding extension: <a href="#id1"><span class="problematic" id="id2">*</span></a>.ckp)</p>
+<dt>file_path<span class="classifier">str</span></dt><dd><p>path to checkpoint file (excluding extension: .ckp)</p>
 </dd>
 </dl>
 </dd>
@@ -7177,44 +7180,87 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 dictionary entries can either be accessed via keywords
 (result[‘selected_vars’]) or via dot-notation
 (result.selected_vars). Contains keys</p>
-<dl class="simple">
+<ul class="simple">
+<li><dl class="simple">
 <dt>Process<span class="classifier">int</span></dt><dd><p>Process that was optimized</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>estimation_method<span class="classifier">String</span></dt><dd><p>Estimation method that was used for optimization</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>T_D<span class="classifier">float</span></dt><dd><p>Estimated optimal value for the temporal depth TD</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>tau_R :</dt><dd><p>Information timescale tau_R, a characteristic timescale of history
 dependence similar to an autocorrelation time.</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>R_tot<span class="classifier">float</span></dt><dd><p>Estimated value for the total history dependence Rtot,</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>AIS_tot<span class="classifier">float</span></dt><dd><p>Estimated value for the total active information storage</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_number_of_bins_d<span class="classifier">int</span></dt><dd><p>Number of bins d for the embedding that yields (R̂tot ,T̂D)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_scaling_k<span class="classifier">int</span></dt><dd><p>Scaling exponent κ for the embedding that yields (R̂tot , T̂D)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_first_bin_size<span class="classifier">int</span></dt><dd><p>Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>history_dependence<span class="classifier">array with floating-point values</span></dt><dd><p>Estimated history dependence for each embedding</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>firing_rate<span class="classifier">float</span></dt><dd><p>Firing rate of the neuron/ spike train</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>recording_length<span class="classifier">float</span></dt><dd><p>Length of the recording (in seconds)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>H_spiking<span class="classifier">float</span></dt><dd><p>Entropy of the spike times</p>
 </dd>
 </dl>
-<p>if analyse_auto_MI was set to True additionally:
-auto_MI : dict</p>
-<blockquote>
-<div><p>numpy array of MI values for each delay</p>
-</div></blockquote>
-<dl class="simple">
+</li>
+</ul>
+<p>if analyse_auto_MI was set to True additionally:</p>
+<ul class="simple">
+<li><dl class="simple">
+<dt>auto_MI<span class="classifier">dict</span></dt><dd><p>numpy array of MI values for each delay</p>
+</dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>auto_MI_delays<span class="classifier">list of int</span></dt><dd><p>list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay</p>
 </dd>
 </dl>
+</li>
+</ul>
 </dd>
 </dl>
 </dd>
@@ -7360,14 +7406,13 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <p>This function will re-use the surrogate table created in the last min-stats
 round if that table is in the analysis_setup. This saves the complete
 calculation of surrogates for this statistic.</p>
-<p>Args:</p>
-<blockquote>
-<div><dl class="simple">
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
 <dt>analysis_setup<span class="classifier">MultivariateTE instance</span></dt><dd><p>information on the current analysis, can have an optional attribute
-‘settings’, a dictionary with parameters for statistical testing:</p>
+settings, a dictionary with parameters for statistical testing:</p>
 <ul class="simple">
 <li><p>n_perm_max_seq : int [optional] - number of permutations
-(default=’n_perm_min_stat’<a href="#id3"><span class="problematic" id="id4">|</span></a>500)</p></li>
+(default=n_perm_min_stat|500)</p></li>
 <li><p>alpha_max_seq : float [optional] - critical alpha level
 (default=0.05)</p></li>
 <li><p>permute_in_time : bool [optional] - generate surrogates by
@@ -7378,8 +7423,7 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dt>data<span class="classifier">Data instance</span></dt><dd><p>raw data</p>
 </dd>
 </dl>
-</div></blockquote>
-<dl class="simple">
+</dd>
 <dt>Returns:</dt><dd><dl class="simple">
 <dt>numpy array, bool</dt><dd><p>statistical significance of each source</p>
 </dd>
@@ -7416,14 +7460,13 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <p>This function will re-use the surrogate table created in the last min-stats
 round if that table is in the analysis_setup. This saves the complete
 calculation of surrogates for this statistic.</p>
-<p>Args:</p>
-<blockquote>
-<div><dl class="simple">
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
 <dt>analysis_setup<span class="classifier">MultivariateTE instance</span></dt><dd><p>information on the current analysis, can have an optional attribute
-‘settings’, a dictionary with parameters for statistical testing:</p>
+settings, a dictionary with parameters for statistical testing:</p>
 <ul class="simple">
 <li><p>n_perm_max_seq : int [optional] - number of permutations
-(default=’n_perm_min_stat’<a href="#id5"><span class="problematic" id="id6">|</span></a>500)</p></li>
+(default=n_perm_min_stat|500)</p></li>
 <li><p>alpha_max_seq : float [optional] - critical alpha level
 (default=0.05)</p></li>
 <li><p>permute_in_time : bool [optional] - generate surrogates by
@@ -7434,8 +7477,7 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 <dt>data<span class="classifier">Data instance</span></dt><dd><p>raw data</p>
 </dd>
 </dl>
-</div></blockquote>
-<dl class="simple">
+</dd>
 <dt>Returns:</dt><dd><dl class="simple">
 <dt>numpy array, bool</dt><dd><p>statistical significance of each source</p>
 </dd>
@@ -7618,7 +7660,7 @@ <h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this
 </dd>
 <dt>float</dt><dd><p>the test’s p-value</p>
 </dd>
-<dt>float</dt><dd><p>the estimated test statisic, i.e., the information transfer from
+<dt>float</dt><dd><p>the estimated test statistic, i.e., the information transfer from
 all sources into the target</p>
 </dd>
 </dl>
@@ -7988,7 +8030,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_helper.html" title="Helper functions"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl package</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_data_class.html b/docs/html/idtxl_data_class.html
index 1dd6c25..8bcb9ce 100644
--- a/docs/html/idtxl_data_class.html
+++ b/docs/html/idtxl_data_class.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>The Data Class &#8212; IDTxl 1.5.1 documentation</title>
+    <title>The Data Class &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="index.html" title="Welcome to IDTxl’s documentation!"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">The Data Class</a></li> 
       </ul>
     </div>  
@@ -939,7 +939,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="index.html" title="Welcome to IDTxl’s documentation!"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">The Data Class</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_estimators.html b/docs/html/idtxl_estimators.html
index 1eb6c1e..901e010 100644
--- a/docs/html/idtxl_estimators.html
+++ b/docs/html/idtxl_estimators.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Information theoretic estimators &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Information theoretic estimators &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_results_class.html" title="The Results Class"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Information theoretic estimators</a></li> 
       </ul>
     </div>  
@@ -1937,7 +1937,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_results_class.html" title="The Results Class"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Information theoretic estimators</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_helper.html b/docs/html/idtxl_helper.html
index a4bb4b2..6779bcc 100644
--- a/docs/html/idtxl_helper.html
+++ b/docs/html/idtxl_helper.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Helper functions &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Helper functions &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_postprocessing.html" title="Postprocessing of inferred networks"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Helper functions</a></li> 
       </ul>
     </div>  
@@ -281,12 +281,16 @@ <h2>utils module<a class="headerlink" href="#utils-module" title="Permalink to t
 <dl class="py class">
 <dt class="sig sig-object py">
 <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">idtxl.idtxl_utils.</span></span><span class="sig-name descname"><span class="pre">timeout</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">timeout_duration</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">exception_message</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'Timeout'</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="_modules/idtxl/idtxl_utils.html#timeout"><span class="viewcode-link"><span class="pre">[source]</span></span></a></dt>
-<dd><p>Context manager for a timeout using threading module.
-args:</p>
-<blockquote>
-<div><p>timeout_duration: float, number of seconds to wait before timeout is triggered
-exception_message: string, message to put in the exception</p>
-</div></blockquote>
+<dd><p>Context manager for a timeout using threading module.</p>
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
+<dt>timeout_duration: float</dt><dd><p>number of seconds to wait before timeout is triggered</p>
+</dd>
+<dt>exception_message<span class="classifier">string</span></dt><dd><p>message to put in the exception</p>
+</dd>
+</dl>
+</dd>
+</dl>
 </dd></dl>
 
 </section>
@@ -420,14 +424,13 @@ <h2>stats module<a class="headerlink" href="#stats-module" title="Permalink to t
 <p>This function will re-use the surrogate table created in the last min-stats
 round if that table is in the analysis_setup. This saves the complete
 calculation of surrogates for this statistic.</p>
-<p>Args:</p>
-<blockquote>
-<div><dl class="simple">
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
 <dt>analysis_setup<span class="classifier">MultivariateTE instance</span></dt><dd><p>information on the current analysis, can have an optional attribute
-‘settings’, a dictionary with parameters for statistical testing:</p>
+settings, a dictionary with parameters for statistical testing:</p>
 <ul class="simple">
 <li><p>n_perm_max_seq : int [optional] - number of permutations
-(default=’n_perm_min_stat’<a href="#id1"><span class="problematic" id="id2">|</span></a>500)</p></li>
+(default=n_perm_min_stat|500)</p></li>
 <li><p>alpha_max_seq : float [optional] - critical alpha level
 (default=0.05)</p></li>
 <li><p>permute_in_time : bool [optional] - generate surrogates by
@@ -438,8 +441,7 @@ <h2>stats module<a class="headerlink" href="#stats-module" title="Permalink to t
 <dt>data<span class="classifier">Data instance</span></dt><dd><p>raw data</p>
 </dd>
 </dl>
-</div></blockquote>
-<dl class="simple">
+</dd>
 <dt>Returns:</dt><dd><dl class="simple">
 <dt>numpy array, bool</dt><dd><p>statistical significance of each source</p>
 </dd>
@@ -476,14 +478,13 @@ <h2>stats module<a class="headerlink" href="#stats-module" title="Permalink to t
 <p>This function will re-use the surrogate table created in the last min-stats
 round if that table is in the analysis_setup. This saves the complete
 calculation of surrogates for this statistic.</p>
-<p>Args:</p>
-<blockquote>
-<div><dl class="simple">
+<dl class="simple">
+<dt>Args:</dt><dd><dl class="simple">
 <dt>analysis_setup<span class="classifier">MultivariateTE instance</span></dt><dd><p>information on the current analysis, can have an optional attribute
-‘settings’, a dictionary with parameters for statistical testing:</p>
+settings, a dictionary with parameters for statistical testing:</p>
 <ul class="simple">
 <li><p>n_perm_max_seq : int [optional] - number of permutations
-(default=’n_perm_min_stat’<a href="#id3"><span class="problematic" id="id4">|</span></a>500)</p></li>
+(default=n_perm_min_stat|500)</p></li>
 <li><p>alpha_max_seq : float [optional] - critical alpha level
 (default=0.05)</p></li>
 <li><p>permute_in_time : bool [optional] - generate surrogates by
@@ -494,8 +495,7 @@ <h2>stats module<a class="headerlink" href="#stats-module" title="Permalink to t
 <dt>data<span class="classifier">Data instance</span></dt><dd><p>raw data</p>
 </dd>
 </dl>
-</div></blockquote>
-<dl class="simple">
+</dd>
 <dt>Returns:</dt><dd><dl class="simple">
 <dt>numpy array, bool</dt><dd><p>statistical significance of each source</p>
 </dd>
@@ -678,7 +678,7 @@ <h2>stats module<a class="headerlink" href="#stats-module" title="Permalink to t
 </dd>
 <dt>float</dt><dd><p>the test’s p-value</p>
 </dd>
-<dt>float</dt><dd><p>the estimated test statisic, i.e., the information transfer from
+<dt>float</dt><dd><p>the estimated test statistic, i.e., the information transfer from
 all sources into the target</p>
 </dd>
 </dl>
@@ -838,7 +838,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_postprocessing.html" title="Postprocessing of inferred networks"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Helper functions</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_network_comparison.html b/docs/html/idtxl_network_comparison.html
index 584408a..0f8b02f 100644
--- a/docs/html/idtxl_network_comparison.html
+++ b/docs/html/idtxl_network_comparison.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Network comparison &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Network comparison &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_process_analysis.html" title="Algorithms for the analysis of node dynamics"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Network comparison</a></li> 
       </ul>
     </div>  
@@ -319,7 +319,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_process_analysis.html" title="Algorithms for the analysis of node dynamics"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Network comparison</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_network_inference.html b/docs/html/idtxl_network_inference.html
index be6534c..164d9ea 100644
--- a/docs/html/idtxl_network_inference.html
+++ b/docs/html/idtxl_network_inference.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Algorithms for network inference &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Algorithms for network inference &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_data_class.html" title="The Data Class"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Algorithms for network inference</a></li> 
       </ul>
     </div>  
@@ -1056,7 +1056,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_data_class.html" title="The Data Class"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Algorithms for network inference</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_postprocessing.html b/docs/html/idtxl_postprocessing.html
index e5f536e..74b6ff4 100644
--- a/docs/html/idtxl_postprocessing.html
+++ b/docs/html/idtxl_postprocessing.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Postprocessing of inferred networks &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Postprocessing of inferred networks &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_estimators.html" title="Information theoretic estimators"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Postprocessing of inferred networks</a></li> 
       </ul>
     </div>  
@@ -657,7 +657,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_estimators.html" title="Information theoretic estimators"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Postprocessing of inferred networks</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_process_analysis.html b/docs/html/idtxl_process_analysis.html
index 7b2c9f1..116a009 100644
--- a/docs/html/idtxl_process_analysis.html
+++ b/docs/html/idtxl_process_analysis.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Algorithms for the analysis of node dynamics &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Algorithms for the analysis of node dynamics &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_network_inference.html" title="Algorithms for network inference"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Algorithms for the analysis of node dynamics</a></li> 
       </ul>
     </div>  
@@ -1753,7 +1753,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_network_inference.html" title="Algorithms for network inference"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Algorithms for the analysis of node dynamics</a></li> 
       </ul>
     </div>
diff --git a/docs/html/idtxl_results_class.html b/docs/html/idtxl_results_class.html
index b1c9e08..9208b19 100644
--- a/docs/html/idtxl_results_class.html
+++ b/docs/html/idtxl_results_class.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>The Results Class &#8212; IDTxl 1.5.1 documentation</title>
+    <title>The Results Class &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -40,7 +40,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_network_comparison.html" title="Network comparison"
              accesskey="P">previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">The Results Class</a></li> 
       </ul>
     </div>  
@@ -705,44 +705,87 @@ <h2>Results node dynamics<a class="headerlink" href="#results-node-dynamics" tit
 dictionary entries can either be accessed via keywords
 (result[‘selected_vars’]) or via dot-notation
 (result.selected_vars). Contains keys</p>
-<dl class="simple">
+<ul class="simple">
+<li><dl class="simple">
 <dt>Process<span class="classifier">int</span></dt><dd><p>Process that was optimized</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>estimation_method<span class="classifier">String</span></dt><dd><p>Estimation method that was used for optimization</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>T_D<span class="classifier">float</span></dt><dd><p>Estimated optimal value for the temporal depth TD</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>tau_R :</dt><dd><p>Information timescale tau_R, a characteristic timescale of history
 dependence similar to an autocorrelation time.</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>R_tot<span class="classifier">float</span></dt><dd><p>Estimated value for the total history dependence Rtot,</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>AIS_tot<span class="classifier">float</span></dt><dd><p>Estimated value for the total active information storage</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_number_of_bins_d<span class="classifier">int</span></dt><dd><p>Number of bins d for the embedding that yields (R̂tot ,T̂D)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_scaling_k<span class="classifier">int</span></dt><dd><p>Scaling exponent κ for the embedding that yields (R̂tot , T̂D)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>opt_first_bin_size<span class="classifier">int</span></dt><dd><p>Size of the first bin τ1 for the embedding that yields (R̂tot , T̂D ),</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>history_dependence<span class="classifier">array with floating-point values</span></dt><dd><p>Estimated history dependence for each embedding</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>firing_rate<span class="classifier">float</span></dt><dd><p>Firing rate of the neuron/ spike train</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>recording_length<span class="classifier">float</span></dt><dd><p>Length of the recording (in seconds)</p>
 </dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>H_spiking<span class="classifier">float</span></dt><dd><p>Entropy of the spike times</p>
 </dd>
 </dl>
-<p>if analyse_auto_MI was set to True additionally:
-auto_MI : dict</p>
-<blockquote>
-<div><p>numpy array of MI values for each delay</p>
-</div></blockquote>
-<dl class="simple">
+</li>
+</ul>
+<p>if analyse_auto_MI was set to True additionally:</p>
+<ul class="simple">
+<li><dl class="simple">
+<dt>auto_MI<span class="classifier">dict</span></dt><dd><p>numpy array of MI values for each delay</p>
+</dd>
+</dl>
+</li>
+<li><dl class="simple">
 <dt>auto_MI_delays<span class="classifier">list of int</span></dt><dd><p>list of delays depending on the given auto_MI_bin_sizes and auto_MI_max_delay</p>
 </dd>
 </dl>
+</li>
+</ul>
 </dd>
 </dl>
 </dd>
@@ -825,7 +868,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_network_comparison.html" title="Network comparison"
              >previous</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">The Results Class</a></li> 
       </ul>
     </div>
diff --git a/docs/html/index.html b/docs/html/index.html
index c491a3c..ffaac49 100644
--- a/docs/html/index.html
+++ b/docs/html/index.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>Welcome to IDTxl’s documentation! &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Welcome to IDTxl’s documentation! &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -36,7 +36,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_data_class.html" title="The Data Class"
              accesskey="N">next</a> |</li>
-        <li class="nav-item nav-item-0"><a href="#">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="#">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Welcome to IDTxl’s documentation!</a></li> 
       </ul>
     </div>  
@@ -206,7 +206,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="idtxl_data_class.html" title="The Data Class"
              >next</a> |</li>
-        <li class="nav-item nav-item-0"><a href="#">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="#">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Welcome to IDTxl’s documentation!</a></li> 
       </ul>
     </div>
diff --git a/docs/html/modules.html b/docs/html/modules.html
index f258837..2fb3f73 100644
--- a/docs/html/modules.html
+++ b/docs/html/modules.html
@@ -6,7 +6,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
 
-    <title>idtxl &#8212; IDTxl 1.5.1 documentation</title>
+    <title>idtxl &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -32,7 +32,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl</a></li> 
       </ul>
     </div>  
@@ -118,7 +118,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">idtxl</a></li> 
       </ul>
     </div>
diff --git a/docs/html/objects.inv b/docs/html/objects.inv
index 351572d735a798d611a9076349e00603b01a4acb..3df3f28c5fe2155bfe4749d860dd0606868ba19a 100644
GIT binary patch
delta 14
VcmX>pdQx<PHM5zX!A84oE&wK11fu`|

delta 14
VcmX>pdQx<PHM6On;YPb|E&wJ~1fu`|

diff --git a/docs/html/py-modindex.html b/docs/html/py-modindex.html
index bcca949..bda581f 100644
--- a/docs/html/py-modindex.html
+++ b/docs/html/py-modindex.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Python Module Index &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Python Module Index &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
@@ -34,7 +34,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="#" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Python Module Index</a></li> 
       </ul>
     </div>  
@@ -223,7 +223,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="#" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Python Module Index</a></li> 
       </ul>
     </div>
diff --git a/docs/html/search.html b/docs/html/search.html
index 0f9bfc4..bee715a 100644
--- a/docs/html/search.html
+++ b/docs/html/search.html
@@ -5,7 +5,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Search &#8212; IDTxl 1.5.1 documentation</title>
+    <title>Search &#8212; IDTxl 1.6.0 documentation</title>
     <link rel="stylesheet" type="text/css" href="_static/pygments.css" />
     <link rel="stylesheet" type="text/css" href="_static/pyramid.css" />
     
