Heron

The heron project is an attempt to use Gaussian Process Regression to improve gravitational wave searches for compact binary coalesences from black holes.

Heron builds a GPR surrogate model trained off data from numerical relativity simulations which can then be used to generate waveforms at locations in the BBH parameter space which have not been sampled by NR. By exaxmining the quality of the GPR’s predictions we can also identify the locations in the parameter space which are most in need of new samples being taken.

The paper draft for this project is in a git repository here.

The detection and analysis of gravitational wave signals is heavily dependent on matched filtering, a process which matches pre-computed template waveforms to signals in noisy data. In the case of gravitational wave detection, problems which can be fruitfully addressed by this technique include the detection of signals from compact binary coalesences (CBCs), such as binary black hole (BBH) and binary neutron star (BNS) mergers, where it is possible to predict the waveform that the event will produce, and the analysis of the features of the signal to identify the properties of the generating physical system—a process known as `parameter estimation’ (PE).

Accurate parameter estimation requires accurate templates; highly accurate templates can be generated using numerical relativity (NR) simulations, however doing so is computationally costly, and as a result only a small number of these high-accuracy templates are available. To provide greater coverage of the physical parameter space it is necessary to form an interpolant to these simulations, which maintains accuracy over the entire waveform.

A number of approaches to this problem exist, including the production of “post-Newtonian approximants” (PN) such as the TaylorT2 approximant, or Inspiral-Merger-Ringdown models, which are calibrated against the NR data, such as the effective one-body-NR (EOBNR) and IMRPhenom families of waveforms, spline interpolant surrogate models. While these models of the waveform can be evaluated with comparitive speed, they are prone to introducing a systematic error into the PE process, as the parameter-space values of the best-matching template generated from them is generally different from those of the true source parameters.

Methods for improving this situation were suggested in [Moore et al], who propose a method for incorporating a measure of the difference between the NR waveform and the approximant which is itself interpolated over the entire parameter space using Gaussian Process Regression (GPR), having been sampled at a number of locations constrained by the existence of NR waveforms.

The approach presented here attempts to by-pass this two-stage interpolation by combining the interpolation of the waveforms and the associated uncertainty, using GPR to interpolate the entire waveform, trained from NR data. GPR assumes that the waveform data are drawn from a Gaussian Process (GP) over the physical parameter space, wWith the mean and the covariance of the estimated from the NR data, and the covariance structure pre-defined by the smoothness conditions on the waveform. This GPR approach is capable not only of producing an interpolated function, but also a Gaussian probability distribution for that waveform, encapsulating the uncertainty in the interpolant at each point in parameter space. This provides a means not only of improving the accuracy of matched filtering, but also provides information about which regions of the parameter space are most poorly understood, and can therefore motivate the placement of future waveform simulations.

This note outlines the theory behind GPR, and briefly discusses the problems which are encountered with thiss technique when addressing a large problem such as this using Gaussian Processes; we then discuss the implementation which we use to by-pass some of these difficulties. We then discuss the training data which was used to produce our GPR surrogate model, and the details of the model itself: GPR is a “non-parametric” regression technique, and as such a minimal number of choices are made about the form of the interpolant, with most learned from the training data, however we did need to make specific choices with regards to the covariance function used to generate the GP. After discussing the details of the model, we cover the process of training that model (selecting the correct hyperparameters for the covariance function), and then address the methods for testing the trained model.

Theory of Gaussian Processes

Gaussian processes are Bayesian models which associate every point an some input space with a probability distribution—specifically a normal distribution, and a collection of input points will form a multi-variate normal distribution. Gaussian processes are a non-parametric supervised machine learning technique\cite{barberBRML2012,mackay2003information}.

To make a prediction, we need to first have a set of prior observations and information about the parameters of the physical system which produced those observations. These combined form the training data for the predictor.

A completely untrained Gaussian process forms the job of a prior probability distribution in a Bayesian analysis; where it is more conventional to consider a prior over a set of, for example, real values, such as a normal distribution, the Gaussian process forms a prior over the functions which might form the regression fit to any observed data. While this prior is intially untrained it still contains information about our preconceptions of the data, for example, whether or not we expect the fit to be analytic, and the average of the functions. By providing training data we can update the Gaussian process, in the same way that the posterior distribution is updated by the addition of new data in a standard Bayesian context, and a posterior on the set of all possible functions to fit the data is produced. We can take the mean of this posterior in the place of the “best fit line” which other techniques produce, and then use the variance to produce an estimate of the uncertainty of the prediction.