@@ -37,7 +37,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Search</a></li> 
       </ul>
     </div>  
@@ -97,7 +97,7 @@ <h3>Navigation</h3>
         <li class="right" >
           <a href="py-modindex.html" title="Python Module Index"
              >modules</a> |</li>
-        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.5.1 documentation</a> &#187;</li>
+        <li class="nav-item nav-item-0"><a href="index.html">IDTxl 1.6.0 documentation</a> &#187;</li>
         <li class="nav-item nav-item-this"><a href="">Search</a></li> 
       </ul>
     </div>
diff --git a/docs/html/searchindex.js b/docs/html/searchindex.js
index 0893c9b..e57c9ab 100644
--- a/docs/html/searchindex.js
+++ b/docs/html/searchindex.js
@@ -1 +1 @@
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\ No newline at end of file

From 4a926b843d918676728ca72e09d62373fd522b56 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 17:00:39 +0100
Subject: [PATCH 21/34] Fix network FDR for empty networks

We previously filtered targets by significant Ombinus tests. Without
this filtering, we may get networks that are empty (i.e., no target has
any incoming links). For empty networks, the Omnibus no. permutations is
None if not explicitely specified by the user. The FDR test crashes if
this case is not explicitely handled when testing for empty networks
When filtering by Omnibus sign., checking the length of the p-values to
be tested was sufficient. Now we also have to check if all Ombines tests
were skipped due to no included source variables by testing if Ombinus
n_perm is None for all p-values.
---
 idtxl/stats.py       |  4 ++--
 test/test_results.py | 22 ++++++++++++----------
 2 files changed, 14 insertions(+), 12 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 7ee3852..de7e84f 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -196,7 +196,7 @@ def network_fdr(settings=None, *results):
             cands = cands + (results_comb._single_target[target].selected_vars_sources)
             n_perm = np.append(n_perm, results_comb.settings.n_perm_max_seq)
 
-    if pval.size == 0:
+    if pval.size == 0 or (n_perm == None).all():  # n_perm is None for empty networks, i.e., no omnibus test was run
         print("No links in final results ...")
         results_comb._add_fdr(
             fdr=None,
@@ -210,7 +210,7 @@ def network_fdr(settings=None, *results):
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
-    if (1 / min(n_perm)) > thresh[0]:
+    if (1 / min(i for i in n_perm if i is not None)) > thresh[0]:
         print(
             "WARNING: Number of permutations ('n_perm_max_seq') for at least one target is too low to allow for "
             f"FDR correction (FDR-threshold: {thresh[0]:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
diff --git a/test/test_results.py b/test/test_results.py
index 616d499..3da5f86 100644
--- a/test/test_results.py
+++ b/test/test_results.py
@@ -1,21 +1,23 @@
 """Test IDTxl results class."""
+import copy as cp
+import itertools as it
 import os
 import pickle
-import pytest
 from tempfile import TemporaryFile
-import itertools as it
-import copy as cp
+
 import numpy as np
-from idtxl.results import AdjacencyMatrix
-from idtxl.multivariate_te import MultivariateTE
-from idtxl.bivariate_te import BivariateTE
-from idtxl.multivariate_mi import MultivariateMI
+import pytest
+from test_estimators_jidt import jpype_missing
+
 from idtxl.bivariate_mi import BivariateMI
-from idtxl.network_comparison import NetworkComparison
+from idtxl.bivariate_te import BivariateTE
 from idtxl.data import Data
 from idtxl.estimators_jidt import JidtDiscreteCMI
-from test_estimators_jidt import jpype_missing
 from idtxl.idtxl_utils import calculate_mi
+from idtxl.multivariate_mi import MultivariateMI
+from idtxl.multivariate_te import MultivariateTE
+from idtxl.network_comparison import NetworkComparison
+from idtxl.results import AdjacencyMatrix
 
 # Use common settings dict that can be used for each test
 settings = {
@@ -457,8 +459,8 @@ def test_adjacency_matrix():
 
 
 if __name__ == "__main__":
-    test_adjacency_matrix()
     test_console_output()
+    test_adjacency_matrix()
     test_results_network_inference()
     test_results_network_comparison()
     test_pickle_results()

From 8c7122a496732c28d3d42f7c9a6568a70aa26c73 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 17:04:07 +0100
Subject: [PATCH 22/34] Ensure network_analysis handles partially empty
 networks

If single links have no significant inputs, i.e., no source variables
are selected, no Omnibus test is performed. If the user did not specify
the no. permutations for the Omnibus test, this parameter is not set.
Ensure that this does not lead to problems when combining results across
targets and one target has no Ombinus test settings, while the other one
does.
---
 test/test_multivariate_te.py | 605 +++++++++++++++++++----------------
 1 file changed, 335 insertions(+), 270 deletions(-)

diff --git a/test/test_multivariate_te.py b/test/test_multivariate_te.py
index 2daaee6..e18c325 100644
--- a/test/test_multivariate_te.py
+++ b/test/test_multivariate_te.py
@@ -1,15 +1,16 @@
 """Provide unit tests for multivariate TE estimation."""
-import pytest
 import itertools as it
+
 import numpy as np
-from idtxl.multivariate_te import MultivariateTE
+import pytest
+from test_checkpointing import _clear_ckp
+from test_estimators_jidt import _get_gauss_data, jpype_missing
+from test_results import _get_discrete_gauss_data
+
 from idtxl.data import Data
 from idtxl.estimators_jidt import JidtDiscreteCMI, JidtKraskovTE
-from test_estimators_jidt import jpype_missing
-from test_results import _get_discrete_gauss_data
-from test_checkpointing import _clear_ckp
 from idtxl.idtxl_utils import calculate_mi
-from test_estimators_jidt import _get_gauss_data
+from idtxl.multivariate_te import MultivariateTE
 
 SEED = 0
 
@@ -22,42 +23,50 @@ def test_gauss_data():
     source = source[1:]
     source_uncorr = source_uncorr[1:]
     target = target[:-1]
-    data = Data(np.hstack((source, source_uncorr, target)),
-                dim_order='sp', normalise=False)
+    data = Data(
+        np.hstack((source, source_uncorr, target)), dim_order="sp", normalise=False
+    )
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'n_perm_max_stat': 21,
-        'n_perm_min_stat': 21,
-        'n_perm_max_seq': 21,
-        'n_perm_omnibus': 21,
-        'max_lag_sources': 2,
-        'min_lag_sources': 1}
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_max_seq": 21,
+        "n_perm_omnibus": 21,
+        "max_lag_sources": 2,
+        "min_lag_sources": 1,
+    }
     nw = MultivariateTE()
-    results = nw.analyse_single_target(
-        settings, data, target=2, sources=[0, 1])
-    te = results.get_single_target(2, fdr=False)['te'][0]
+    results = nw.analyse_single_target(settings, data, target=2, sources=[0, 1])
+    te = results.get_single_target(2, fdr=False)["te"][0]
     sources = results.get_target_sources(2, fdr=False)
 
     # Assert that only the correlated source was detected.
-    assert len(sources) == 1, 'Wrong no. inferred sources: {0}.'.format(
-        len(sources))
-    assert sources[0] == 0, 'Wrong inferred source: {0}.'.format(sources[0])
+    assert len(sources) == 1, "Wrong no. inferred sources: {0}.".format(len(sources))
+    assert sources[0] == 0, "Wrong inferred source: {0}.".format(sources[0])
     # Compare BivarateMI() estimate to JIDT estimate. Mimick realisations used
     # internally by the algorithm.
-    est = JidtKraskovTE({
-        'history_target': 1,
-        'history_source': 1,
-        'source_target_delay': 1,
-        'normalise': False})
+    est = JidtKraskovTE(
+        {
+            "history_target": 1,
+            "history_source": 1,
+            "source_target_delay": 1,
+            "normalise": False,
+        }
+    )
     jidt_cmi = est.estimate(source=source, target=target)
-    print('Estimated MI: {0:0.6f}, estimated MI using JIDT core estimator: '
-          '{1:0.6f} (expected: {2:0.6f}).'.format(te, jidt_cmi, expected_mi))
+    print(
+        "Estimated MI: {0:0.6f}, estimated MI using JIDT core estimator: "
+        "{1:0.6f} (expected: {2:0.6f}).".format(te, jidt_cmi, expected_mi)
+    )
     assert np.isclose(te, jidt_cmi, atol=0.005), (
-        'Estimated MI {0:0.6f} differs from JIDT estimate {1:0.6f} (expected: '
-        'MI {2:0.6f}).'.format(te, jidt_cmi, expected_mi))
-    assert np.isclose(te, expected_mi, atol=0.05), (
-        'Estimated TE {0:0.6f} differs from expected TE {1:0.6f}.'.format(
-            te, expected_mi))
+        "Estimated MI {0:0.6f} differs from JIDT estimate {1:0.6f} (expected: "
+        "MI {2:0.6f}).".format(te, jidt_cmi, expected_mi)
+    )
+    assert np.isclose(
+        te, expected_mi, atol=0.05
+    ), "Estimated TE {0:0.6f} differs from expected TE {1:0.6f}.".format(
+        te, expected_mi
+    )
 
 
 @jpype_missing
@@ -67,64 +76,83 @@ def test_return_local_values():
     data = Data(seed=SEED)
     data.generate_mute_data(500, 5)
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'noise_level': 0,
-        'local_values': True,  # request calculation of local values
-        'n_perm_max_stat': 21,
-        'n_perm_min_stat': 21,
-        'n_perm_max_seq': 21,
-        'n_perm_omnibus': 21,
-        'max_lag_sources': max_lag,
-        'min_lag_sources': 4,
-        'max_lag_target': max_lag}
+        "cmi_estimator": "JidtKraskovCMI",
+        "noise_level": 0,
+        "local_values": True,  # request calculation of local values
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_max_seq": 21,
+        "n_perm_omnibus": 21,
+        "max_lag_sources": max_lag,
+        "min_lag_sources": 4,
+        "max_lag_target": max_lag,
+    }
     target = 3
     sources = [0, 4]
     te = MultivariateTE()
-    results = te.analyse_single_target(
-        settings, data, target=target, sources=sources)
-    settings['local_values'] = False
+    results = te.analyse_single_target(settings, data, target=target, sources=sources)
+    settings["local_values"] = False
     results_avg = te.analyse_single_target(
-        settings, data, target=target, sources=sources)
+        settings, data, target=target, sources=sources
+    )
 
     # Test if any sources were inferred. If not, return (this may happen
     # sometimes due to too few samples, however, a higher no. samples is not
     # feasible for a unit test).
-    if results.get_single_target(target, fdr=False)['te'] is None:
+    if results.get_single_target(target, fdr=False)["te"] is None:
         return
-    if results_avg.get_single_target(target, fdr=False)['te'] is None:
+    if results_avg.get_single_target(target, fdr=False)["te"] is None:
         return
 
-    lte = results.get_single_target(target, fdr=False)['te']
+    lte = results.get_single_target(target, fdr=False)["te"]
     n_sources = len(results.get_target_sources(target, fdr=False))
-    assert type(lte) is np.ndarray, (
-        'LTE estimation did not return an array of values: {0}'.format(lte))
-    assert lte.shape[0] == n_sources, (
-        'Wrong dim (no. sources) in LTE estimate: {0}'.format(lte.shape))
-    assert lte.shape[1] == data.n_realisations_samples((0, max_lag)), (
-        'Wrong dim (no. samples) in LTE estimate: {0}'.format(lte.shape))
-    assert lte.shape[2] == data.n_replications, (
-        'Wrong dim (no. replications) in LTE estimate: {0}'.format(lte.shape))
+    assert (
+        type(lte) is np.ndarray
+    ), "LTE estimation did not return an array of values: {0}".format(lte)
+    assert (
+        lte.shape[0] == n_sources
+    ), "Wrong dim (no. sources) in LTE estimate: {0}".format(lte.shape)
+    assert lte.shape[1] == data.n_realisations_samples(
+        (0, max_lag)
+    ), "Wrong dim (no. samples) in LTE estimate: {0}".format(lte.shape)
+    assert (
+        lte.shape[2] == data.n_replications
+    ), "Wrong dim (no. replications) in LTE estimate: {0}".format(lte.shape)
 
     # Check if average and mean local values are the same. Test each source
     # separately. Inferred sources and variables may differ between the two
     # calls to analyse_single_target() due to low number of surrogates used in
     # unit testing.
-    te_single_link = results_avg.get_single_target(target, fdr=False)['te']
+    te_single_link = results_avg.get_single_target(target, fdr=False)["te"]
     sources_local = results.get_target_sources(target, fdr=False)
     sources_avg = results_avg.get_target_sources(target, fdr=False)
     for s in list(set(sources_avg).intersection(sources_local)):
         i1 = np.where(sources_avg == s)[0][0]
         i2 = np.where(sources_local == s)[0][0]
         # Skip comparison if inferred variables differ between links.
-        vars_local = [v for v in results_avg.get_single_target(target, fdr=False).selected_vars_sources if v[0] == s]
-        vars_avg = [v for v in results.get_single_target(target, fdr=False).selected_vars_sources if v[0] == s]
+        vars_local = [
+            v
+            for v in results_avg.get_single_target(
+                target, fdr=False
+            ).selected_vars_sources
+            if v[0] == s
+        ]
+        vars_avg = [
+            v
+            for v in results.get_single_target(target, fdr=False).selected_vars_sources
+            if v[0] == s
+        ]
         if vars_local != vars_avg:
             continue
-        print('Compare average ({0:.4f}) and local values ({1:.4f}).'.format(
-            te_single_link[i1], np.mean(lte[i2, :, :])))
+        print(
+            "Compare average ({0:.4f}) and local values ({1:.4f}).".format(
+                te_single_link[i1], np.mean(lte[i2, :, :])
+            )
+        )
         assert np.isclose(te_single_link[i1], np.mean(lte[i2, :, :]), rtol=0.00005), (
-            'Single link average MI ({0:.6f}) and mean LMI ({1:.6f}) '
-            ' deviate.'.format(te_single_link[i1], np.mean(lte[i2, :, :])))
+            "Single link average MI ({0:.6f}) and mean LMI ({1:.6f}) "
+            " deviate.".format(te_single_link[i1], np.mean(lte[i2, :, :]))
+        )
 
 
 @jpype_missing
@@ -132,18 +160,18 @@ def test_multivariate_te_init():
     """Test instance creation for MultivariateTE class."""
     # Test error on missing estimator
     settings = {
-        'n_perm_max_stat': 21,
-        'n_perm_omnibus': 30,
-        'max_lag_sources': 7,
-        'min_lag_sources': 2,
-        'max_lag_target': 5}
+        "n_perm_max_stat": 21,
+        "n_perm_omnibus": 30,
+        "max_lag_sources": 7,
+        "min_lag_sources": 2,
+        "max_lag_target": 5,
+    }
     nw = MultivariateTE()
     with pytest.raises(AssertionError):
-        nw.analyse_single_target(
-            settings=settings, data=Data(seed=SEED), target=1)
+        nw.analyse_single_target(settings=settings, data=Data(seed=SEED), target=1)
 
     # Test setting of min and max lags
-    settings['cmi_estimator'] = 'JidtKraskovCMI'
+    settings["cmi_estimator"] = "JidtKraskovCMI"
     data = Data(seed=SEED)
     data.generate_mute_data(n_samples=10, n_replications=5)
 
@@ -151,70 +179,70 @@ def test_multivariate_te_init():
     nw.analyse_single_target(settings=settings, data=data, target=1)
 
     # Valid: max lag sources smaller than max lag target
-    settings['max_lag_sources'] = 3
+    settings["max_lag_sources"] = 3
     nw.analyse_single_target(settings=settings, data=data, target=1)
 
     # Invalid: min lag sources bigger than max lag
-    settings['min_lag_sources'] = 8
-    settings['max_lag_sources'] = 7
-    settings['max_lag_target'] = 5
+    settings["min_lag_sources"] = 8
+    settings["max_lag_sources"] = 7
+    settings["max_lag_target"] = 5
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
 
     # Invalid: taus bigger than lags
-    settings['min_lag_sources'] = 2
-    settings['max_lag_sources'] = 4
-    settings['max_lag_target'] = 5
-    settings['tau_sources'] = 10
+    settings["min_lag_sources"] = 2
+    settings["max_lag_sources"] = 4
+    settings["max_lag_target"] = 5
+    settings["tau_sources"] = 10
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['tau_sources'] = 1
-    settings['tau_target'] = 10
+    settings["tau_sources"] = 1
+    settings["tau_target"] = 10
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
 
     # Invalid: negative lags or taus
-    settings['min_lag_sources'] = 1
-    settings['max_lag_target'] = 5
-    settings['max_lag_sources'] = -7
-    settings['tau_target'] = 1
+    settings["min_lag_sources"] = 1
+    settings["max_lag_target"] = 5
+    settings["max_lag_sources"] = -7
+    settings["tau_target"] = 1
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['max_lag_sources'] = 7
-    settings['min_lag_sources'] = -4
+    settings["max_lag_sources"] = 7
+    settings["min_lag_sources"] = -4
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['min_lag_sources'] = 4
-    settings['max_lag_target'] = -1
+    settings["min_lag_sources"] = 4
+    settings["max_lag_target"] = -1
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['max_lag_target'] = 5
-    settings['tau_sources'] = -1
+    settings["max_lag_target"] = 5
+    settings["tau_sources"] = -1
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['tau_sources'] = 1
-    settings['tau_target'] = -1
+    settings["tau_sources"] = 1
+    settings["tau_target"] = -1
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
 
     # Invalid: lags or taus are no integers
-    settings['tau_target'] = 1
-    settings['min_lag_sources'] = 1.5
+    settings["tau_target"] = 1
+    settings["min_lag_sources"] = 1.5
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['min_lag_sources'] = 1
-    settings['max_lag_sources'] = 1.5
+    settings["min_lag_sources"] = 1
+    settings["max_lag_sources"] = 1.5
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['max_lag_sources'] = 7
-    settings['tau_sources'] = 1.5
+    settings["max_lag_sources"] = 7
+    settings["tau_sources"] = 1.5
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['tau_sources'] = 1
-    settings['tau_target'] = 1.5
+    settings["tau_sources"] = 1
+    settings["tau_target"] = 1.5
     with pytest.raises(RuntimeError):
         nw.analyse_single_target(settings=settings, data=data, target=1)
-    settings['tau_target'] = 1
+    settings["tau_target"] = 1
 
     # Invalid: sources or target is no int
     with pytest.raises(RuntimeError):  # no int
@@ -226,17 +254,13 @@ def test_multivariate_te_init():
     with pytest.raises(RuntimeError):  # wrong type
         nw.analyse_single_target(settings=settings, data=data, target={})
     with pytest.raises(RuntimeError):  # negative
-        nw.analyse_single_target(settings=settings, data=data, target=0,
-                                 sources=-1)
-    with pytest.raises(RuntimeError):   # negative
-        nw.analyse_single_target(settings=settings, data=data, target=0,
-                                 sources=[-1])
+        nw.analyse_single_target(settings=settings, data=data, target=0, sources=-1)
+    with pytest.raises(RuntimeError):  # negative
+        nw.analyse_single_target(settings=settings, data=data, target=0, sources=[-1])
     with pytest.raises(RuntimeError):  # not in data
-        nw.analyse_single_target(settings=settings, data=data, target=0,
-                                 sources=20)
+        nw.analyse_single_target(settings=settings, data=data, target=0, sources=20)
     with pytest.raises(RuntimeError):  # not in data
-        nw.analyse_single_target(settings=settings, data=data, target=0,
-                                 sources=[20])
+        nw.analyse_single_target(settings=settings, data=data, target=0, sources=[20])
 
 
 @jpype_missing
@@ -247,75 +271,85 @@ def test_multivariate_te_one_realisation_per_replication():
     # This is easyer to assert/verify later. We also test data.get_realisations
     # this way.
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'n_perm_max_stat': 21,
-        'max_lag_target': 5,
-        'max_lag_sources': 5,
-        'min_lag_sources': 4}
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "max_lag_target": 5,
+        "max_lag_sources": 5,
+        "min_lag_sources": 4,
+    }
     target = 0
     data = Data(normalise=False, seed=SEED)
     n_repl = 10
     n_procs = 2
-    n_points = n_procs * (settings['max_lag_sources'] + 1) * n_repl
-    data.set_data(np.arange(n_points).reshape(
-        n_procs, settings['max_lag_sources'] + 1, n_repl), 'psr')
+    n_points = n_procs * (settings["max_lag_sources"] + 1) * n_repl
+    data.set_data(
+        np.arange(n_points).reshape(n_procs, settings["max_lag_sources"] + 1, n_repl),
+        "psr",
+    )
     nw_0 = MultivariateTE()
-    nw_0._initialise(settings, data, 'all', target)
-    assert (not nw_0.selected_vars_full)
-    assert (not nw_0.selected_vars_sources)
-    assert (not nw_0.selected_vars_target)
-    assert ((nw_0._replication_index == np.arange(n_repl)).all())
-    assert (nw_0._current_value == (target, max(
-           settings['max_lag_sources'], settings['max_lag_target'])))
-    assert (nw_0._current_value_realisations[:, 0] ==
-            data.data[target, -1, :]).all()
+    nw_0._initialise(settings, data, "all", target)
+    assert not nw_0.selected_vars_full
+    assert not nw_0.selected_vars_sources
+    assert not nw_0.selected_vars_target
+    assert (nw_0._replication_index == np.arange(n_repl)).all()
+    assert nw_0._current_value == (
+        target,
+        max(settings["max_lag_sources"], settings["max_lag_target"]),
+    )
+    assert (nw_0._current_value_realisations[:, 0] == data.data[target, -1, :]).all()
 
 
 @jpype_missing
 def test_faes_method():
     """Check if the Faes method is working."""
-    settings = {'cmi_estimator': 'JidtKraskovCMI',
-                'add_conditionals': 'faes',
-                'max_lag_sources': 5,
-                'min_lag_sources': 3,
-                'max_lag_target': 7}
+    settings = {
+        "cmi_estimator": "JidtKraskovCMI",
+        "add_conditionals": "faes",
+        "max_lag_sources": 5,
+        "min_lag_sources": 3,
+        "max_lag_target": 7,
+    }
     nw_1 = MultivariateTE()
     data = Data(seed=SEED)
     data.generate_mute_data()
     sources = [1, 2, 3]
     target = 0
     nw_1._initialise(settings, data, sources, target)
-    assert (nw_1._selected_vars_sources ==
-            [i for i in it.product(sources, [nw_1.current_value[1]])]), (
-                'Did not add correct additional conditioning vars.')
+    assert nw_1._selected_vars_sources == [
+        i for i in it.product(sources, [nw_1.current_value[1]])
+    ], "Did not add correct additional conditioning vars."
 
 
 @jpype_missing
 def test_add_conditional_manually():
     """Enforce the conditioning on additional variables."""
-    settings = {'cmi_estimator': 'JidtKraskovCMI',
-                'max_lag_sources': 5,
-                'min_lag_sources': 3,
-                'max_lag_target': 7}
+    settings = {
+        "cmi_estimator": "JidtKraskovCMI",
+        "max_lag_sources": 5,
+        "min_lag_sources": 3,
+        "max_lag_target": 7,
+    }
     nw = MultivariateTE()
     data = Data(seed=SEED)
     data.generate_mute_data()
 
     # Add a conditional with a lag bigger than the max_lag requested above
-    settings['add_conditionals'] = (8, 0)
+    settings["add_conditionals"] = (8, 0)
     with pytest.raises(IndexError):
         nw._initialise(settings, data, sources=[1, 2], target=0)
 
     # Add valid conditionals and test if they were added
-    settings['add_conditionals'] = [(0, 1), (1, 3)]
+    settings["add_conditionals"] = [(0, 1), (1, 3)]
     nw._initialise(settings=settings, data=data, target=0, sources=[1, 2])
     # Get list of conditionals after intialisation and convert absolute samples
     # back to lags for comparison.
     cond_list = nw._idx_to_lag(nw.selected_vars_full)
-    assert settings['add_conditionals'][0] in cond_list, (
-        'First enforced conditional is missing from results.')
-    assert settings['add_conditionals'][1] in cond_list, (
-        'Second enforced conditional is missing from results.')
+    assert (
+        settings["add_conditionals"][0] in cond_list
+    ), "First enforced conditional is missing from results."
+    assert (
+        settings["add_conditionals"][1] in cond_list
+    ), "Second enforced conditional is missing from results."
 
 
 @jpype_missing
@@ -328,31 +362,30 @@ def test_check_source_set():
     data = Data(seed=SEED)
     data.generate_mute_data(100, 5)
     nw_0 = MultivariateTE()
-    nw_0.settings = {'verbose': True}
+    nw_0.settings = {"verbose": True}
     # Add list of sources.
     sources = [1, 2, 3]
     nw_0._check_source_set(sources, data.n_processes)
-    assert nw_0.source_set == sources, 'Sources were not added correctly.'
+    assert nw_0.source_set == sources, "Sources were not added correctly."
 
     # Assert that initialisation fails if the target is also in the source list
     sources = [0, 1, 2, 3]
     nw_0.target = 0
     with pytest.raises(RuntimeError):
-        nw_0._check_source_set(sources=[0, 1, 2, 3],
-                               n_processes=data.n_processes)
+        nw_0._check_source_set(sources=[0, 1, 2, 3], n_processes=data.n_processes)
 
     # Test if a single source, no list is added correctly.
     sources = 1
     nw_0._check_source_set(sources, data.n_processes)
-    assert (type(nw_0.source_set) is list)
+    assert type(nw_0.source_set) is list
 
     # Test if 'all' is handled correctly
     nw_0.target = 0
-    nw_0._check_source_set('all', data.n_processes)
-    assert nw_0.source_set == [1, 2, 3, 4], 'Sources were not added correctly.'
+    nw_0._check_source_set("all", data.n_processes)
+    assert nw_0.source_set == [1, 2, 3, 4], "Sources were not added correctly."
 
     # Test invalid inputs.
-    with pytest.raises(RuntimeError):   # sources greater than no. procs
+    with pytest.raises(RuntimeError):  # sources greater than no. procs
         nw_0._check_source_set(8, data.n_processes)
     with pytest.raises(RuntimeError):  # negative value as source
         nw_0._check_source_set(-3, data.n_processes)
@@ -366,32 +399,35 @@ def test_define_candidates():
     max_lag_target = 10
     current_val = (target, 10)
     procs = [target]
-    samples = np.arange(current_val[1] - 1, current_val[1] - max_lag_target,
-                        -tau_target)
+    samples = np.arange(
+        current_val[1] - 1, current_val[1] - max_lag_target, -tau_target
+    )
     # Test if candidates that are added manually to the conditioning set are
     # removed from the candidate set.
     nw = MultivariateTE()
     nw.current_value = current_val
     settings = [
-        {'add_conditionals': None},
-        {'add_conditionals': (2, 3)},
-        {'add_conditionals': [(2, 3), (4, 1)]},
-        {'add_conditionals': [(1, 9)]},
-        {'add_conditionals': [(1, 9), (2, 3), (4, 1)]}]
+        {"add_conditionals": None},
+        {"add_conditionals": (2, 3)},
+        {"add_conditionals": [(2, 3), (4, 1)]},
+        {"add_conditionals": [(1, 9)]},
+        {"add_conditionals": [(1, 9), (2, 3), (4, 1)]},
+    ]
     for s in settings:
         nw.settings = s
         candidates = nw._define_candidates(procs, samples)
-        assert (1, 9) in candidates, 'Sample missing from candidates: (1, 9).'
-        assert (1, 6) in candidates, 'Sample missing from candidates: (1, 6).'
-        assert (1, 3) in candidates, 'Sample missing from candidates: (1, 3).'
-        if s['add_conditionals'] is not None:
-            if type(s['add_conditionals']) is tuple:
-                cond_ind = nw._lag_to_idx([s['add_conditionals']])
+        assert (1, 9) in candidates, "Sample missing from candidates: (1, 9)."
+        assert (1, 6) in candidates, "Sample missing from candidates: (1, 6)."
+        assert (1, 3) in candidates, "Sample missing from candidates: (1, 3)."
+        if s["add_conditionals"] is not None:
+            if isinstace(s["add_conditionals"], tuple):
+                cond_ind = nw._lag_to_idx([s["add_conditionals"]])
             else:
-                cond_ind = nw._lag_to_idx(s['add_conditionals'])
+                cond_ind = nw._lag_to_idx(s["add_conditionals"])
             for c in cond_ind:
-                assert c not in candidates, (
-                    'Sample added erronously to candidates: {}.'.format(c))
+                assert (
+                    c not in candidates
+                ), f"Sample added erronously to candidates: {c}."
 
 
 @jpype_missing
@@ -401,109 +437,135 @@ def test_analyse_network():
     data = Data(seed=SEED)
     data.generate_mute_data(10, 5)
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'n_perm_max_stat': 21,
-        'n_perm_min_stat': 21,
-        'n_perm_max_seq': 21,
-        'n_perm_omnibus': 21,
-        'max_lag_sources': 5,
-        'min_lag_sources': 4,
-        'max_lag_target': 5}
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_max_seq": 21,
+        "n_perm_omnibus": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 4,
+        "max_lag_target": 5,
+    }
     nw_0 = MultivariateTE()
 
     # Test all to all analysis
-    results = nw_0.analyse_network(
-        settings, data, targets='all', sources='all')
+    results = nw_0.analyse_network(settings, data, targets="all", sources="all")
     targets_analysed = results.targets_analysed
     sources = np.arange(n_processes)
-    assert all(np.array(targets_analysed) == np.arange(n_processes)), (
-                'Network analysis did not run on all targets.')
+    assert all(
+        np.array(targets_analysed) == np.arange(n_processes)
+    ), "Network analysis did not run on all targets."
     for t in results.targets_analysed:
         s = np.array(list(set(sources) - set([t])))
-        assert all(np.array(results._single_target[t].sources_tested) == s), (
-                    'Network analysis did not run on all sources for target '
-                    '{0}'. format(t))
+        assert all(
+            np.array(results._single_target[t].sources_tested) == s
+        ), f"Network analysis did not run on all sources for target {t}"
     # Test analysis for subset of targets
     target_list = [1, 2, 3]
-    results = nw_0.analyse_network(
-        settings, data, targets=target_list, sources='all')
+    results = nw_0.analyse_network(settings, data, targets=target_list, sources="all")
     targets_analysed = results.targets_analysed
-    assert all(np.array(targets_analysed) == np.array(target_list)), (
-                'Network analysis did not run on correct subset of targets.')
+    assert all(
+        np.array(targets_analysed) == np.array(target_list)
+    ), "Network analysis did not run on correct subset of targets."
     for t in results.targets_analysed:
         s = np.array(list(set(sources) - set([t])))
-        assert all(np.array(results._single_target[t].sources_tested) == s), (
-                    'Network analysis did not run on all sources for target '
-                    '{0}'. format(t))
+        assert all(
+            np.array(results._single_target[t].sources_tested) == s
+        ), f"Network analysis did not run on all sources for target {t}"
 
     # Test analysis for subset of sources
     source_list = [1, 2, 3]
     target_list = [0, 4]
-    results = nw_0.analyse_network(settings, data, targets=target_list,
-                                   sources=source_list)
+    results = nw_0.analyse_network(
+        settings, data, targets=target_list, sources=source_list
+    )
 
     targets_analysed = results.targets_analysed
-    assert all(np.array(targets_analysed) == np.array(target_list)), (
-                'Network analysis did not run for all targets.')
+    assert all(
+        np.array(targets_analysed) == np.array(target_list)
+    ), "Network analysis did not run for all targets."
     for t in results.targets_analysed:
-        assert all(results._single_target[t].sources_tested ==
-                   np.array(source_list)), (
-                        'Network analysis did not run on the correct subset '
-                        'of sources for target {0}'.format(t))
+        assert all(
+            results._single_target[t].sources_tested == np.array(source_list)
+        ), f"Network analysis did not run on the correct subset of sources for target {t}"
+
+
+def test_analyse_network_empty_target():
+    # Test analyze network when one target has no selected sources such that
+    # for the Omnibus Test the defaults are not set. Ensure, combining results
+    # works if one target has no incoming links.
+    data = Data(seed=SEED)
+    data.generate_mute_data(104, 10)
+    np.random.seed(SEED)
+    data._data[0, :, :] = np.random.rand(104, 10)
+    settings = {
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_max_seq": 21,
+        # "n_perm_omnibus": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 4,
+        "max_lag_target": 5,
+    }
+    # For the first target, no sources are selected and hence Omnibus defaults
+    # are not set. If later a target has inputs, the settings of the combined
+    # results structure will contain the Omnibus settings from that target's
+    # analysis.
+    nw = MultivariateTE()
+    nw.analyse_network(settings, data, targets=[4, 2], sources=[[0], [0, 1, 3]])
 
 
 @jpype_missing
 def test_permute_time():
     """Create surrogates by permuting data in time instead of over replic."""
     # Test if perm type is set to default
-    default = 'random'
+    default = "random"
     data = Data(seed=SEED)
     data.generate_mute_data(10, 5)
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'n_perm_max_stat': 21,
-        'n_perm_max_seq': 21,
-        'n_perm_omnibus': 21,
-        'max_lag_sources': 5,
-        'min_lag_sources': 4,
-        'max_lag_target': 5,
-        'permute_in_time': True}
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_max_seq": 21,
+        "n_perm_omnibus": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 4,
+        "max_lag_target": 5,
+        "permute_in_time": True,
+    }
     nw_0 = MultivariateTE()
-    results = nw_0.analyse_network(
-        settings, data, targets='all', sources='all')
-    assert results.settings.perm_type == default, (
-        'Perm type was not set to default.')
+    results = nw_0.analyse_network(settings, data, targets="all", sources="all")
+    assert results.settings.perm_type == default, "Perm type was not set to default."
 
 
 def test_discrete_input():
     """Test multivariate TE estimation from discrete data."""
     # Generate Gaussian test data
     covariance = 0.4
-    data = _get_discrete_gauss_data(covariance=covariance,
-                                    n=10000,
-                                    delay=1,
-                                    normalise=False,
-                                    seed=SEED)
-    corr_expected = covariance / (
-        1 * np.sqrt(covariance**2 + (1-covariance)**2))
+    data = _get_discrete_gauss_data(
+        covariance=covariance, n=10000, delay=1, normalise=False, seed=SEED
+    )
+    corr_expected = covariance / (1 * np.sqrt(covariance**2 + (1 - covariance) ** 2))
     expected_mi = calculate_mi(corr_expected)
     settings = {
-        'cmi_estimator': 'JidtDiscreteCMI',
-        'discretise_method': 'none',
-        'n_discrete_bins': 5,  # alphabet size of the variables analysed
-        'n_perm_max_stat': 21,
-        'n_perm_omnibus': 30,
-        'n_perm_max_seq': 30,
-        'min_lag_sources': 1,
-        'max_lag_sources': 2,
-        'max_lag_target': 1}
+        "cmi_estimator": "JidtDiscreteCMI",
+        "discretise_method": "none",
+        "n_discrete_bins": 5,  # alphabet size of the variables analysed
+        "n_perm_max_stat": 21,
+        "n_perm_omnibus": 30,
+        "n_perm_max_seq": 30,
+        "min_lag_sources": 1,
+        "max_lag_sources": 2,
+        "max_lag_target": 1,
+    }
     nw = MultivariateTE()
     res = nw.analyse_single_target(settings=settings, data=data, target=1)
-    assert np.isclose(
-        res._single_target[1].omnibus_te, expected_mi, atol=0.05), (
-            'Estimated TE for discrete variables is not correct. Expected: '
-            '{0}, Actual results: {1}.'.format(
-                expected_mi, res._single_target[1].omnibus_te))
+    assert np.isclose(res._single_target[1].omnibus_te, expected_mi, atol=0.05), (
+        "Estimated TE for discrete variables is not correct. Expected: "
+        "{0}, Actual results: {1}.".format(
+            expected_mi, res._single_target[1].omnibus_te
+        )
+    )
 
 
 def test_include_target_candidates():
@@ -535,64 +597,67 @@ def test_checkpoint():
     n_processes = 5  # the MuTE network has 5 nodes
     data = Data(seed=SEED)
     data.generate_mute_data(10, 5)
-    filename_ckp = './my_checkpoint'
+    filename_ckp = "./my_checkpoint"
     settings = {
-        'cmi_estimator': 'JidtKraskovCMI',
-        'n_perm_max_stat': 21,
-        'n_perm_min_stat': 21,
-        'n_perm_max_seq': 21,
-        'n_perm_omnibus': 21,
-        'max_lag_sources': 5,
-        'min_lag_sources': 4,
-        'max_lag_target': 5,
-        'write_ckp': True,
-        'filename_ckp': filename_ckp}
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_max_seq": 21,
+        "n_perm_omnibus": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 4,
+        "max_lag_target": 5,
+        "write_ckp": True,
+        "filename_ckp": filename_ckp,
+    }
     nw_0 = MultivariateTE()
 
     # Test all to all analysis
-    results = nw_0.analyse_network(
-        settings, data, targets='all', sources='all')
+    results = nw_0.analyse_network(settings, data, targets="all", sources="all")
     targets_analysed = results.targets_analysed
     sources = np.arange(n_processes)
-    assert all(np.array(targets_analysed) == np.arange(n_processes)), (
-                'Network analysis did not run on all targets.')
+    assert all(
+        np.array(targets_analysed) == np.arange(n_processes)
+    ), "Network analysis did not run on all targets."
     for t in results.targets_analysed:
         s = np.array(list(set(sources) - set([t])))
-        assert all(np.array(results._single_target[t].sources_tested) == s), (
-                    'Network analysis did not run on all sources for target '
-                    '{0}'. format(t))
+        assert all(
+            np.array(results._single_target[t].sources_tested) == s
+        ), "Network analysis did not run on all sources for target " "{0}".format(t)
     # Test analysis for subset of targets
     target_list = [1, 2, 3]
-    results = nw_0.analyse_network(
-        settings, data, targets=target_list, sources='all')
+    results = nw_0.analyse_network(settings, data, targets=target_list, sources="all")
     targets_analysed = results.targets_analysed
-    assert all(np.array(targets_analysed) == np.array(target_list)), (
-                'Network analysis did not run on correct subset of targets.')
+    assert all(
+        np.array(targets_analysed) == np.array(target_list)
+    ), "Network analysis did not run on correct subset of targets."
     for t in results.targets_analysed:
         s = np.array(list(set(sources) - set([t])))
-        assert all(np.array(results._single_target[t].sources_tested) == s), (
-                    'Network analysis did not run on all sources for target '
-                    '{0}'. format(t))
+        assert all(
+            np.array(results._single_target[t].sources_tested) == s
+        ), "Network analysis did not run on all sources for target " "{0}".format(t)
 