The possibility of using Gaussian Processes in the analysis of data from gravitational wave detectors has been proposed by Moore and Gair\cite{2014PhRvL.113y1101M,2016PhRvD..93f4001M} who propose its use to calculate the uncertainties in current generation post-Newtonian approximants to numerical relativity simulations, and to incorporate this into the current matched-filtering analyses which are conducted on triggers from gravitational wave detectors.

Gaussian processes trained with $N$ data require the ability to both store and invert an $N× N$ matrix of covariances between observations; this can be a considerable computational challenge, and there are a number of approaches to get around this problem, including sparse Gaussian processes, where a limit on the parameter-space distance between training points is set, and the covariance of points outside this radius are ignored\cite{EPFL-CONF-161319}, and heirarchical methods\cite{hodlr}.

Gaussian processes can be extended from the case of a single-dimensional input predicting a single-dimensional output to the ability to predict a multi-dimensional output from a multi-dimensional input\cite{Alvarez2011,Alvarez2011a,Bonilla2007}.

Covariance Functions

The covariance function defines the similarity of a pair of data points, according to some relationship with suitable properties. The similarity of input data is assumed to be related to the similarity of the output, and therefore the more similar two inputs are the more likely their outputs are to be similar.

As such, the form of the covariance function represents prior knowledge about the data, and can encode understanding of effects such as periodicity within the data.

Stationary Covariance Function: A stationary covariance function is a

function $f(\vec{x} - \vec{x}’)$, and which is thus invariant to translations in the input space.

Isotropic Covariance Function: If a covariance function is a function of

the form $f(|\vec{x} - \vec{x}’|)$ then it is isotropic, and invariant under all rigid motions.

One of the most frequently encountered covariance functions in the literature is the exponential squared covariance functions \cite{rasmussen2006gaussian}

$$\label{eq:squaredexponentialkernel} k\mathrm{SE}(r) = exp( - \frac{r^2}{2 l^2} )$$

for $r$ the Euclidean distance of a datum from the centre of the parameter space, and $l$ is a scale factor associated with the axis along which the data are defined.

Kernel algebra

It is possible to define new kernels from the standard set through a series of defined operations.

Consider two covariance functions, $f_1$ and $f_2$, then

Kernel Addition $f = f_1 + f_2$ is a covariance function.

Kernel product $f = f_1 f_2$ is a covariance function.

Non-stationarity of the parameter space

Gaussian Process implementation

Computational Complexity

One severe disadvantage of Gaussian Processes as a data analysis tool are their high computational complexity. Producing a prediction from a GP requires inverting the covariance matrix; matrix inversion is an $\mathcal{O}(N^3)$ process in time, and scales with $\mathcal{O}(N^2)$ in memory use. This effectively limits the number of training points which can be input to a GP to fewer than $10^4$. A number of approaches have been developed in the literature to address this short-coming by utilising computationally tractable approximations to either the matrix inversion or the Gaussian process.

These approaches can be grouped into three broad categories; sparse Gaussian processes, which use a modified covariance function to force the covariance matrix to have a near-diagonal structure; hierarchical approaches, which do not modify the covariancec function, but approximate the off-diagonal terms’ influence on the inversion; and local expert approaches, in which the parameter space is divided into many sub-spaces, and each sub-space is modelled using an independent Gaussian process.

Sparse Gaussian proceses

Hierarchical Gaussian processes

Gaussian process local experts

Local expert approaches attempt to improve the computational performance of GPs by diving the parameter space of the model into multiple sub-spaces. In a conventional GP the training data, $\mathcal{D} = \{ (x^n, y^n), n=1,…,N \} = \mathcal{X} ∪ \mathcal{Y},$ is used in its entirity to train a single GP. If these data were instead divided into $M$ subsets, of size $K$, we can train $M$ separate GPs, which will each provide an independent prediction for any given point in the parameter space. The network structure which is established by this subdivision of the parameter space is known as a gating network.