     # Test analysis for subset of sources
     source_list = [1, 2, 3]
     target_list = [0, 4]
-    results = nw_0.analyse_network(settings, data, targets=target_list,
-                                   sources=source_list)
+    results = nw_0.analyse_network(
+        settings, data, targets=target_list, sources=source_list
+    )
 
     targets_analysed = results.targets_analysed
-    assert all(np.array(targets_analysed) == np.array(target_list)), (
-                'Network analysis did not run for all targets.')
+    assert all(
+        np.array(targets_analysed) == np.array(target_list)
+    ), "Network analysis did not run for all targets."
     for t in results.targets_analysed:
-        assert all(results._single_target[t].sources_tested ==
-                   np.array(source_list)), (
-                        'Network analysis did not run on the correct subset '
-                        'of sources for target {0}'.format(t))
+        assert all(
+            results._single_target[t].sources_tested == np.array(source_list)
+        ), f"Network analysis did not run on the correct subset of sources for target {t}"
 
     _clear_ckp(filename_ckp)
 
 
-if __name__ == '__main__':
+if __name__ == "__main__":
+    test_analyse_network_empty_target()
     test_return_local_values()
     test_discrete_input()
     test_analyse_network()

From 4fc866d0cf165ba538cb5958e36711496792a37c Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 17:06:13 +0100
Subject: [PATCH 23/34] Fix formatting and pylint errors

---
 idtxl/results.py | 40 +++++++++++++++++-----------------------
 1 file changed, 17 insertions(+), 23 deletions(-)

diff --git a/idtxl/results.py b/idtxl/results.py
index ce33470..b4ffacf 100644
--- a/idtxl/results.py
+++ b/idtxl/results.py
@@ -171,16 +171,13 @@ def _check_result(self, process, settings):
         # Check if new result process is part of the network
         if process > (self.data_properties.n_nodes - 1):
             raise RuntimeError(
-                "Can not add single result - process {0} is not"
-                " in no. nodes in the data ({1}).".format(
-                    process, self.data_properties.n_nodes
-                )
+                f"Can not add single result - process {process} is not"
+                f" in no. nodes in the data ({self.data_properties.n_nodes})."
             )
         # Don't add duplicate processes
         if self._is_duplicate_process(process):
             raise RuntimeError(
-                "Can not add single result - results for target"
-                " or process {0} already exist.".format(process)
+                f"Can not add single result - results for target or process {process} already exist."
             )
         # Don't add results with conflicting settings
         if utils.conflicting_entries(self.settings, settings):
@@ -192,8 +189,7 @@ def _is_duplicate_process(self, process):
         # Test if process is already present in object
         if process in self._processes_analysed:
             return True
-        else:
-            return False
+        return False
 
     def combine_results(self, *results):
         """Combine multiple (partial) results objects.
@@ -226,15 +222,14 @@ def combine_results(self, *results):
             processes = r._processes_analysed
             if utils.conflicting_entries(self.settings, r.settings):
                 raise RuntimeError(
-                    "Can not combine results - analysis " "settings are not equal."
+                    "Can not combine results - analysis settings are not equal."
                 )
             for p in processes:
                 # Remove potential partial FDR-corrected results. These are no
                 # longer valid for the combined network.
                 if self._is_duplicate_process(p):
                     raise RuntimeError(
-                        "Can not combine results - results for "
-                        "process {0} already exist.".format(p)
+                        f"Can not combine results - results for process {p} already exist."
                     )
                 try:
                     del r.fdr_corrected
@@ -247,11 +242,10 @@ def combine_results(self, *results):
                 except AttributeError:
                     try:
                         results_to_add = r._single_process[p]
-                    except AttributeError:
+                    except AttributeError as e:
                         raise AttributeError(
-                            "Did not find any method attributes to combine "
-                            "(.single_proces or ._single_target)."
-                        )
+                            "Did not find any method attributes to combine (.single_process or ._single_target)."
+                        ) from e
                 self._add_single_result(p, results_to_add, r.settings)
 
 
@@ -366,21 +360,21 @@ def get_single_process(self, process, fdr=True):
         if fdr:
             try:
                 return self._single_process_fdr[process]
-            except AttributeError:
+            except AttributeError as e:
                 raise RuntimeError(
                     f"No FDR-corrected results for process {process}. Set fdr=False for uncorrected results."
-                )
-            except KeyError:
+                ) from e
+            except KeyError as e:
                 raise RuntimeError(
                     f"No FDR-corrected results for process {process}. Set fdr=False for uncorrected results."
-                )
+                ) from e
         else:
             try:
                 return self._single_process[process]
-            except AttributeError:
-                raise RuntimeError("No results have been added.")
-            except KeyError:
-                raise RuntimeError("No results for process {0}.".format(process))
+            except AttributeError as e:
+                raise RuntimeError("No results have been added.") from e
+            except KeyError as e:
+                raise RuntimeError(f"No results for process {process}.") from e
 
     def get_significant_processes(self, fdr=True):
         """Return statistically-significant processes.

From ac1ecee959ac5d2eba825bec4303281c539dd899 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 29 Jan 2024 17:19:43 +0100
Subject: [PATCH 24/34] Fix typo

---
 test/test_multivariate_te.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/test_multivariate_te.py b/test/test_multivariate_te.py
index e18c325..fafeb8e 100644
--- a/test/test_multivariate_te.py
+++ b/test/test_multivariate_te.py
@@ -420,7 +420,7 @@ def test_define_candidates():
         assert (1, 6) in candidates, "Sample missing from candidates: (1, 6)."
         assert (1, 3) in candidates, "Sample missing from candidates: (1, 3)."
         if s["add_conditionals"] is not None:
-            if isinstace(s["add_conditionals"], tuple):
+            if isinstance(s["add_conditionals"], tuple):
                 cond_ind = nw._lag_to_idx([s["add_conditionals"]])
             else:
                 cond_ind = nw._lag_to_idx(s["add_conditionals"])

From 44218dd8cde5c461bd594d1cb7787874c11e7d91 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Fri, 2 Feb 2024 12:10:08 +0100
Subject: [PATCH 25/34] Refactor bivariate sequential max stats

---
 idtxl/stats.py | 76 ++++++++++++++++++++++++--------------------------
 1 file changed, 37 insertions(+), 39 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index de7e84f..44407ca 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -196,7 +196,9 @@ def network_fdr(settings=None, *results):
             cands = cands + (results_comb._single_target[target].selected_vars_sources)
             n_perm = np.append(n_perm, results_comb.settings.n_perm_max_seq)
 
-    if pval.size == 0 or (n_perm == None).all():  # n_perm is None for empty networks, i.e., no omnibus test was run
+    if (
+        pval.size == 0 or (n_perm == None).all()
+    ):  # n_perm is None for empty networks, i.e., no omnibus test was run
         print("No links in final results ...")
         results_comb._add_fdr(
             fdr=None,
@@ -752,14 +754,13 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
     alpha = analysis_setup.settings["alpha_max_seq"]
     _check_permute_in_time(analysis_setup, data, n_permutations)
     permute_in_time = analysis_setup.settings["permute_in_time"]
+    assert analysis_setup.selected_vars_sources, "No sources to test."
 
     if analysis_setup.settings["verbose"]:
         print(
             f"sequential maximum statistic, n_perm: {n_permutations}, testing {len(analysis_setup.selected_vars_sources)} selected sources"
         )
 
-    assert analysis_setup.selected_vars_sources, "No sources to test."
-
     # Check if target variables were selected to distinguish between TE and MI
     # analysis.
     if len(analysis_setup._selected_vars_target) == 0:
@@ -783,20 +784,18 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
         source_vars = [
             s for s in analysis_setup.selected_vars_sources if s[0] == source
         ]
-        source_vars_idx = [
-            i
-            for i, s in enumerate(analysis_setup.selected_vars_sources)
-            if s[0] == source
-        ]
 
         # Determine length of conditioning set and allocate memory.
-        idx_conditional = source_vars.copy()
         if conditional_realisations_target is not None:
-            idx_conditional += analysis_setup.selected_vars_target
+            size_conditioning_set = len(source_vars) + len(
+                analysis_setup.selected_vars_target
+            )
+        else:
+            size_conditioning_set = len(source_vars)
         conditional_realisations = np.empty(
             (
                 data.n_realisations(analysis_setup.current_value) * len(source_vars),
-                len(idx_conditional) - 1,
+                size_conditioning_set - 1,
             )
         ).astype(data.data_type)
         candidate_realisations = np.empty(
@@ -812,28 +811,33 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
         surr_table = np.zeros((len(source_vars), n_permutations))
         # Collect data for each candidate and the corresponding conditioning set.
         for idx_c, candidate in enumerate(source_vars):
-            temp_cond = data.get_realisations(
+            realizations_remaining_source_vars = data.get_realisations(
                 analysis_setup.current_value,
                 set(source_vars).difference(set([candidate])),
             )[0]
-            temp_cand = data.get_realisations(
+            realizations_current_candidate = data.get_realisations(
                 analysis_setup.current_value, [candidate]
             )[0]
             # The following may happen if either the requested conditioning is
             # 'none' or if the conditioning set that is tested consists only of
             # a single candidate.
-            if temp_cond is None:
+            if realizations_remaining_source_vars is None:
                 conditional_realisations = conditional_realisations_target
                 re_use = ["var2", "conditional"]
             else:
                 re_use = ["var2"]
                 if conditional_realisations_target is None:
-                    conditional_realisations[i_1:i_2,] = temp_cond
+                    conditional_realisations[
+                        i_1:i_2,
+                    ] = realizations_remaining_source_vars
                 else:
                     conditional_realisations[i_1:i_2,] = np.hstack(
-                        (temp_cond, conditional_realisations_target)
+                        (
+                            realizations_remaining_source_vars,
+                            conditional_realisations_target,
+                        )
                     )
-            candidate_realisations[i_1:i_2,] = temp_cand
+            candidate_realisations[i_1:i_2,] = realizations_current_candidate
             i_1 = i_2
             i_2 += data.n_realisations(analysis_setup.current_value)
 
@@ -851,7 +855,7 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                         analysis_setup.current_value, [candidate]
                     )[0],
                     var2=analysis_setup._current_value_realisations,
-                    conditional=temp_cond,
+                    conditional=realizations_remaining_source_vars,
                 )
             else:
                 analysis_setup.settings["analytical_surrogates"] = False
@@ -870,7 +874,7 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                         re_use=["var2", "conditional"],
                         var1=surr_candidate_realisations,
                         var2=analysis_setup._current_value_realisations,
-                        conditional=temp_cond,
+                        conditional=realizations_remaining_source_vars,
                     )
                 except ex.AlgorithmExhaustedError as aee:
                     # The aglorithm cannot continue here, so
@@ -898,7 +902,7 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                 conditional=conditional_realisations,
             )
         except ex.AlgorithmExhaustedError as aee:
-            # The aglorithm cannot continue here, so
+            # The algorithm cannot continue here, so
             #  we'll terminate the max sequential stats test,
             #  and declare all not significant
             print(
@@ -907,21 +911,18 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             )
             # For now we don't need a stack trace:
             # traceback.print_tb(aee.__traceback__)
-            # Return (signficance, pvalue, TEs):
+            # Return (significance, pvalue, TEs):
             return (
                 np.zeros(len(analysis_setup.selected_vars_sources)).astype(bool),
                 np.ones(len(analysis_setup.selected_vars_sources)),
                 np.zeros(len(analysis_setup.selected_vars_sources)),
             )
 
-        selected_vars_order = utils.argsort_descending(individual_stat_source)
-        individual_stat_source_sorted = utils.sort_descending(individual_stat_source)
-        max_distribution = _sort_table_max(surr_table)
-        significance_source = np.zeros(individual_stat_source.shape[0], dtype=bool)
-        pvalue_source = np.ones(individual_stat_source.shape[0])
-
         # Compare each original value with the distribution of the same rank,
         # starting with the highest value.
+        individual_stat_source_sorted = utils.sort_descending(individual_stat_source)
+        source_vars_sorted = [source_vars[i] for i in utils.argsort_descending(individual_stat_source)]
+        max_distribution = _sort_table_max(surr_table)
         for c in range(individual_stat_source.shape[0]):
             s, p = _find_pvalue(
                 individual_stat_source_sorted[c],
@@ -929,24 +930,21 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                 alpha,
                 tail="one_bigger",
             )
-            significance_source[c] = s
-            pvalue_source[c] = p
-
             if not s:  # break as soon as a candidate is no longer significant
                 if analysis_setup.settings["verbose"]:
                     print(
                         f"\nStopping sequential max stats at candidate with rank {c}."
                     )
                 break
+            # Ensure results are written in the original order of selected
+            # source variables
+            source_var_idx = analysis_setup.selected_vars_sources.index(
+                source_vars_sorted[c]
+            )
+            significance[source_var_idx] = s
+            pvalue[source_var_idx] = p
+            individual_stat[source_var_idx] = individual_stat_source_sorted[c]
 
-        # Get back original order of variables within the current source. Write
-        # re-ordered results into global results array at the respective
-        # variable locations.
-        significance_source[selected_vars_order] = significance_source.copy()
-        pvalue_source[selected_vars_order] = pvalue_source.copy()
-        significance[source_vars_idx] = significance_source
-        pvalue[source_vars_idx] = pvalue_source
-        individual_stat[source_vars_idx] = individual_stat_source
     return significance, pvalue, individual_stat
 
 
@@ -1462,7 +1460,7 @@ def _sort_table_min(table):
 def _sort_table_max(table):
     """Sort each column in a table in descending order."""
     table_sorted = np.empty(table.shape)
-    for permutation in range(0, table.shape[1]):
+    for permutation in range(table.shape[1]):
         table_sorted[:, permutation] = utils.sort_descending(table[:, permutation])
     return table_sorted
 

From 208dde198ae495ee7ce7f3ed0042146284e5fee3 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Fri, 2 Feb 2024 14:30:58 +0100
Subject: [PATCH 26/34] Add unit tests for (bivariate) sequential max stats

---
 test/test_stats.py | 105 ++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 104 insertions(+), 1 deletion(-)

diff --git a/test/test_stats.py b/test/test_stats.py
index 872cd65..7d03a15 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -1,5 +1,7 @@
 """Unit tests for stats module."""
 # pylint: disable=protected-access
+import copy as cp
+
 import numpy as np
 import pytest
 from test_estimators_jidt import _get_gauss_data
@@ -49,7 +51,14 @@ def test_max_statistic_sequential():
         setup.current_value, [setup.current_value]
     )[0]
 
-    setup._selected_vars_realisations = np.hstack(
+    setup._selected_vars_realisations = data.get_realisations(  # use actual realisation for target var and first source var
+        setup.current_value,
+        [setup.selected_vars_target[0]] + setup.selected_vars_sources,
+    )[
+        0
+    ]
+    setup_permuted = cp.deepcopy(setup)
+    setup_permuted._selected_vars_realisations = np.hstack(
         (
             data.get_realisations(  # use actual realisation for target var and first source var
                 setup.current_value,
@@ -62,7 +71,100 @@ def test_max_statistic_sequential():
             ),  # use random data as realizations for second source var
         )
     )
+    for s, expected in zip([setup, setup_permuted], [[True, True], [True, False]]):
+        [sign, p, te] = stats.max_statistic_sequential(analysis_setup=s, data=data)
+        print(p)
+        print(sign)
+        print(te)
+        assert len(p) == len(te)
+        assert len(p) == len(sign)
+        assert len(p) == len(setup.selected_vars_sources)
+        assert np.array_equal(sign, expected), "Incorrect sources inferred"
+
+    setup = MultivariateTE()
+    setup._initialise(settings, data, sources=[0, 1, 4], target=3)
+    setup.current_value = (3, 4)
+    setup.selected_vars_target = [(1, 1)]
+    setup.selected_vars_sources = [(0, 1), (0, 2), (1, 1), (4, 1)]
+    setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
+    setup._current_value_realisations = data.get_realisations(
+        setup.current_value, [setup.current_value]
+    )[0]
+    setup._selected_vars_realisations = data.get_realisations(
+        setup.current_value, setup.selected_vars_full
+    )[0]
+
     [sign, p, te] = stats.max_statistic_sequential(analysis_setup=setup, data=data)
+    assert len(p) == len(te)
+    assert len(p) == len(sign)
+    assert len(p) == len(setup.selected_vars_sources)
+    print(p)
+    print(setup.selected_vars_sources)
+    print(sign)
+    print(te, "\n\n")
+
+    for permuted_var in [0, 4]:
+        np.random.seed(0)
+        data.generate_mute_data(104, 10)
+        data._data[permuted_var, :, :] = np.ones((104, 10))
+        setup._current_value_realisations = data.get_realisations(
+            setup.current_value, [setup.current_value]
+        )[0]
+        setup._selected_vars_realisations = data.get_realisations(
+            setup.current_value, setup.selected_vars_full
+        )[0]
+        [sign, p, te] = stats.max_statistic_sequential(analysis_setup=setup, data=data)
+        for i, var in enumerate(setup.selected_vars_sources):
+            if var[0] == permuted_var:
+                assert not sign[
+                    i
+                ], "Seq. max. stats returned sign result for random data"
+                assert (
+                    p[i] > setup.settings["alpha_max_seq"]
+                ), f"{p[i]} not smaller than critical alpha {setup.settings['alpha_max_seq']}"
+
+
+def test_max_statistic_sequential_bivariate():
+    np.random.seed(0)
+    data = Data()
+    data.generate_mute_data(104, 10)
+    settings = {
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_omnibus": 21,
+        "n_perm_max_seq": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 1,
+        "max_lag_target": 5,
+    }
+    setup = MultivariateTE()
+    setup._initialise(settings, data, sources=[0, 2], target=1)
+    setup.current_value = (1, 4)
+    setup.selected_vars_target = [(1, 1)]
+    setup.selected_vars_sources = [(0, 1), (2, 1)]
+    setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
+
+    setup._current_value_realisations = data.get_realisations(
+        setup.current_value, [setup.current_value]
+    )[0]
+
+    setup._selected_vars_realisations = np.hstack(
+        (
+            data.get_realisations(  # use actual realisation for target var and first source var
+                setup.current_value,
+                [setup.selected_vars_target[0], setup.selected_vars_sources[0]],
+            )[
+                0
+            ],
+            np.random.rand(
+                data.n_realisations(setup.current_value), 1
+            ),  # use random data as realizations for second source var
+        )
+    )
+    [sign, p, te] = stats.max_statistic_sequential_bivariate(
+        analysis_setup=setup, data=data
+    )
     print(p)
     print(sign)
     print(te)
@@ -508,4 +610,5 @@ def test_analytical_surrogates():
     # test_omnibus_test()
     # test_max_statistic()
     # test_min_statistic()
+    test_max_statistic_sequential_bivariate()
     test_max_statistic_sequential()

From 47773cffb8fd8c47608cb32447bf1617163bfba9 Mon Sep 17 00:00:00 2001
From: "David A. Ehrlich" <davidalexander.ehrlich@uni-goettingen.de>
Date: Fri, 2 Feb 2024 14:00:06 +0100
Subject: [PATCH 27/34] Fix bug in _perform_fdr_correction by replacing with
 statsmodels

---
 idtxl/stats.py | 61 +++++++++++++++++++++-----------------------------
 1 file changed, 25 insertions(+), 36 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 44407ca..8f6b8ce 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -3,6 +3,7 @@
 import copy as cp
 
 import numpy as np
+from statsmodels.stats.multitest import fdrcorrection
 
 from . import idtxl_exceptions as ex
 from . import idtxl_utils as utils
@@ -72,16 +73,16 @@ def ais_fdr(settings=None, *results):
         results_comb._add_fdr(fdr=None, alpha=alpha, constant=constant)
         return results_comb
 
-    sign, thresh = _perform_fdr_corretion(
+    sign, min_thresh = _perform_fdr_correction(
         pval, constant, alpha, len(results_comb.processes_analysed)
     )
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
-    if (1 / min(n_perm)) > thresh[0]:
+    if (1 / min(n_perm)) > min_thresh:
         print(
             "WARNING: Number of permutations ('n_perm_max_seq') for at least one target is too low to allow for "
-            f"FDR correction (FDR-threshold: {thresh[0]:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
+            f"FDR correction (FDR-threshold: {min_thresh:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
         )
         results_comb._add_fdr(fdr=None, alpha=alpha, constant=constant)
         return results_comb
@@ -208,14 +209,14 @@ def network_fdr(settings=None, *results):
         )
         return results_comb
 
-    sign, thresh = _perform_fdr_corretion(pval, constant, alpha, n_tests)
+    sign, min_thresh = _perform_fdr_correction(pval, constant, alpha, n_tests)
 
     # If the number of permutations for calculating p-values for individual
     # variables is too low, return without performing any correction.
-    if (1 / min(i for i in n_perm if i is not None)) > thresh[0]:
+    if (1 / min(i for i in n_perm if i is not None)) > min_thresh:
         print(
             "WARNING: Number of permutations ('n_perm_max_seq') for at least one target is too low to allow for "
-            f"FDR correction (FDR-threshold: {thresh[0]:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
+            f"FDR correction (FDR-threshold: {min_thresh:.4f}, min. theoretically possible p-value: {1 / min(n_perm)})."
         )
         results_comb._add_fdr(
             fdr=None,
@@ -254,13 +255,16 @@ def network_fdr(settings=None, *results):
     return results_comb
 
 
-def _perform_fdr_corretion(pval, constant, alpha, n_tests):
+def _perform_fdr_correction(pval, constant, alpha, n_tests):
     """Calculate sequential threshold for FDR-correction.
 
     Calculate sequential thresholds for FDR-correction of p-values. The
     constant defines how the threshold is calculated. See Genovese (2002) for
     details.
 
+    Internally uses the statsmodels implementation of the Benjamini-Hochberg
+    and Benjamini-Yekutieli procedures for FDR-correction.
+
     References:
 
     - Genovese, C.R., Lazar, N.A., & Nichols, T. (2002). Thresholding of
@@ -283,36 +287,21 @@ def _perform_fdr_corretion(pval, constant, alpha, n_tests):
     Returns:
         array of bools
             significance of p-values in the order of the input array
-        array of floats
-            FDR-thresholds for each p-value in increasing order
+        float
+            smallest threshold for significance
     """
-    # Sort all p-values in ascending order.
-    sort_idx = np.argsort(pval)
-    pval_sorted = np.sort(pval)
-
-    # Calculate threshold
-    if constant == 2:  # pick the requested constant (see Genovese, p.872)
-        if n_tests < 1000:
-            const = np.sum(1 / np.arange(1, n_tests + 1))
-        else:
-            const = np.log(n_tests) + np.e  # aprx. harmonic sum with Euler's number
-    elif constant == 1:
-        # This is less strict than the other one and corresponds to a
-        # Bonoferroni-correction for the first p-value, however, it makes more
-        # strict assumptions on the distribution of p-values, while constant 2
-        # works for any joint distribution of the p-values.
-        const = 1
-
-    # Calculate threshold for each p-value.
-    thresh = (np.arange(1, len(pval_sorted) + 1) / n_tests) * alpha / const
-
-    # Compare data to threshold.
-    sign = pval_sorted <= thresh
-    if np.invert(sign).any():
-        first_false = np.where(np.invert(sign))[0][0]
-        sign[first_false:] = False  # avoids false positives due to equal pvals
-    sign[sort_idx] = sign.copy()  # restore original ordering of significance values
-    return sign, thresh
+    
+    # Convert constant to statsmodels "method" parameter
+    method = "indep" if constant == 1 else "negcorr"
+    sign, _ = fdrcorrection(pval, alpha=alpha, method=method)
+
+    # Compute smallest threshold to check for sufficiency of permutations
+    if constant == 1:
+        min_thresh = alpha / n_tests
+    else:
+        min_thresh = alpha / (n_tests * np.sum(1 / np.arange(1, n_tests + 1)))
+
+    return sign, min_thresh
 
 
 def omnibus_test(analysis_setup, data):

From 4f9ab51344739ce1b202186cc24db1730778d120 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Fri, 2 Feb 2024 14:48:07 +0100
Subject: [PATCH 28/34] Adapt unit tests to fixed FDR correction

---
 idtxl/stats.py     |  6 ++++--
 test/test_stats.py | 12 ++++++------
 2 files changed, 10 insertions(+), 8 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index 8f6b8ce..fcf8a19 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -290,7 +290,7 @@ def _perform_fdr_correction(pval, constant, alpha, n_tests):
         float
             smallest threshold for significance
     """
-    
+
     # Convert constant to statsmodels "method" parameter
     method = "indep" if constant == 1 else "negcorr"
     sign, _ = fdrcorrection(pval, alpha=alpha, method=method)
@@ -910,7 +910,9 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
         # Compare each original value with the distribution of the same rank,
         # starting with the highest value.
         individual_stat_source_sorted = utils.sort_descending(individual_stat_source)
-        source_vars_sorted = [source_vars[i] for i in utils.argsort_descending(individual_stat_source)]
+        source_vars_sorted = [
+            source_vars[i] for i in utils.argsort_descending(individual_stat_source)
+        ]
         max_distribution = _sort_table_max(surr_table)
         for c in range(individual_stat_source.shape[0]):
             s, p = _find_pvalue(
diff --git a/test/test_stats.py b/test/test_stats.py
index 7d03a15..16460c7 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -242,7 +242,7 @@ def test_network_fdr():
         "sources_tested": [0, 2],
         "omnibus_pval": 0.031,
         "omnibus_sign": True,
-        "selected_sources_pval": np.array([0.00001, 0.00014, 0.01]),
+        "selected_sources_pval": np.array([0.00001, 0.00014, 0.049]),
         "selected_sources_te": np.array([1.8, 1.75, 0.75]),
     }
     target_1["selected_vars_full"] = (
@@ -366,12 +366,12 @@ def test_fdr_sorting():
     pval_sorted = np.arange(n_tests) / 300
     constant = 2
     alpha = 0.05
-    sign, _ = stats._perform_fdr_corretion(pval_sorted, constant, alpha, n_tests)
+    sign, _ = stats._perform_fdr_correction(pval_sorted, constant, alpha, n_tests)
     assert sign[:3].all()
 
     permutation = np.random.permutation(n_tests)
     pval_unsorted = pval_sorted[permutation].copy()
-    sign_unsorted, _ = stats._perform_fdr_corretion(
+    sign_unsorted, _ = stats._perform_fdr_correction(
         pval_unsorted, constant, alpha, n_tests
     )
     assert sign_unsorted.sum() == 3
@@ -600,7 +600,7 @@ def test_analytical_surrogates():
     # test_ais_fdr()
     # test_analytical_surrogates()
     # test_data_type()
-    # test_network_fdr()
+    test_network_fdr()
     # test_fdr_sorting()
     # test_find_pvalue()
     # test_find_table_max()
@@ -610,5 +610,5 @@ def test_analytical_surrogates():
     # test_omnibus_test()
     # test_max_statistic()
     # test_min_statistic()
-    test_max_statistic_sequential_bivariate()
-    test_max_statistic_sequential()
+    # test_max_statistic_sequential_bivariate()
+    # test_max_statistic_sequential()

From 8461992a968598ae7c95000207e844717a3da6c9 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Fri, 2 Feb 2024 15:37:42 +0100
Subject: [PATCH 29/34] Fix output of TE/MI values if biv. seq. max stats exits
 early

If a source was not significant, the computation of p-values was stopped
and also the TE/MI values were not written into the output array
further. Ensure that always all TE/MI values are returned, also the
non-significant ones.
---
 idtxl/stats.py | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index fcf8a19..e65215d 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -907,6 +907,11 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                 np.zeros(len(analysis_setup.selected_vars_sources)),
             )
 
+        # Assign estimated TE/MI values to output structure. These are returned
+        # regardless of whether they are significant.
+        for s, stat in zip(source_vars, individual_stat_source):
+            individual_stat[analysis_setup.selected_vars_sources.index(s)] = stat
+
         # Compare each original value with the distribution of the same rank,
         # starting with the highest value.
         individual_stat_source_sorted = utils.sort_descending(individual_stat_source)
@@ -934,7 +939,6 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             )
             significance[source_var_idx] = s
             pvalue[source_var_idx] = p
-            individual_stat[source_var_idx] = individual_stat_source_sorted[c]
 
     return significance, pvalue, individual_stat
 

From ec6ee52e752921f7a62f5244e83f913e03d5d774 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 5 Feb 2024 15:13:10 +0100
Subject: [PATCH 30/34] Fix typos in comments and log output

---
 idtxl/bivariate_te.py | 6 +++---
 idtxl/stats.py        | 8 ++++----
 2 files changed, 7 insertions(+), 7 deletions(-)

diff --git a/idtxl/bivariate_te.py b/idtxl/bivariate_te.py
index b380b6e..18a4e5b 100644
--- a/idtxl/bivariate_te.py
+++ b/idtxl/bivariate_te.py
@@ -7,9 +7,9 @@
 Note:
     Written for Python 3.4+
 """
-from .network_inference import NetworkInferenceTE, NetworkInferenceBivariate
-from .stats import network_fdr
+from .network_inference import NetworkInferenceBivariate, NetworkInferenceTE
 from .results import ResultsNetworkInference
+from .stats import network_fdr
 
 
 class BivariateTE(NetworkInferenceTE, NetworkInferenceBivariate):
@@ -157,7 +157,7 @@ def analyse_network(self, settings, data, targets="all", sources="all"):
         )
         for t, target in enumerate(targets):
             if settings["verbose"]:
-                print("\n####### analysing target with index {t} from list {targets}")
+                print(f"\n####### analysing target with index {t} from list {targets}")
             res_single = self.analyse_single_target(settings, data, target, sources[t])
             results.combine_results(res_single)
 
diff --git a/idtxl/stats.py b/idtxl/stats.py
index e65215d..f59a0e8 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -575,9 +575,9 @@ def max_statistic_sequential(analysis_setup, data):
             candidate_realisations_current,
         ] = analysis_setup._separate_realisations(idx_conditional, candidate)
 
-        # The following may happen if either the requested conditing is 'none'
-        # or if the conditiong set that is tested consists only of a single
-        # candidate.
+        # The following may happen if either the requested conditioning is
+        # 'none' or if the conditioning set that is tested consists only of a
+        # single candidate.
         if conditional_realisations_current is None:
             conditional_realisations = None
             re_use = ["var2", "conditional"]
@@ -1496,7 +1496,7 @@ def _find_pvalue(statistic, distribution, alpha, tail):
         alpha = alpha / 2
     else:
         raise ValueError(
-            f"Unkown value for tail: {tail}, should be one, one_bigger, one_smaller, or two"
+            f"Unknown value for tail: {tail}, should be one, one_bigger, one_smaller, or two"
         )
 
     # If the statistic is larger than all values in the test distribution, set

From dd8d0355134deeb6c10f8379777ab6fa9887146b Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 5 Feb 2024 15:13:54 +0100
Subject: [PATCH 31/34] Add unit tests for (bivariate) seq. max stats

---
 test/test_stats.py | 105 +++++++++++++++++++++++++--------------------
 1 file changed, 59 insertions(+), 46 deletions(-)

diff --git a/test/test_stats.py b/test/test_stats.py
index 16460c7..0f56abd 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -13,6 +13,8 @@
 from idtxl.multivariate_te import MultivariateTE
 from idtxl.results import ResultsNetworkInference, ResultsSingleProcessAnalysis
 
+SEED = 0
+
 
 def test_omnibus_test():
     print("Write test for omnibus test.")
@@ -27,9 +29,9 @@ def test_min_statistic():
 
 
 def test_max_statistic_sequential():
-    np.random.seed(0)
-    data = Data()
-    data.generate_mute_data(104, 10)
+    np.random.seed(SEED)
+    data = Data(seed=SEED)
+    data.generate_mute_data(1000, 1)
     settings = {
         "cmi_estimator": "JidtKraskovCMI",
         "n_perm_max_stat": 21,
@@ -41,10 +43,10 @@ def test_max_statistic_sequential():
         "max_lag_target": 5,
     }
     setup = MultivariateTE()
-    setup._initialise(settings, data, sources=[0, 2], target=1)
-    setup.current_value = (1, 4)
-    setup.selected_vars_target = [(1, 1)]
-    setup.selected_vars_sources = [(0, 1), (2, 1)]
+    setup._initialise(settings, data, sources=[0, 1], target=3)
+    setup.current_value = (3, 4)
+    setup.selected_vars_target = [(3, 1)]
+    setup.selected_vars_sources = [(0, 1), (0, 2), (1, 1)]
     setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
 
     setup._current_value_realisations = data.get_realisations(
@@ -58,28 +60,39 @@ def test_max_statistic_sequential():
         0
     ]
     setup_permuted = cp.deepcopy(setup)
+    data_permuted = cp.deepcopy(data)
+    # data_permuted._data[1, :, :] = np.random.rand(1, 1000, 1)
+    # data_permuted._data[1, :, :] = np.ones((1, 1000, 1))
     setup_permuted._selected_vars_realisations = np.hstack(
         (
             data.get_realisations(  # use actual realisation for target var and first source var
                 setup.current_value,
-                [setup.selected_vars_target[0], setup.selected_vars_sources[0]],
+                [setup.selected_vars_target[0]] + list(setup.selected_vars_sources[:2]),
             )[
                 0
             ],
-            np.random.rand(
-                data.n_realisations(setup.current_value), 1
-            ),  # use random data as realizations for second source var
+            # data_permuted._data[
+            #     1, : data.n_realisations(setup.current_value), :
+            # ],  # use random data as realizations for second source var
+            # np.ones((data.n_realisations(setup.current_value), 1)),
+            np.random.rand(data.n_realisations(setup.current_value), 1),
         )
     )
-    for s, expected in zip([setup, setup_permuted], [[True, True], [True, False]]):
-        [sign, p, te] = stats.max_statistic_sequential(analysis_setup=s, data=data)
+    for s, d, expected in zip(
+        [setup, setup_permuted],
+        [data, data_permuted],
+        [[True, True, True], [True, True, False]],
+    ):
+        [sign, p, te] = stats.max_statistic_sequential(analysis_setup=s, data=d)
+        print(te)
         print(p)
         print(sign)
-        print(te)
         assert len(p) == len(te)
         assert len(p) == len(sign)
         assert len(p) == len(setup.selected_vars_sources)
-        assert np.array_equal(sign, expected), "Incorrect sources inferred"
+        assert np.array_equal(
+            sign, expected
+        ), f"Incorrect sources inferred, expected: {expected}"
 
     setup = MultivariateTE()
     setup._initialise(settings, data, sources=[0, 1, 4], target=3)
@@ -125,9 +138,9 @@ def test_max_statistic_sequential():
 
 
 def test_max_statistic_sequential_bivariate():
-    np.random.seed(0)
-    data = Data()
-    data.generate_mute_data(104, 10)
+    np.random.seed(SEED)
+    data = Data(seed=SEED)
+    data.generate_mute_data(1000, 1)
     settings = {
         "cmi_estimator": "JidtKraskovCMI",
         "n_perm_max_stat": 21,
@@ -139,38 +152,38 @@ def test_max_statistic_sequential_bivariate():
         "max_lag_target": 5,
     }
     setup = MultivariateTE()
-    setup._initialise(settings, data, sources=[0, 2], target=1)
-    setup.current_value = (1, 4)
-    setup.selected_vars_target = [(1, 1)]
-    setup.selected_vars_sources = [(0, 1), (2, 1)]
+    setup._initialise(settings, data, sources=[0, 1], target=3)
+    setup.current_value = (3, 4)
+    setup.selected_vars_target = [(3, 1)]
+    setup.selected_vars_sources = [(0, 1), (0, 2), (1, 1)]
     setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
-
     setup._current_value_realisations = data.get_realisations(
         setup.current_value, [setup.current_value]
     )[0]
+    setup._selected_vars_realisations = data.get_realisations(
+        setup.current_value, setup.selected_vars_full
+    )[0]
 
-    setup._selected_vars_realisations = np.hstack(
-        (
-            data.get_realisations(  # use actual realisation for target var and first source var
-                setup.current_value,
-                [setup.selected_vars_target[0], setup.selected_vars_sources[0]],
-            )[
-                0
-            ],
-            np.random.rand(
-                data.n_realisations(setup.current_value), 1
-            ),  # use random data as realizations for second source var
+    # Bivariate sequential max stats collects source variable realizations from
+    # data object for running the test.
+    data_permuted = cp.deepcopy(data)
+    data_permuted._data[1, :, :] = np.random.randn(1, 1000, 1)
+
+    for d, expected in zip(
+        [data, data_permuted], [[True, True, True], [True, True, False]]
+    ):
+        [sign, p, te] = stats.max_statistic_sequential_bivariate(
+            analysis_setup=setup, data=d
         )
-    )
-    [sign, p, te] = stats.max_statistic_sequential_bivariate(
-        analysis_setup=setup, data=data
-    )
-    print(p)
-    print(sign)
-    print(te)
-    assert len(p) == len(te)
-    assert len(p) == len(sign)
-    assert len(p) == len(setup.selected_vars_sources)
+        print(te)
+        print(p)
+        print(sign)
+        assert len(p) == len(te)
+        assert len(p) == len(sign)
+        assert len(p) == len(setup.selected_vars_sources)
+        assert np.array_equal(
+            sign, expected
+        ), f"Incorrect sources inferred, expected: {expected}"
 
     setup = MultivariateTE()
     setup._initialise(settings, data, sources=[0, 1, 4], target=3)
@@ -600,7 +613,7 @@ def test_analytical_surrogates():
     # test_ais_fdr()
     # test_analytical_surrogates()
     # test_data_type()
-    test_network_fdr()
+    # test_network_fdr()
     # test_fdr_sorting()
     # test_find_pvalue()
     # test_find_table_max()
@@ -611,4 +624,4 @@ def test_analytical_surrogates():
     # test_max_statistic()
     # test_min_statistic()
     # test_max_statistic_sequential_bivariate()
-    # test_max_statistic_sequential()
+    test_max_statistic_sequential()

From bfdbc1534f2d69b3b3fd75a4eb7656a2eb9d82be Mon Sep 17 00:00:00 2001
From: PWollstadtHRI <Patricia.Wollstadt@honda-ri.de>
Date: Mon, 12 Feb 2024 13:19:30 +0100
Subject: [PATCH 32/34] Fix inclusion of knn submodule during installation

---
 idtxl/knn/__init__.py | 0
 setup.py              | 2 +-
 2 files changed, 1 insertion(+), 1 deletion(-)
 create mode 100644 idtxl/knn/__init__.py

diff --git a/idtxl/knn/__init__.py b/idtxl/knn/__init__.py
new file mode 100644
index 0000000..e69de29
diff --git a/setup.py b/setup.py
index 6197c07..7adedd6 100644
--- a/setup.py
+++ b/setup.py
@@ -6,7 +6,7 @@
 
 setup(
     name="idtxl",
-    packages=["idtxl"],
+    packages=["idtxl", "idtxl/knn"],
     include_package_data=True,
     version="1.6",
     description="Information Dynamics Toolkit xl",

From 975cfc736a0e58caddb8737ce4bc6be2e6c41ea1 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 25 Mar 2024 14:39:40 +0100
Subject: [PATCH 33/34] Fix conditioning in biv. sequential max stats

Use the correct conditioning set for estimating the surrogate values.
Before, only the remaining source vars instead of the remaining source
vars and the target vars were used.