Early approaches to using local experts in GPs used kd-trees\cite{shen2005fast} to sub-divide the parameter space, and then modelled each subspace with its own GP. The GPs were trained together, with each having the same kernel hyper-parameters. Final predictions were then produced as a weighted sum of the individual GPs’ predictions. While this approach was somewhat effective, it enforced a stationary structure on the covariance matrix, and the paper does not treat the combination of the prediction uncertainties.

Approaches which follow the work of \cite{Jacobs:1991:AML:1351011.1351018} on mixtures of local experts have had some more promise, allowing each GP to have its own set of hyper-parameters, allowing greater freedom in modelling heteroscedastic and non-stationary data.

Deciding on the number of sub-models is a non-trivial problem; one approach is to model the parameter space using an infinite mixture model (IMM)\cite{rasmussen2002infinite}, in which the gating network is effectively a Dirichlet process over the training data. The predictions from each sub-model are then summed to find the global prediction. While this approach offers greater flexibility for modelling more complex underlying functions, it does little to improve the speed of GP predictions. Additional IMM approaches are proposed by \cite{meeds2006alternative}, and a comparable, variational approach is taken by \cite{yuan2009variational}.

All of these approaches have the difficulty of requiring the gating network to assign a weight (often called a responsibility to each sub-model’s prediction when calculating the global prediction, adding an additional layer of inference, which normally requires an MCMC sampler to perform. Product-of-experts models avoid this complication by multiplying the sub-model predictions, but these models have either turned out to be excessively confident,\cite{2014arXiv1412.3078N}, or excessively conservative\cite{2014arXiv1410.7827C}.

These problems have lead to the development of the Bayesian Committee Machine (BCM)\cite{tresp2000bayesian}, which assigns a weight to each sub-model’s prediction which is equal to the inverse of the prediction’s covariance, in order that sub-models which better observe the predicted region are given a greater weight in the global prediction. This approach can suffer as a result of models which contains week experts, and so the robust Bayesian Committee Machine\cite{deisenroth2015distributed} has been proposed to provide a more robust framework for Gaussian process regression with many experts. This approach also allows for the computation of the model’s prediction to be highly-parallelised, with the potential for each sub-model being evaluated on separate compute nodes, and combined together by another process running on another node.

Training Data

The data set being used for initial testing of heron is from the Georgia tech catalogue.

Spin-free catalogue

107	GT0448	1.000000	0.0	0.0	1673.544207
120	GT0905	1.000000	0.0	0.0	2108.784247
154	GT0717	0.909091	0.0	0.0	2218.726677
40	GT0370	0.869565	0.0	0.0	1933.067607
47	GT0898	0.833333	0.0	0.0	1482.263604
1	GT0371	0.769231	-0.0	-0.0	1928.511674
38	GT0730	0.714286	0.0	0.0	2211.053781
50	GT0372	0.689655	-0.0	-0.0	1924.333143
67	GT0477	0.666667	-0.0	-0.0	1677.553681
109	GT0374	0.625000	0.0	0.0	1931.951263
135	GT0615	0.588235	0.0	0.0	2202.255101
164	GT0727	0.571429	0.0	0.0	1677.225057
22	GT0618	0.555556	0.0	0.0	2198.091865
43	GT0376	0.526316	0.0	0.0	1869.599581
8	GT0446	0.500000	-0.0	-0.0	1669.850828
114	GT0378	0.487805	0.0	0.0	1866.705100
72	GT0617	0.476190	0.0	0.0	2183.817521
74	GT0757	0.444444	0.0	0.0	1662.185370
83	GT0621	0.434783	0.0	0.0	2173.230497
11	GT0380	0.425532	0.0	0.0	1858.792677
113	GT0622	0.416667	0.0	0.0	2172.392097
103	GT0565	0.400000	0.0	0.0	1665.310903
92	GT0651	0.384615	0.0	0.0	2163.291158
130	GT0650	0.370370	0.0	0.0	2157.575325
0	GT0653	0.357143	0.0	0.0	2153.499315
55	GT0654	0.344828	0.0	0.0	2149.934886
46	GT0453	0.333333	-0.0	-0.0	1655.664652
95	GT0758	0.322581	0.0	0.0	2137.921256
77	GT0759	0.312500	0.0	0.0	2136.142217
15	GT0760	0.303030	0.0	0.0	2133.003306
16	GT0761	0.294118	0.0	0.0	2132.212652
101	GT0748	0.277778	0.0	0.0	2120.306087
6	GT0778	0.270270	0.0	0.0	2120.665927
10	GT0777	0.256410	0.0	0.0	2113.319336
163	GT0454	0.250000	0.0	0.0	1589.495511
73	GT0797	0.243902	0.0	0.0	2106.858417
126	GT0798	0.238095	0.0	0.0	2099.965688
78	GT0791	0.232558	0.0	0.0	2101.702718
21	GT0795	0.227273	0.0	0.0	2101.245038
27	GT0656	0.222222	0.0	0.0	2094.168849
84	GT0577	0.200000	0.0	0.0	1801.025328
112	GT0604	0.166667	0.0	0.0	1783.645872
91	GT0818	0.142857	0.0	0.0	1958.621094
49	GT0568	0.100000	0.0	0.0	2296.274219

Catalogue

	Row	GTID	GT-Tag	Mass Ratio (q)	a1x	a1y	a1z	a2x	a2y	a2z	Lx	Ly	Lz	mf	af	Momega
	389	GT0830	D09_q1.00_a0.35_Q20	1	0	0	0.35	0	0	0.35	0	-0	0.9073	0.9395	0.7903	0.06416
	51	GT0355	D11_a0.2_q1.00_m103_As	1	0	0	0.2	0	0	-0.2	0	-0	0.9911	0.9515	0.6861	0.04918
	52	GT0356	D11_a0.4_q1.00_m103_As	1	0	0	0.3999	0	0	-0.4001	0	-0	0.9911	0.9514	0.6857	0.0492
	53	GT0357	D11_a0.6_q1.00_m103_As	1	0	0	0.5999	0	0	-0.6001	0	-0	0.9911	0.9512	0.6849	0.04919
	54	GT0358	D11_a0.8_q1.00_m103_As	1	0	0	0.7999	0	0	-0.8001	0	-0	0.9911	0.9509	0.6839	0.04922
	123	GT0476	D11_q1.00_a-0.2_m200	1	0	0	-0.2	0	0	-0.2	0	-0	1.003	0.9564	0.624	0.04975
	195	GT0564	D11_q1.00_a-0.4_m200	1	0	0	-0.4	0	0	-0.4	0	-0	1.015	0.9605	0.56	0.05046
	183	GT0546	D11_q1.00_a0.00_0.20_m160	1	0	0	0	0	0	0.2	0	-0	0.9853	0.9487	0.7164	0.0489
	184	GT0547	D11_q1.00_a0.00_0.40_m160	1	0	0	0	0	0	0.4	0	-0	0.9796	0.9454	0.7459	0.04866
	185	GT0548	D11_q1.00_a0.00_0.60_m160	1	0	0	0	0	0	0.6	0	-0	0.9739	0.9416	0.7747	0.04823
	188	GT0551	D11_q1.00_a0.00_0.80_m160	1	0	0	0	0	0	0.8	0	-0	0.9683	0.9372	0.8024	0.04832
	101	GT0448	D11_q1.00_a0.0_m240	1	0	0	0	0	0	0	0	-0	0.9911	0.9516	0.6862	0.04914
	202	GT0574	D11_q1.00_a0.1_m200	1	0	0	0.1	0	0	0.1	0	-0	0.9853	0.9487	0.7165	0.04889
	186	GT0549	D11_q1.00_a0.20_0.40_m160	1	0	0	0.2	0	0	0.4	0	-0	0.9739	0.9417	0.7751	0.0484
	189	GT0554	D11_q1.00_a0.20_0.60_m160	1	0	0	0.2	0	0	0.6	0	-0	0.9683	0.9374	0.8033	0.048
	190	GT0555	D11_q1.00_a0.20_0.80_m160	1	0	0	0.2	0	0	0.8	0	-0	0.9627	0.9324	0.8304	0.04813
	85	GT0420	D11_q1.00_a0.2_m200	1	0	0	0.2	0	0	0.2	0	-0	0.9796	0.9455	0.7462	0.04862
	205	GT0579	D11_q1.00_a0.3_m200	1	0	0	0.3	0	0	0.3	0	-0	0.9739	0.9418	0.7753	0.04837
	86	GT0422	D11_q1.00_a0.4_m200	1	0	0	0.4	0	0	0.4	0	-0	0.9683	0.9376	0.8036	0.04811
	206	GT0580	D11_q1.00_a0.5_m200	1	0	0	0.5	0	0	0.5	0	-0	0.9627	0.9327	0.8313	0.04784
	208	GT0583	D11_q1.00_a0.7_m200	1	0	0	0.7	0	0	0.7	0	-0	0.9516	0.92	0.884	0.0471
	87	GT0424	D11_q1.00_a0.8_m200	1	0	0	0.8	0	0	0.8	0	-0	0.9461	0.9117	0.9086	0.04772
	209	GT0584	D12_q1.00_a0.60_m200	1	0	0	0.6	0	0	0.6	0	-0	0.9888	0.9269	0.858	0.04204