Also, fix stats function for empty conditioning sets, which may occurr
for bivariate MI estimation.

Add and update unit tests to reflect fixes.
---
 idtxl/stats.py     | 190 +++++++++++++++++++++++++--------------------
 test/test_stats.py |  61 ++++++++++++++-
 2 files changed, 163 insertions(+), 88 deletions(-)

diff --git a/idtxl/stats.py b/idtxl/stats.py
index f59a0e8..d5ac5b9 100644
--- a/idtxl/stats.py
+++ b/idtxl/stats.py
@@ -1,4 +1,5 @@
 """Provide statistics functions."""
+
 # pylint: disable=protected-access
 import copy as cp
 
@@ -594,15 +595,15 @@ def max_statistic_sequential(analysis_setup, data):
             and permute_in_time
         ):
             # Generate the surrogates analytically
-            surr_table[
-                idx_c, :
-            ] = analysis_setup._cmi_estimator.estimate_surrogates_analytic(
-                n_perm=n_permutations,
-                var1=data.get_realisations(analysis_setup.current_value, [candidate])[
-                    0
-                ],
-                var2=analysis_setup._current_value_realisations,
-                conditional=conditional_realisations_current,
+            surr_table[idx_c, :] = (
+                analysis_setup._cmi_estimator.estimate_surrogates_analytic(
+                    n_perm=n_permutations,
+                    var1=data.get_realisations(
+                        analysis_setup.current_value, [candidate]
+                    )[0],
+                    var2=analysis_setup._current_value_realisations,
+                    conditional=conditional_realisations_current,
+                )
             )
         else:
             analysis_setup.settings["analytical_surrogates"] = False
@@ -750,8 +751,56 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
             f"sequential maximum statistic, n_perm: {n_permutations}, testing {len(analysis_setup.selected_vars_sources)} selected sources"
         )
 
+    def _allocate_memory_for_current_source(n_source_vars):
+        # Determine length of conditioning set and allocate memory.
+        if conditional_realisations_target is None:
+            if n_source_vars - 1 == 0:
+                conditional_realisations = None
+            else:
+                conditional_realisations = np.empty(
+                    (
+                        data.n_realisations(analysis_setup.current_value)
+                        * n_source_vars,
+                        n_source_vars - 1,
+                    )
+                ).astype(data.data_type)
+        else:
+            conditional_realisations = np.empty(
+                (
+                    data.n_realisations(analysis_setup.current_value) * n_source_vars,
+                    n_source_vars - 1 + len(analysis_setup.selected_vars_target),
+                )
+            ).astype(data.data_type)
+
+        candidate_realisations = np.empty(
+            (data.n_realisations(analysis_setup.current_value) * n_source_vars, 1)
+        ).astype(data.data_type)
+        return candidate_realisations, conditional_realisations
+
+    def _get_current_cond_set(source_vars, candidate):
+        # Return conditioning set for current source variable.
+        realizations_remaining_source_vars = data.get_realisations(
+            analysis_setup.current_value,
+            set(source_vars).difference(set([candidate])),
+        )[0]
+        # The following may happen if either the requested conditioning is
+        # None or if the conditioning set that is tested consists only of
+        # a single candidate.
+        if realizations_remaining_source_vars is None:
+            # Returns None for no targets, i.e., an empty cond set
+            return conditional_realisations_target, ["var2", "conditional"]
+        if conditional_realisations_target is None:
+            return realizations_remaining_source_vars, ["var2"]
+        return np.hstack(
+            (
+                realizations_remaining_source_vars,
+                conditional_realisations_target,
+            )
+        ), ["var2"]
+
     # Check if target variables were selected to distinguish between TE and MI
-    # analysis.
+    # analysis. We here only ever analyse results for a single target, so no
+    # need to separate target sources by target.
     if len(analysis_setup._selected_vars_target) == 0:
         conditional_realisations_target = None
     else:
@@ -768,65 +817,35 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
     significance = np.zeros(len(analysis_setup.selected_vars_sources)).astype(bool)
     pvalue = np.ones(len(analysis_setup.selected_vars_sources))
     individual_stat = np.zeros(len(analysis_setup.selected_vars_sources))
+
     for source in significant_sources:
+
         # Find selected past variables for current source
         source_vars = [
             s for s in analysis_setup.selected_vars_sources if s[0] == source
         ]
+        candidate_realisations, conditional_realisations = (
+            _allocate_memory_for_current_source(len(source_vars))
+        )
 
-        # Determine length of conditioning set and allocate memory.
-        if conditional_realisations_target is not None:
-            size_conditioning_set = len(source_vars) + len(
-                analysis_setup.selected_vars_target
-            )
-        else:
-            size_conditioning_set = len(source_vars)
-        conditional_realisations = np.empty(
-            (
-                data.n_realisations(analysis_setup.current_value) * len(source_vars),
-                size_conditioning_set - 1,
-            )
-        ).astype(data.data_type)
-        candidate_realisations = np.empty(
-            (data.n_realisations(analysis_setup.current_value) * len(source_vars), 1)
-        ).astype(data.data_type)
-
-        # Calculate TE/MI for each candidate in the conditional source set,
-        # i.e., calculate the conditional MI between each candidate and the
-        # current value, conditional on all selected variables in the
-        # conditioning set. Then sort the estimated TE/MI values.
+        # Calculate TE/MI for each candidate variable in the current source
+        # set, i.e., calculate the conditional MI between each candidate and
+        # the current value, conditional on all variables in the conditioning
+        # set. Then sort the estimated TE/MI values.
         i_1 = 0
         i_2 = data.n_realisations(analysis_setup.current_value)
         surr_table = np.zeros((len(source_vars), n_permutations))
         # Collect data for each candidate and the corresponding conditioning set.
         for idx_c, candidate in enumerate(source_vars):
-            realizations_remaining_source_vars = data.get_realisations(
-                analysis_setup.current_value,
-                set(source_vars).difference(set([candidate])),
-            )[0]
-            realizations_current_candidate = data.get_realisations(
+
+            conditional_realisations_current, re_use = _get_current_cond_set(
+                source_vars, candidate
+            )
+            if conditional_realisations is not None:
+                conditional_realisations[i_1:i_2,] = conditional_realisations_current
+            candidate_realisations[i_1:i_2,] = data.get_realisations(
                 analysis_setup.current_value, [candidate]
             )[0]
-            # The following may happen if either the requested conditioning is
-            # 'none' or if the conditioning set that is tested consists only of
-            # a single candidate.
-            if realizations_remaining_source_vars is None:
-                conditional_realisations = conditional_realisations_target
-                re_use = ["var2", "conditional"]
-            else:
-                re_use = ["var2"]
-                if conditional_realisations_target is None:
-                    conditional_realisations[
-                        i_1:i_2,
-                    ] = realizations_remaining_source_vars
-                else:
-                    conditional_realisations[i_1:i_2,] = np.hstack(
-                        (
-                            realizations_remaining_source_vars,
-                            conditional_realisations_target,
-                        )
-                    )
-            candidate_realisations[i_1:i_2,] = realizations_current_candidate
             i_1 = i_2
             i_2 += data.n_realisations(analysis_setup.current_value)
 
@@ -836,15 +855,15 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                 and permute_in_time
             ):
                 # Generate the surrogates analytically
-                surr_table[
-                    idx_c, :
-                ] = analysis_setup._cmi_estimator.estimate_surrogates_analytic(
-                    n_perm=n_permutations,
-                    var1=data.get_realisations(
-                        analysis_setup.current_value, [candidate]
-                    )[0],
-                    var2=analysis_setup._current_value_realisations,
-                    conditional=realizations_remaining_source_vars,
+                surr_table[idx_c, :] = (
+                    analysis_setup._cmi_estimator.estimate_surrogates_analytic(
+                        n_perm=n_permutations,
+                        var1=data.get_realisations(
+                            analysis_setup.current_value, [candidate]
+                        )[0],
+                        var2=analysis_setup._current_value_realisations,
+                        conditional=conditional_realisations_current,
+                    )
                 )
             else:
                 analysis_setup.settings["analytical_surrogates"] = False
@@ -856,22 +875,21 @@ def max_statistic_sequential_bivariate(analysis_setup, data):
                     analysis_setup.settings,
                 )
                 try:
-                    surr_table[
-                        idx_c, :
-                    ] = analysis_setup._cmi_estimator.estimate_parallel(
-                        n_chunks=n_permutations,
-                        re_use=["var2", "conditional"],
-                        var1=surr_candidate_realisations,
-                        var2=analysis_setup._current_value_realisations,
-                        conditional=realizations_remaining_source_vars,
+                    surr_table[idx_c, :] = (
+                        analysis_setup._cmi_estimator.estimate_parallel(
+                            n_chunks=n_permutations,
+                            re_use=["var2", "conditional"],
+                            var1=surr_candidate_realisations,
+                            var2=analysis_setup._current_value_realisations,
+                            conditional=conditional_realisations_current,
+                        )
                     )
                 except ex.AlgorithmExhaustedError as aee:
-                    # The aglorithm cannot continue here, so
-                    #  we'll terminate the max sequential stats test,
-                    #  and declare all not significant
+                    # The algorithm cannot continue here, so we'll terminate the max sequential
+                    # stats test, and declare all not significant
                     print(
                         f"AlgorithmExhaustedError encountered in estimations: {aee.message}. "
-                        "Stopping sequential max stats at candidate with rank 0"
+                        "Stopping sequential max stats at candidate with rank 0."
                     )
                     return (
                         np.zeros(len(analysis_setup.selected_vars_sources)).astype(
@@ -1404,15 +1422,15 @@ def _create_surrogate_table(
         ):
             # Generate the surrogates analytically
             analysis_setup.settings["analytical_surrogates"] = True
-            surr_table[
-                idx_c, :
-            ] = analysis_setup._cmi_estimator.estimate_surrogates_analytic(
-                n_perm=n_perm,
-                var1=data.get_realisations(analysis_setup.current_value, [candidate])[
-                    0
-                ],
-                var2=current_value_realisations,
-                conditional=conditional,
+            surr_table[idx_c, :] = (
+                analysis_setup._cmi_estimator.estimate_surrogates_analytic(
+                    n_perm=n_perm,
+                    var1=data.get_realisations(
+                        analysis_setup.current_value, [candidate]
+                    )[0],
+                    var2=current_value_realisations,
+                    conditional=conditional,
+                )
             )
         else:
             analysis_setup.settings["analytical_surrogates"] = False
diff --git a/test/test_stats.py b/test/test_stats.py
index 0f56abd..d634035 100644
--- a/test/test_stats.py
+++ b/test/test_stats.py
@@ -1,4 +1,5 @@
 """Unit tests for stats module."""
+
 # pylint: disable=protected-access
 import copy as cp
 
@@ -151,6 +152,8 @@ def test_max_statistic_sequential_bivariate():
         "min_lag_sources": 1,
         "max_lag_target": 5,
     }
+
+    # Test bivariate TE
     setup = MultivariateTE()
     setup._initialise(settings, data, sources=[0, 1], target=3)
     setup.current_value = (3, 4)
@@ -185,6 +188,7 @@ def test_max_statistic_sequential_bivariate():
             sign, expected
         ), f"Incorrect sources inferred, expected: {expected}"
 
+    # Ensure no false positives on random data.
     setup = MultivariateTE()
     setup._initialise(settings, data, sources=[0, 1, 4], target=3)
     setup.current_value = (3, 4)
@@ -232,6 +236,58 @@ def test_max_statistic_sequential_bivariate():
                 ), f"{p[i]} not smaller than critical alpha {setup.settings['alpha_max_seq']}"
 
 
+def test_max_statistic_sequential_bivariate_mi():
+    # Test bivariate sequential max stats when analyzing MI results. Here, the
+    # conditioning set is different than for TE estimation.
+    np.random.seed(SEED)
+    data = Data(seed=SEED)
+    data.generate_mute_data(1000, 1)
+    settings = {
+        "cmi_estimator": "JidtKraskovCMI",
+        "n_perm_max_stat": 21,
+        "n_perm_min_stat": 21,
+        "n_perm_omnibus": 21,
+        "n_perm_max_seq": 21,
+        "max_lag_sources": 5,
+        "min_lag_sources": 1,
+        "max_lag_target": 5,
+    }
+    # Test bivariate MI
+    setup = MultivariateTE()
+    setup._initialise(settings, data, sources=[0, 1], target=3)
+    setup.current_value = (3, 4)
+    setup.selected_vars_target = []
+    setup.selected_vars_sources = [(0, 1), (0, 2), (3, 1)]
+    setup.selected_vars_full = setup.selected_vars_target + setup.selected_vars_sources
+    setup._current_value_realisations = data.get_realisations(
+        setup.current_value, [setup.current_value]
+    )[0]
+    setup._selected_vars_realisations = data.get_realisations(
+        setup.current_value, setup.selected_vars_full
+    )[0]
+
+    # Bivariate sequential max stats collects source variable realizations from
+    # data object for running the test.
+    data_permuted = cp.deepcopy(data)
+    data_permuted._data[3, :, :] = np.random.randn(1, 1000, 1)
+
+    for d, expected in zip(
+        [data, data_permuted], [[True, True, True], [True, True, False]]
+    ):
+        [sign, p, te] = stats.max_statistic_sequential_bivariate(
+            analysis_setup=setup, data=d
+        )
+        print(te)
+        print(p)
+        print(sign)
+        assert len(p) == len(te)
+        assert len(p) == len(sign)
+        assert len(p) == len(setup.selected_vars_sources)
+        assert np.array_equal(
+            sign, expected
+        ), f"Incorrect sources inferred, expected: {expected}"
+
+
 def test_network_fdr():
     # Simulate results for a 3-node network, analyzed all-to-all. Set the
     # omnibus p-value for targets 1 and 2 such that they does not survive
@@ -623,5 +679,6 @@ def test_analytical_surrogates():
     # test_omnibus_test()
     # test_max_statistic()
     # test_min_statistic()
-    # test_max_statistic_sequential_bivariate()
-    test_max_statistic_sequential()
+    test_max_statistic_sequential_bivariate_mi()
+    test_max_statistic_sequential_bivariate()
+    # test_max_statistic_sequential()

From 1e5089dc88745afb7c93cef3ad6d6a2fc29985c6 Mon Sep 17 00:00:00 2001
From: Patricia Wollstadt <patricia.wollstadt@gmx.de>
Date: Mon, 25 Mar 2024 14:52:17 +0100
Subject: [PATCH 34/34] Fix formatting and pylint errors

---
 demos/demo_active_information_storage.py  |   3 +-
 demos/demo_bivariate_mi.py                |  15 +-
 demos/demo_bivariate_pid.py               |  76 ++--
 demos/demo_bivariate_te.py                |  15 +-
 demos/demo_core_estimators.py             |  14 +-
 demos/demo_hd_estimator.py                |  36 +-
 demos/demo_multivariate_mi.py             |  17 +-
 demos/demo_multivariate_pid.py            | 423 +++++++++++++++++-----
 demos/demo_multivariate_te.py             |  17 +-
 demos/demo_multivariate_te_mpi.py         |  42 ++-
 demos/demo_significant_subgraph_mining.py |  58 +--
 idtxl/estimator.py                        |  63 ++--
 idtxl/network_inference.py                |  11 +-
 13 files changed, 522 insertions(+), 268 deletions(-)

diff --git a/demos/demo_active_information_storage.py b/demos/demo_active_information_storage.py
index c0f6d2f..b214a26 100644
--- a/demos/demo_active_information_storage.py
+++ b/demos/demo_active_information_storage.py
@@ -8,8 +8,7 @@
 
 # b) Initialise analysis object and define settings
 network_analysis = ActiveInformationStorage()
-settings = {'cmi_estimator':  'JidtGaussianCMI',
-            'max_lag': 5}
+settings = {"cmi_estimator": "JidtGaussianCMI", "max_lag": 5}
 
 # c) Run analysis
 results = network_analysis.analyse_network(settings=settings, data=data)
diff --git a/demos/demo_bivariate_mi.py b/demos/demo_bivariate_mi.py
index 25fe509..afb2c40 100644
--- a/demos/demo_bivariate_mi.py
+++ b/demos/demo_bivariate_mi.py
@@ -1,8 +1,9 @@
 # Import classes
+import matplotlib.pyplot as plt
+
 from idtxl.bivariate_mi import BivariateMI
 from idtxl.data import Data
 from idtxl.visualise_graph import plot_network
-import matplotlib.pyplot as plt
 
 # a) Generate test data
 data = Data()
@@ -10,14 +11,16 @@
 
 # b) Initialise analysis object and define settings
 network_analysis = BivariateMI()
-settings = {'cmi_estimator': 'JidtGaussianCMI',
-            'max_lag_sources': 5,
-            'min_lag_sources': 1}
+settings = {
+    "cmi_estimator": "JidtGaussianCMI",
+    "max_lag_sources": 5,
+    "min_lag_sources": 1,
+}
 
 # c) Run analysis
 results = network_analysis.analyse_network(settings=settings, data=data)
 
 # d) Plot inferred network to console and via matplotlib
-results.print_edge_list(weights='max_te_lag', fdr=False)
-plot_network(results=results, weights='max_te_lag', fdr=False)
+results.print_edge_list(weights="max_te_lag", fdr=False)
+plot_network(results=results, weights="max_te_lag", fdr=False)
 plt.show()
diff --git a/demos/demo_bivariate_pid.py b/demos/demo_bivariate_pid.py
index f8ede06..ec40795 100644
--- a/demos/demo_bivariate_pid.py
+++ b/demos/demo_bivariate_pid.py
@@ -1,5 +1,6 @@
 # Import classes
 import numpy as np
+
 from idtxl.bivariate_pid import BivariatePID
 from idtxl.data import Data
 
@@ -9,46 +10,61 @@
 x = np.random.randint(0, alph, n)
 y = np.random.randint(0, alph, n)
 z = np.logical_xor(x, y).astype(int)
-data = Data(np.vstack((x, y, z)), 'ps', normalise=False)
+data = Data(np.vstack((x, y, z)), "ps", normalise=False)
 