	67	GT0389	D6.2_q1_a-0.6_m100	1	0	0	-0.6	0	0	-0.6	0	-0	0.8844	0.9651	0.4876	0.1168
	70	GT0392	D6.2_q1_a-0.8_m100	1	0	0	-0.8	0	0	-0.8	0	-0	0.9012	0.9675	0.4122	0.12
	414	GT0860	D8_q1.0_a0.6_0.6_th90_0_r100_res140_CE-all	1	0.1093	0.4806	0.3421	0.4599	-0.2874	0.2567	0	-0	0.8756	0.9399	0.7827	0.07765
	415	GT0861	D8_q1.0_a0.6_0.6_th90_135_r100_res140_CE-all	1	-0.1588	-0.4137	-0.4045	0.2971	0.521	-0.02011	0	-0	0.9141	0.9567	0.6201	0.08503
	416	GT0862	D8_q1.0_a0.6_0.6_th90_180_r100_res140_CE-all	1	0.5423	-0.2546	-0.0338	0.05283	0.1879	-0.5674	0	-0	0.9185	0.958	0.6071	0.0727
	417	GT0863	D8_q1.0_a0.6_0.6_th90_225_r100_res140_CE-all	1	-0.5122	0.2699	-0.1574	-0.5064	-0.1753	-0.27	0	-0	0.9077	0.9555	0.6629	0.0867
	418	GT0864	D8_q1.0_a0.6_0.6_th90_270_r100_res140_CE-all	1	-0.269	-0.5361	-0.01391	-0.269	-0.5361	-0.01391	-0	0	0.8951	0.9493	0.7364	0.08353
	419	GT0865	D8_q1.0_a0.6_0.6_th90_315_r100_res140_CE-all	1	-0.09605	0.4669	0.3644	0.2076	0.562	0.03321	0	-0	0.8821	0.9432	0.782	0.07936
	420	GT0866	D8_q1.0_a0.6_0.6_th90_45_r100_res140_CE-all	1	-0.3544	0.2327	0.4245	0.4938	-0.3409	-0.0005569	-0	0	0.8836	0.9447	0.7486	0.07808
	421	GT0867	D8_q1.0_a0.6_0.6_th90_90_r100_res140_CE-all	1	0.5993	0.02989	-0	-0.5993	-0.02989	-0	0	-0	0.8988	0.9493	0.6789	0.08106
	128	GT0482	EK_D6.2_a0.6_th030_M77	1	0.5196	0.3	0	-0.5196	-0.3	0	0	0	0.8473	0.9509	0.6842	0.1012
	129	GT0483	EK_D6.2_a0.6_th045_M77	1	0.4243	0.4243	0	-0.4243	-0.4243	0	0	0	0.8473	0.9504	0.6826	0.1012
	130	GT0484	EK_D6.2_a0.6_th060_M77	1	0.3	0.5196	0	-0.3	-0.5196	0	0	0	0.8473	0.9499	0.6811	0.1013
	131	GT0485	EK_D6.2_a0.6_th090_M77	1	3.674E-17	0.6	0	-3.674E-17	-0.6	0	0	0	0.8473	0.9494	0.6798	0.1014
	132	GT0486	EK_D6.2_a0.6_th120_M77	1	-0.3	0.5196	0	0.3	-0.5196	0	0	0	0.8473	0.9498	0.6817	0.1013
	133	GT0487	EK_D6.2_a0.6_th150_M77	1	-0.5196	0.3	0	0.5196	-0.3	0	0	0	0.8473	0.9507	0.6846	0.1013
	134	GT0489	EK_D6.2_a0.6_th210_M77	1	-0.5196	-0.3	0	0.5196	0.3	0	0	0	0.8473	0.9509	0.6842	0.1012
	135	GT0490	EK_D6.2_a0.6_th240_M77	1	-0.3	-0.5196	0	0.3	0.5196	0	0	0	0.8473	0.9499	0.6811	0.1013