 # b) Initialise analysis object and define settings for both PID estimators
 pid = BivariatePID()
-settings_tartu = {'pid_estimator': 'TartuPID', 'lags_pid': [0, 0]}
+settings_tartu = {"pid_estimator": "TartuPID", "lags_pid": [0, 0]}
 settings_sydney = {
-    'alph_s1': alph,
-    'alph_s2': alph,
-    'alph_t': alph,
-    'max_unsuc_swaps_row_parm': 60,
-    'num_reps': 63,
-    'max_iters': 1000,
-    'pid_estimator': 'SydneyPID',
-    'lags_pid': [0, 0]}
+    "alph_s1": alph,
+    "alph_s2": alph,
+    "alph_t": alph,
+    "max_unsuc_swaps_row_parm": 60,
+    "num_reps": 63,
+    "max_iters": 1000,
+    "pid_estimator": "SydneyPID",
+    "lags_pid": [0, 0],
+}
 
 # c) Run Tartu estimator
 results_tartu = pid.analyse_single_target(
-    settings=settings_tartu, data=data, target=2, sources=[0, 1])
+    settings=settings_tartu, data=data, target=2, sources=[0, 1]
+)
 
 # d) Run Sydney estimator
 pid = BivariatePID()
 results_sydney = pid.analyse_single_target(
-    settings=settings_sydney, data=data, target=2, sources=[0, 1])
+    settings=settings_sydney, data=data, target=2, sources=[0, 1]
+)
 
 # e) Print results to console
-print('\nLogical XOR')
-print('Estimator            Sydney\t\tTartu\t\tExpected\n')
-print('Uni s1               {0:.4f}\t\t{1:.4f}\t\t{2:.2f}'.format(
-    results_sydney.get_single_target(2)['unq_s1'],
-    results_tartu.get_single_target(2)['unq_s1'],
-    0))
-print('Uni s2               {0:.4f}\t\t{1:.4f}\t\t{2:.2f}'.format(
-    results_sydney.get_single_target(2)['unq_s2'],
-    results_tartu.get_single_target(2)['unq_s2'],
-    0))
-print('Shared s1_s2         {0:.4f}\t\t{1:.4f}\t\t{2:.2f}'.format(
-    results_sydney.get_single_target(2)['shd_s1_s2'],
-    results_tartu.get_single_target(2)['shd_s1_s2'],
-    0))
-print('Synergy s1_s2        {0:.4f}\t\t{1:.4f}\t\t{2:.2f}'.format(
-    results_sydney.get_single_target(2)['syn_s1_s2'],
-    results_tartu.get_single_target(2)['syn_s1_s2'],
-    1))
+print("\nLogical XOR")
+print("Estimator            Sydney\t\tTartu\t\tExpected\n")
+print(
+    "Uni s1               {0:.4f}\t\t{1:.4f}\t\t{2:.2f}".format(
+        results_sydney.get_single_target(2)["unq_s1"],
+        results_tartu.get_single_target(2)["unq_s1"],
+        0,
+    )
+)
+print(
+    "Uni s2               {0:.4f}\t\t{1:.4f}\t\t{2:.2f}".format(
+        results_sydney.get_single_target(2)["unq_s2"],
+        results_tartu.get_single_target(2)["unq_s2"],
+        0,
+    )
+)
+print(
+    "Shared s1_s2         {0:.4f}\t\t{1:.4f}\t\t{2:.2f}".format(
+        results_sydney.get_single_target(2)["shd_s1_s2"],
+        results_tartu.get_single_target(2)["shd_s1_s2"],
+        0,
+    )
+)
+print(
+    "Synergy s1_s2        {0:.4f}\t\t{1:.4f}\t\t{2:.2f}".format(
+        results_sydney.get_single_target(2)["syn_s1_s2"],
+        results_tartu.get_single_target(2)["syn_s1_s2"],
+        1,
+    )
+)
diff --git a/demos/demo_bivariate_te.py b/demos/demo_bivariate_te.py
index 25bb8b4..892b8ff 100644
--- a/demos/demo_bivariate_te.py
+++ b/demos/demo_bivariate_te.py
@@ -1,8 +1,9 @@
 # Import classes
+import matplotlib.pyplot as plt
+
 from idtxl.bivariate_te import BivariateTE
 from idtxl.data import Data
 from idtxl.visualise_graph import plot_network
-import matplotlib.pyplot as plt
 
 # a) Generate test data
 data = Data()
@@ -10,14 +11,16 @@
 
 # b) Initialise analysis object and define settings
 network_analysis = BivariateTE()
-settings = {'cmi_estimator': 'JidtGaussianCMI',
-            'max_lag_sources': 5,
-            'min_lag_sources': 1}
+settings = {
+    "cmi_estimator": "JidtGaussianCMI",
+    "max_lag_sources": 5,
+    "min_lag_sources": 1,
+}
 
 # c) Run analysis
 results = network_analysis.analyse_network(settings=settings, data=data)
 
 # d) Plot inferred network to console and via matplotlib
-results.print_edge_list(weights='max_te_lag', fdr=False)
-plot_network(results=results, weights='max_te_lag', fdr=False)
+results.print_edge_list(weights="max_te_lag", fdr=False)
+plot_network(results=results, weights="max_te_lag", fdr=False)
 plt.show()
diff --git a/demos/demo_core_estimators.py b/demos/demo_core_estimators.py
index 3d4d695..911c7f0 100644
--- a/demos/demo_core_estimators.py
+++ b/demos/demo_core_estimators.py
@@ -1,22 +1,24 @@
 """Demonstrate the usage of IDTxl's core estimators."""
+
 import numpy as np
-from idtxl.estimators_python import PythonKraskovCMI
+
 from idtxl.estimators_jidt import (
     JidtDiscreteAIS,
     JidtDiscreteCMI,
     JidtDiscreteMI,
     JidtDiscreteTE,
-    JidtKraskovAIS,
-    JidtKraskovCMI,
-    JidtKraskovMI,
-    JidtKraskovTE,
     JidtGaussianAIS,
     JidtGaussianCMI,
     JidtGaussianMI,
     JidtGaussianTE,
+    JidtKraskovAIS,
+    JidtKraskovCMI,
+    JidtKraskovMI,
+    JidtKraskovTE,
 )
-from idtxl.estimators_opencl import OpenCLKraskovMI, OpenCLKraskovCMI
+from idtxl.estimators_opencl import OpenCLKraskovCMI, OpenCLKraskovMI
 from idtxl.estimators_pid import SydneyPID, TartuPID
+from idtxl.estimators_python import PythonKraskovCMI
 from idtxl.idtxl_utils import calculate_mi
 
 # Generate Gaussian test data
diff --git a/demos/demo_hd_estimator.py b/demos/demo_hd_estimator.py
index a2728d5..352322d 100644
--- a/demos/demo_hd_estimator.py
+++ b/demos/demo_hd_estimator.py
@@ -12,15 +12,17 @@
 
 >>> test_fast_emb.py
 """
+
 # Import classes
 import numpy as np
+
 from idtxl.data_spiketime import Data_spiketime
 from idtxl.embedding_optimization_ais_Rudelt import OptimizationRudelt
 
 # a) Generate test data
 print("\nTest optimization_Rudelt using BBC estimator on Rudelt data")
-data = Data_spiketime()             # initialise empty data object
-data.load_Rudelt_data()             # load Rudelt spike time data
+data = Data_spiketime()  # initialise empty data object
+data.load_Rudelt_data()  # load Rudelt spike time data
 
 # b) Initialise analysis object and define settings
 # Run optimization with the Bayesian bias criterion (BBC) (alternatively, set
@@ -29,20 +31,22 @@
 # 'embedding_number_of_bins_set', and 'embedding_scaling_exponent_set' to
 # reduce the run time of this demo. Use defaults if unsure about the settings.
 settings = {
-    'embedding_past_range_set': [0.005, 0.31548, 5.0],
-    'embedding_number_of_bins_set': [1, 3, 5],
-    'embedding_scaling_exponent_set':
-        {'number_of_scalings': 3,
-         'min_first_bin_size': 0.005,
-         'min_step_for_scaling': 0.01},
-    'number_of_bootstraps_R_max': 10,
-    'number_of_bootstraps_R_tot': 10,
-    'auto_MI_bin_size_set': [0.01, 0.025, 0.05, 0.25],
-    'estimation_method': 'bbc',
-    'debug': True,
-    'visualization': True,
-    'output_path': '.',
-    'output_prefix': 'systemtest_optimizationRudelt_image1'}
+    "embedding_past_range_set": [0.005, 0.31548, 5.0],
+    "embedding_number_of_bins_set": [1, 3, 5],
+    "embedding_scaling_exponent_set": {
+        "number_of_scalings": 3,
+        "min_first_bin_size": 0.005,
+        "min_step_for_scaling": 0.01,
+    },
+    "number_of_bootstraps_R_max": 10,
+    "number_of_bootstraps_R_tot": 10,
+    "auto_MI_bin_size_set": [0.01, 0.025, 0.05, 0.25],
+    "estimation_method": "bbc",
+    "debug": True,
+    "visualization": True,
+    "output_path": ".",
+    "output_prefix": "systemtest_optimizationRudelt_image1",
+}
 hd_estimator = OptimizationRudelt(settings)
 
 # c) Run analysis and save a plot of the results to current directory.
diff --git a/demos/demo_multivariate_mi.py b/demos/demo_multivariate_mi.py
index 0bde929..311827b 100644
--- a/demos/demo_multivariate_mi.py
+++ b/demos/demo_multivariate_mi.py
@@ -1,8 +1,9 @@
 # Import classes
-from idtxl.multivariate_mi import MultivariateMI
+import matplotlib.pyplot as plt
+
 from idtxl.data import Data
+from idtxl.multivariate_mi import MultivariateMI
 from idtxl.visualise_graph import plot_network
-import matplotlib.pyplot as plt
 
 # a) Generate test data
 data = Data()
@@ -10,14 +11,16 @@
 
 # b) Initialise analysis object and define settings
 network_analysis = MultivariateMI()
-settings = {'cmi_estimator': 'JidtGaussianCMI',
-            'max_lag_sources': 5,
-            'min_lag_sources': 1}
+settings = {
+    "cmi_estimator": "JidtGaussianCMI",
+    "max_lag_sources": 5,
+    "min_lag_sources": 1,
+}
 
 # c) Run analysis
 results = network_analysis.analyse_network(settings=settings, data=data)
 
 # d) Plot inferred network to console and via matplotlib
-results.print_edge_list(weights='max_te_lag', fdr=False)
-plot_network(results=results, weights='max_te_lag', fdr=False)
+results.print_edge_list(weights="max_te_lag", fdr=False)
+plot_network(results=results, weights="max_te_lag", fdr=False)
 plt.show()
diff --git a/demos/demo_multivariate_pid.py b/demos/demo_multivariate_pid.py
index ee119f4..ef4ef8f 100644
--- a/demos/demo_multivariate_pid.py
+++ b/demos/demo_multivariate_pid.py
@@ -1,7 +1,8 @@
 # Import classes
 import numpy as np
-from idtxl.multivariate_pid import MultivariatePID
+
 from idtxl.data import Data
+from idtxl.multivariate_pid import MultivariatePID
 
 # a) Generate test data
 n = 100
@@ -9,32 +10,55 @@
 x = np.random.randint(0, alph, n)
 y = np.random.randint(0, alph, n)
 z = np.logical_xor(x, y).astype(int)
-data = Data(np.vstack((x, y, z)), 'ps', normalise=False)
+data = Data(np.vstack((x, y, z)), "ps", normalise=False)
 
 # b) Initialise analysis object and define settings for SxPID estimators
 pid = MultivariatePID()
-settings_SxPID = {'pid_estimator': 'SxPID', 'lags_pid': [0, 0]}
+settings_SxPID = {"pid_estimator": "SxPID", "lags_pid": [0, 0]}
 
 # c) Run Goettingen estimator
 results_SxPID = pid.analyse_single_target(
-    settings=settings_SxPID, data=data, target=2, sources=[0, 1])
+    settings=settings_SxPID, data=data, target=2, sources=[0, 1]
+)
 
 
 # e) Print results to console
-print('\nLogical XOR')
-print('Estimator            SxPID\t\tExpected\n')
-print('Uni s1               {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),)][2],
-    .5896))
-print('Uni s2               {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((2,),)][2],
-    0.5896))
-print('Shared s1_s2         {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),(2,),)][2],
-    -0.5896))
-print('Synergy s1_s2        {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,2,),)][2],
-    0.415))
+print("\nLogical XOR")
+print("Estimator            SxPID\t\tExpected\n")
+print(
+    "Uni s1               {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][((1,),)][2], 0.5896
+    )
+)
+print(
+    "Uni s2               {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][((2,),)][2], 0.5896
+    )
+)
+print(
+    "Shared s1_s2         {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (1,),
+                (2,),
+            )
+        ][2],
+        -0.5896,
+    )
+)
+print(
+    "Synergy s1_s2        {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+            )
+        ][2],
+        0.415,
+    )
+)
 
 # Some special Examples
 
@@ -42,104 +66,309 @@
 x = np.asarray([0, 1, 0, 2])
 y = np.asarray([1, 0, 2, 0])
 z = np.asarray([1, 1, 2, 2])
-data = Data(np.vstack((x, y, z)), 'ps', normalise=False)
+data = Data(np.vstack((x, y, z)), "ps", normalise=False)
 
 pid = MultivariatePID()
-settings_SxPID = {'pid_estimator': 'SxPID', 'lags_pid': [0, 0]}
+settings_SxPID = {"pid_estimator": "SxPID", "lags_pid": [0, 0]}
 
 results_SxPID = pid.analyse_single_target(
-    settings=settings_SxPID, data=data, target=2, sources=[0, 1])
-
-print('\nLogical PwUnq')
-print('Estimator            SxPID\t\tExpected\n')
-print('Uni s1               {0:.4f}\t\t{1:.1f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),)][2],
-    .5))
-print('Uni s2               {0:.4f}\t\t{1:.1f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((2,),)][2],
-    0.5))
-print('Shared s1_s2         {0:.4f}\t\t{1:.1f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),(2,),)][2],
-    0.))
-print('Synergy s1_s2        {0:.4f}\t\t{1:.1f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,2,),)][2],
-    0.))
+    settings=settings_SxPID, data=data, target=2, sources=[0, 1]
+)
+
+print("\nLogical PwUnq")
+print("Estimator            SxPID\t\tExpected\n")
+print(
+    "Uni s1               {0:.4f}\t\t{1:.1f}".format(
+        results_SxPID.get_single_target(2)["avg"][((1,),)][2], 0.5
+    )
+)
+print(
+    "Uni s2               {0:.4f}\t\t{1:.1f}".format(
+        results_SxPID.get_single_target(2)["avg"][((2,),)][2], 0.5
+    )
+)
+print(
+    "Shared s1_s2         {0:.4f}\t\t{1:.1f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (1,),
+                (2,),
+            )
+        ][2],
+        0.0,
+    )
+)
+print(
+    "Synergy s1_s2        {0:.4f}\t\t{1:.1f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+            )
+        ][2],
+        0.0,
+    )
+)
 
 # Redundancy Error
 x = np.asarray([0, 0, 0, 1, 1, 1, 0, 1])
 y = np.asarray([0, 0, 0, 1, 1, 1, 1, 0])
 z = np.asarray([0, 0, 0, 1, 1, 1, 0, 1])
-data = Data(np.vstack((x, y, z)), 'ps', normalise=False)
+data = Data(np.vstack((x, y, z)), "ps", normalise=False)
 
 pid = MultivariatePID()
-settings_SxPID = {'pid_estimator': 'SxPID', 'lags_pid': [0, 0]}
+settings_SxPID = {"pid_estimator": "SxPID", "lags_pid": [0, 0]}
 
 results_SxPID = pid.analyse_single_target(
-    settings=settings_SxPID, data=data, target=2, sources=[0, 1])
-
-print('\nLogical RndErr')
-print('Estimator            SxPID\t\tExpected\n')
-print('Uni s1               {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),)][2],
-    .4433))
-print('Uni s2               {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((2,),)][2],
-    -0.368))
-print('Shared s1_s2         {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,),(2,),)][2],
-    0.5567))
-print('Synergy s1_s2        {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(2)['avg'][((1,2,),)][2],
-    0.368))
+    settings=settings_SxPID, data=data, target=2, sources=[0, 1]
+)
+
+print("\nLogical RndErr")
+print("Estimator            SxPID\t\tExpected\n")
+print(
+    "Uni s1               {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][((1,),)][2], 0.4433
+    )
+)
+print(
+    "Uni s2               {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][((2,),)][2], -0.368
+    )
+)
+print(
+    "Shared s1_s2         {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (1,),
+                (2,),
+            )
+        ][2],
+        0.5567,
+    )
+)
+print(
+    "Synergy s1_s2        {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(2)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+            )
+        ][2],
+        0.368,
+    )
+)
 
 # Three bits hash
 s1 = np.asarray([0, 0, 0, 0, 1, 1, 1, 1])
 s2 = np.asarray([0, 0, 1, 1, 0, 0, 1, 1])
 s3 = np.asarray([0, 1, 0, 1, 0, 1, 0, 1])
-z  = np.asarray([0, 1, 1, 0, 1, 0, 0, 1])
-data = Data(np.vstack((s1, s2, s3, z)), 'ps', normalise=False)
+z = np.asarray([0, 1, 1, 0, 1, 0, 0, 1])
+data = Data(np.vstack((s1, s2, s3, z)), "ps", normalise=False)
 
 pid = MultivariatePID()
-settings_SxPID = {'pid_estimator': 'SxPID', 'lags_pid': [0, 0, 0]}
+settings_SxPID = {"pid_estimator": "SxPID", "lags_pid": [0, 0, 0]}
 
 results_SxPID = pid.analyse_single_target(
-    settings=settings_SxPID, data=data, target=3, sources=[0, 1, 2])
-
-print('\nLogical PwUnq')
-print('Estimator                                                SxPID\t\tExpected\n')
-print('Uni s1                                                   {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,),)][2], 0.3219))
-print('Uni s2                                                   {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((2,),)][2], 0.3219))
-print('Uni s3                                                   {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((3,),)][2], 0.3219))
-print('Synergy s1_s2_s3                                         {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,2,3),)][2], 0.2451))
-print('Synergy s1_s2                                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,2,),)][2], 0.1699))
-print('Synergy s1_s3                                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,3,),)][2], 0.1699))
-print('Synergy s2_s3                                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((2,3,),)][2], 0.1699))
-print('Shared s1_s2_s3                                          {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,),(2,),(3,),)][2], 0.1926))
-print('Shared of (s1, s2)                                       {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,), (2,),)][2], -0.1926))
-print('Shared of (s1, s2)                                       {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,), (3,),)][2], -0.1926))
-print('Shared of (s2, s3)                                       {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((2,), (3,),)][2], -0.1926))
-print('Shared of (Synergy s1_s2, Synergy s1_s3)                 {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,2,), (1,3),)][2], 0.0931))
-print('Shared of (Synergy s1_s2, Synergy s2_s3)                 {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,2,), (2,3),)][2], 0.0931))
-print('Shared of (Synergy s1_s3, Synergy s2_s3)                 {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,3,), (2,3),)][2], 0.0931))
-print('Shared of (Synergy s1_s2, Synergy s1_s3, Synergy s2_s3)  {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,2,), (1,3), (2,3),)][2], -0.2268))
-print('Shared of (s1, Synergy s2_s3)                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((1,), (2,3),)][2], -0.1292))
-print('Shared of (s2, Synergy s1_s3)                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((2,), (1,3),)][2], -0.1292))
-print('Shared of (s3, Synergy s1_s2)                            {0:.4f}\t\t{1:.4f}'.format(
-    results_SxPID.get_single_target(3)['avg'][((3,), (1,2),)][2], -0.1292))
+    settings=settings_SxPID, data=data, target=3, sources=[0, 1, 2]
+)
+
+print("\nLogical PwUnq")
+print("Estimator                                                SxPID\t\tExpected\n")
+print(
+    "Uni s1                                                   {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][((1,),)][2], 0.3219
+    )
+)
+print(
+    "Uni s2                                                   {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][((2,),)][2], 0.3219
+    )
+)
+print(
+    "Uni s3                                                   {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][((3,),)][2], 0.3219
+    )
+)
+print(
+    "Synergy s1_s2_s3                                         {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][((1, 2, 3),)][2], 0.2451
+    )
+)
+print(
+    "Synergy s1_s2                                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+            )
+        ][2],
+        0.1699,
+    )
+)
+print(
+    "Synergy s1_s3                                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    3,
+                ),
+            )
+        ][2],
+        0.1699,
+    )
+)
+print(
+    "Synergy s2_s3                                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    2,
+                    3,
+                ),
+            )
+        ][2],
+        0.1699,
+    )
+)
+print(
+    "Shared s1_s2_s3                                          {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (1,),
+                (2,),
+                (3,),
+            )
+        ][2],
+        0.1926,
+    )
+)
+print(
+    "Shared of (s1, s2)                                       {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (1,),
+                (2,),
+            )
+        ][2],
+        -0.1926,
+    )
+)
+print(
+    "Shared of (s1, s2)                                       {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (1,),
+                (3,),
+            )
+        ][2],
+        -0.1926,
+    )
+)
+print(
+    "Shared of (s2, s3)                                       {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (2,),
+                (3,),
+            )
+        ][2],
+        -0.1926,
+    )
+)
+print(
+    "Shared of (Synergy s1_s2, Synergy s1_s3)                 {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+                (1, 3),
+            )
+        ][2],
+        0.0931,
+    )
+)
+print(
+    "Shared of (Synergy s1_s2, Synergy s2_s3)                 {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+                (2, 3),
+            )
+        ][2],
+        0.0931,
+    )
+)
+print(
+    "Shared of (Synergy s1_s3, Synergy s2_s3)                 {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    3,
+                ),
+                (2, 3),
+            )
+        ][2],
+        0.0931,
+    )
+)
+print(
+    "Shared of (Synergy s1_s2, Synergy s1_s3, Synergy s2_s3)  {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (
+                    1,
+                    2,
+                ),
+                (1, 3),
+                (2, 3),
+            )
+        ][2],
+        -0.2268,
+    )
+)
+print(
+    "Shared of (s1, Synergy s2_s3)                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (1,),
+                (2, 3),
+            )
+        ][2],
+        -0.1292,
+    )
+)
+print(
+    "Shared of (s2, Synergy s1_s3)                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (2,),
+                (1, 3),
+            )
+        ][2],
+        -0.1292,
+    )
+)
+print(
+    "Shared of (s3, Synergy s1_s2)                            {0:.4f}\t\t{1:.4f}".format(
+        results_SxPID.get_single_target(3)["avg"][
+            (
+                (3,),
+                (1, 2),
+            )
+        ][2],
+        -0.1292,
+    )
+)
diff --git a/demos/demo_multivariate_te.py b/demos/demo_multivariate_te.py
index 5fc1f0a..b9f9e0e 100644
--- a/demos/demo_multivariate_te.py
+++ b/demos/demo_multivariate_te.py
@@ -1,8 +1,9 @@
 # Import classes
-from idtxl.multivariate_te import MultivariateTE
+import matplotlib.pyplot as plt
+
 from idtxl.data import Data
+from idtxl.multivariate_te import MultivariateTE
 from idtxl.visualise_graph import plot_network
-import matplotlib.pyplot as plt
 
 # a) Generate test data
 data = Data()
@@ -10,14 +11,16 @@
 
 # b) Initialise analysis object and define settings
 network_analysis = MultivariateTE()
-settings = {'cmi_estimator': 'JidtGaussianCMI',
-            'max_lag_sources': 5,
-            'min_lag_sources': 1}
+settings = {
+    "cmi_estimator": "JidtGaussianCMI",
+    "max_lag_sources": 5,
+    "min_lag_sources": 1,
+}
 
 # c) Run analysis
 results = network_analysis.analyse_network(settings=settings, data=data)
 
 # d) Plot inferred network to console and via matplotlib
-results.print_edge_list(weights='max_te_lag', fdr=False)
-plot_network(results=results, weights='max_te_lag', fdr=False)
+results.print_edge_list(weights="max_te_lag", fdr=False)
+plot_network(results=results, weights="max_te_lag", fdr=False)
 plt.show()
diff --git a/demos/demo_multivariate_te_mpi.py b/demos/demo_multivariate_te_mpi.py
index e88dfdb..9af5ef2 100644
--- a/demos/demo_multivariate_te_mpi.py
+++ b/demos/demo_multivariate_te_mpi.py
@@ -1,13 +1,15 @@
 # Import classes
-from idtxl.multivariate_te import MultivariateTE
-from idtxl.data import Data
-from idtxl.visualise_graph import plot_network
-import matplotlib.pyplot as plt
-import time
 import sys
+import time
 
+import matplotlib.pyplot as plt
 from mpi4py import MPI
 
+from idtxl.data import Data
+from idtxl.multivariate_te import MultivariateTE
+from idtxl.visualise_graph import plot_network
+
+
 def main(args):
     """Demo of how to use the MPIEstimator for network analysis
     Call this script using
@@ -15,16 +17,16 @@ def main(args):
     for systems supporting MPI Spawn (MPI version >= 2) or
     mpiexec -n <number of workers + 1> python -m mpi4py.futures demo/demo_multivariate_te_mpi.py <number of workers>
     for legacy MPI 1 implementations.
-    
+
     Call
     python demos/demo_multivariate_te_mpi.py 0
     for a comparison without MPI.
 
     """
-    assert MPI.COMM_WORLD.Get_rank() == 0 
+    assert MPI.COMM_WORLD.Get_rank() == 0
 
     max_workers = int(args[1])
-    print(f'Running TE Test with {max_workers} MPI workers.')
+    print(f"Running TE Test with {max_workers} MPI workers.")
 
     # a) Generate test data
     data = Data(seed=12345)
@@ -32,23 +34,25 @@ def main(args):
 
     # b) Initialise analysis object and define settings
     network_analysis = MultivariateTE()
-    settings = {'cmi_estimator': 'JidtGaussianCMI',
-                'max_lag_sources': 5,
-                'min_lag_sources': 1,
-                'MPI': max_workers > 0,
-                'max_workers': max_workers
-                }
+    settings = {
+        "cmi_estimator": "JidtGaussianCMI",
+        "max_lag_sources": 5,
+        "min_lag_sources": 1,
+        "MPI": max_workers > 0,
+        "max_workers": max_workers,
+    }
 
     # c) Run analysis
     start = time.time()
     results = network_analysis.analyse_network(settings=settings, data=data)
     end = time.time()
-    print(f'On {max_workers} workers, the task took {end - start} seconds.')
+    print(f"On {max_workers} workers, the task took {end - start} seconds.")
 
     # d) Plot inferred network to console and via matplotlib
-    results.print_edge_list(weights='max_te_lag', fdr=False)
-    plot_network(results=results, weights='max_te_lag', fdr=False)
+    results.print_edge_list(weights="max_te_lag", fdr=False)
+    plot_network(results=results, weights="max_te_lag", fdr=False)
     plt.show()
 
-if __name__ == '__main__':
-    main(sys.argv)
\ No newline at end of file
+
+if __name__ == "__main__":
+    main(sys.argv)
diff --git a/demos/demo_significant_subgraph_mining.py b/demos/demo_significant_subgraph_mining.py
index a7f810f..6455f72 100644
--- a/demos/demo_significant_subgraph_mining.py
+++ b/demos/demo_significant_subgraph_mining.py
@@ -11,21 +11,21 @@
 data_b = Data()
 data_b.generate_mute_data(100, 5)
 settings = {
-    'cmi_estimator': 'JidtGaussianCMI',
-    'n_perm_max_stat': 50,
-    'n_perm_min_stat': 50,
-    'n_perm_omnibus': 200,
-    'n_perm_max_seq': 50,
-    'max_lag_target': 5,
-    'max_lag_sources': 5,
-    'min_lag_sources': 1,
-    'permute_in_time': True
-    }
+    "cmi_estimator": "JidtGaussianCMI",
+    "n_perm_max_stat": 50,
+    "n_perm_min_stat": 50,
+    "n_perm_omnibus": 200,
+    "n_perm_max_seq": 50,
+    "max_lag_target": 5,
+    "max_lag_sources": 5,
+    "min_lag_sources": 1,
+    "permute_in_time": True,
+}
 # Run analysis with different targets to simulate different results for both
 # analyses.
 network_analysis = MultivariateTE()
-res_a = network_analysis.analyse_network(settings, data_a, sources='all')
-res_b = network_analysis.analyse_network(settings, data_b,  sources='all')
+res_a = network_analysis.analyse_network(settings, data_a, sources="all")
+res_b = network_analysis.analyse_network(settings, data_b, sources="all")
 
 # Perform significant subgraph mining on inferred networks using different
 # correction methods. Use the IDTxl format as input, which looks also at the
@@ -36,13 +36,13 @@
     [res_b.get_source_variables(fdr=False) for n in range(sample_size)],
     design="between",
     alpha=0.05,
-    data_format="idtxl"
-    )
+    data_format="idtxl",
+)
 res_ssm_between = SSM_between.enumerate_significant_subgraphs(
     method="Hommel",  # correction method: "Tarone", "Hommel", or "Westfall-Young"
     verbose=True,
     num_perm=10000,
-    max_depth=np.infty
+    max_depth=np.infty,
 )
 # List of subgraphs, each entry is a tuple comprising a list of edges and the
 # p-value for this subgraph.
@@ -53,14 +53,14 @@
     [res_b.get_source_variables(fdr=False) for n in range(sample_size)],
     design="within",
     alpha=0.05,
-    data_format="idtxl"
-    )
+    data_format="idtxl",
+)
 res_ssm_within = SSM_within.enumerate_significant_subgraphs(
     method="Westfall-Young",  # correction method: "Tarone", "Hommel", or "Westfall-Young"
     wy_algorithm="simple_depth_first",
     verbose=True,
     num_perm=10000,
-    max_depth=np.infty
+    max_depth=np.infty,
 )
 print(res_ssm_within)
 
@@ -69,29 +69,35 @@
     [res_b.get_source_variables(fdr=False) for n in range(sample_size)],
     design="within",
     alpha=0.05,
-    data_format="idtxl"
-    )
+    data_format="idtxl",
+)
 res_ssm_within = SSM_within.enumerate_significant_subgraphs(
     method="Tarone",  # correction method: "Tarone", "Hommel", or "Westfall-Young"
     verbose=True,
     num_perm=10000,
-    max_depth=np.infty
+    max_depth=np.infty,
 )
 print(res_ssm_within)
 
 # Use adjacency matrices as input. This ignores the time dimension of the
 # inferred network.
 SSM_within = SignificantSubgraphMining(
-    [res_a.get_adjacency_matrix(weights='binary', fdr=False).edge_matrix for n in range(sample_size)],
-    [res_b.get_adjacency_matrix(weights='binary', fdr=False).edge_matrix for n in range(sample_size)],
+    [
+        res_a.get_adjacency_matrix(weights="binary", fdr=False).edge_matrix
+        for n in range(sample_size)
+    ],
+    [
+        res_b.get_adjacency_matrix(weights="binary", fdr=False).edge_matrix
+        for n in range(sample_size)
+    ],
     design="within",
     alpha=0.05,
-    data_format="adjacency"
-    )
+    data_format="adjacency",
+)
 res_ssm_within = SSM_within.enumerate_significant_subgraphs(
     method="Tarone",  # correction method: "Tarone", "Hommel", or "Westfall-Young"
     verbose=True,
     num_perm=10000,
-    max_depth=np.infty
+    max_depth=np.infty,
 )
 print(res_ssm_within)
diff --git a/idtxl/estimator.py b/idtxl/estimator.py
index 46ba078..e531bf5 100644
--- a/idtxl/estimator.py
+++ b/idtxl/estimator.py
@@ -1,11 +1,12 @@
 """Provide estimator base class for information theoretic measures."""
-import os
+
 import importlib
 import inspect
-from pprint import pprint
+import os
 from abc import ABCMeta, abstractmethod
+from pprint import pprint
+
 import numpy as np
-from . import idtxl_exceptions as ex
 
 MODULE_EXTENSIONS = ".py"  # ('.py', '.pyc', '.pyo')
 ESTIMATOR_PREFIX = "estimators_"
@@ -54,10 +55,10 @@ def _find_estimator(est):
         # constraint may be relaxed in the future.
         if not np.issubclass_(est, Estimator):
             raise RuntimeError(
-                "Provided class should implement abstract class" " Estimator."
+                "Provided class should implement abstract class Estimator."
             )
         return est
-    elif type(est) is str:
+    if isinstance(est, str):
         module_list = _package_contents()
         estimator = None
         for m in module_list:
@@ -67,11 +68,10 @@ def _find_estimator(est):
             except AttributeError:
                 pass
         if not estimator:
-            raise RuntimeError("Estimator {0} not found.".format(est))
+            raise RuntimeError(f"Estimator {est} not found.")
     else:
         raise TypeError(
-            "Please provide an estimator class or the name of an "
-            "estimator as string."
+            "Please provide an estimator class or the name of an estimator as string."
         )
 
 
@@ -197,7 +197,7 @@ def _check_settings(self, settings=None):
         """
         if settings is None:
             return {}
-        elif type(settings) is not dict:
+        elif not isinstance(settings, dict):
             raise TypeError("settings should be a dictionary.")
         else:
             return settings
@@ -206,20 +206,15 @@ def _check_number_of_points(self, n_points):
         """Sanity check for number of points going into the estimator."""
         if (n_points - 1) <= int(self.settings["kraskov_k"]):
             raise RuntimeError(
-                "Insufficient number of points ({0}) for the "
-                "requested number of nearest neighbours "
-                "(kraskov_k: {1}).".format(n_points, self.settings["kraskov_k"])
+                f"Insufficient number of points ({n_points}) for the requested number of nearest neighbours "
+                f"(kraskov_k: {self.settings['kraskov_k']})."
             )
         if (n_points - 1) <= (
             int(self.settings["kraskov_k"]) + int(self.settings["theiler_t"])
         ):
             raise RuntimeError(
-                "Insufficient number of points ({0}) for the "
-                "requested number of nearest neighbours "
-                "(kraskov_k: {1}) and Theiler-correction "
-                "(theiler_t: {2}).".format(
-                    n_points, self.settings["kraskov_k"], self.settings["theiler_t"]
-                )
+                f"Insufficient number of points ({n_points}) for the requested number of nearest neighbours (kraskov_k:"
+                f" {self.settings['kraskov_k']}) and Theiler-correction (theiler_t: {self.settings['theiler_t']})."
             )
 
     def _ensure_one_dim_input(self, var):
@@ -239,7 +234,7 @@ def _ensure_one_dim_input(self, var):
             else:
                 raise TypeError("2D input arrays must have shape[1] == 1.")
         elif len(var.shape) > 2:
-            raise TypeError("Input arrays must be 1D or 2D with shape[1] == " "1.")
+            raise TypeError("Input arrays must be 1D or 2D with shape[1] == 1.")
         return var
 
     def _ensure_two_dim_input(self, var):
@@ -334,20 +329,14 @@ def estimate_parallel(self, n_chunks=1, re_use=None, **data):
                     n_samples_total.append(data[v].shape[0])
             assert (
                 np.array(n_samples_total) == n_samples_total[0]
-            ).all(), "No. realisations should be the same for all variables: " "{0}".format(
-                n_samples_total
-            )
+            ).all(), f"No. realisations should be the same for all variables: {n_samples_total}"
             n_samples_total = n_samples_total[0]
             assert (
                 n_samples_total is not None
             ), "All variables provided for estimation are empty."
             assert n_samples_total % n_chunks == 0, (
-                "No. chunks ({0}) does not match data length ({1}). Remainder:"
-                " {2}.".format(
-                    n_chunks,
-                    data[slice_vars[0]].shape[0],
-                    data[slice_vars[0]].shape[0] % n_chunks,
-                )
+                f"No. chunks ({n_chunks}) does not match data length ({data[slice_vars[0]].shape[0]}). "
+                f"Remainder: {data[slice_vars[0]].shape[0] % n_chunks}."
             )
 
             # Cut data into chunks and call estimator serially on each chunk.
@@ -357,12 +346,9 @@ def estimate_parallel(self, n_chunks=1, re_use=None, **data):
             results = np.empty((n_chunks))
             for i in range(n_chunks):
                 chunk_data = {}
-                # Slice data into single chunks
-                for (
-                    v
-                ) in (
-                    slice_vars
-                ):  # NOTE: I am consciously not creating a deep copy here to save memory
+                # Slice data into single chunks. NOTE: I am consciously not
+                # creating a deep copy here to save memory
+                for v in slice_vars:
                     if data[v] is not None:
                         chunk_data[v] = data[v][idx_1:idx_2, :]
                     else:
@@ -370,12 +356,9 @@ def estimate_parallel(self, n_chunks=1, re_use=None, **data):
                 # Collect data that is reused over chunks.
                 for v in re_use:
                     if data[v] is not None:
-                        assert data[v].shape[0] == chunk_size, (
-                            "No. samples in variable {0} ({1}) is not equal "
-                            "to chunk size ({2}).".format(
-                                v, data[v].shape[0], chunk_size
-                            )
-                        )
+                        assert (
+                            data[v].shape[0] == chunk_size
+                        ), f"No. samples in variable {v} ({data[v].shape[0]}) is not equal to chunk size ({chunk_size})."
                     chunk_data[v] = data[v]
                 results[i] = self.estimate(**chunk_data)
                 idx_1 = idx_2
diff --git a/idtxl/network_inference.py b/idtxl/network_inference.py
index 7f888b3..1103896 100755
--- a/idtxl/network_inference.py
+++ b/idtxl/network_inference.py
@@ -1,4 +1,5 @@
 """Parent class for all network inference."""
+
 import numpy as np
 
 from . import idtxl_exceptions as ex
@@ -1080,7 +1081,7 @@ def _prune_candidates(self, data):
                 self.current_value, remaining_candidates
             )[0]
             try:
-                [significant, p, surr_table] = stats.min_statistic(
+                significant, p, surr_table = stats.min_statistic(
                     self,
                     data,
                     self.selected_vars_sources,
@@ -1131,9 +1132,7 @@ def _test_final_conditional(self, data):
         else:
             if self.settings["verbose"]:
                 print(
-                    "selected variables: {0}".format(
-                        self._idx_to_lag(self.selected_vars_full)
-                    )
+                    f"selected variables: {self._idx_to_lag(self.selected_vars_full)}"
                 )
             try:
                 [s, p, stat] = stats.omnibus_test(self, data)
@@ -1141,8 +1140,8 @@ def _test_final_conditional(self, data):
                 # The algorithm cannot continue here, so
                 #  we'll set the results to zero
                 print(
-                    "AlgorithmExhaustedError encountered in estimations: {}. "
-                    "Halting current estimation set.".format(aee.message)
+                    "AlgorithmExhaustedError encountered in estimations: "
+                    f"{aee.message}. Halting current estimation set."
                 )
                 # For now we don't need a stack trace:
                 # traceback.print_tb(aee.__traceback__)