	136	GT0491	EK_D6.2_a0.6_th270_M77	1	-1.102E-16	-0.6	0	1.102E-16	0.6	0	0	0	0.8473	0.9494	0.6798	0.1014
	137	GT0492	EK_D6.2_a0.6_th300_M77	1	0.3	-0.5196	0	-0.3	0.5196	0	0	0	0.8473	0.9498	0.6817	0.1013
	138	GT0493	EK_D6.2_a0.6_th330_M77	1	0.5196	-0.3	0	-0.5196	0.3	0	0	0	0.8473	0.9507	0.6846	0.1013
	139	GT0500	LZ_D6.2_a0.6_th034_ph000_M77	1	0.3355	0	0.4974	-0.3355	0	0.4974	0	0	0.7979	NaN	NaN	0.0977
	140	GT0501	LZ_D6.2_a0.6_th034_ph030_M77	1	0.2906	0.1678	0.4974	-0.2906	-0.1678	0.4974	0	0	0.7979	NaN	NaN	0.09769
	141	GT0502	LZ_D6.2_a0.6_th034_ph045_M77	1	0.2372	0.2372	0.4974	-0.2372	-0.2372	0.4974	0	0	0.7979	NaN	NaN	0.0977
	142	GT0503	LZ_D6.2_a0.6_th034_ph060_M77	1	0.1678	0.2906	0.4974	-0.1678	-0.2906	0.4974	0	0	0.7979	NaN	NaN	0.09772
	143	GT0504	LZ_D6.2_a0.6_th034_ph090_M77	1	2.054E-17	0.3355	0.4974	-2.054E-17	-0.3355	0.4974	0	0	0.7979	0.932	0.8268	0.09774
	144	GT0505	LZ_D6.2_a0.6_th034_ph120_M77	1	-0.1678	0.2906	0.4974	0.1678	-0.2906	0.4974	0	0	0.7979	0.9326	0.8281	0.09775
	145	GT0506	LZ_D6.2_a0.6_th034_ph150_M77	1	-0.2906	0.1678	0.4974	0.2906	-0.1678	0.4974	0	0	0.7979	0.9329	0.8286	0.09773
	146	GT0507	LZ_D6.2_a0.6_th034_ph180_M77	1	-0.3355	4.109E-17	0.4974	0.3355	-4.109E-17	0.4974	0	0	0.7979	NaN	NaN	0.0977
	147	GT0508	LZ_D6.2_a0.6_th034_ph210_M77	1	-0.2906	-0.1678	0.4974	0.2906	0.1678	0.4974	0	0	0.7979	NaN	NaN	0.09769
	148	GT0509	LZ_D6.2_a0.6_th034_ph240_M77	1	-0.1678	-0.2906	0.4974	0.1678	0.2906	0.4974	0	0	0.7979	NaN	NaN	0.09772
	149	GT0510	LZ_D6.2_a0.6_th034_ph270_M77	1	-6.163E-17	-0.3355	0.4974	6.163E-17	0.3355	0.4974	0	0	0.7979	0.932	0.8268	0.09774
	150	GT0511	LZ_D6.2_a0.6_th034_ph300_M77	1	0.1678	-0.2906	0.4974	-0.1678	0.2906	0.4974	0	0	0.7979	0.9326	0.8281	0.09775
	151	GT0512	LZ_D6.2_a0.6_th034_ph330_M77	1	0.2906	-0.1678	0.4974	-0.2906	0.1678	0.4974	0	0	0.7979	0.9329	0.8286	0.09773
	158	GT0520	SS_D6.2_a0.6_th015_M103	1	0.1553	0	0.5796	-0.1553	0	0.5796	0	0	0.7899	0.9283	0.8523	0.09727
	159	GT0521	SS_D6.2_a0.6_th030_M103	1	0.3	0	0.5196	-0.3	0	0.5196	0	0	0.7958	NaN	NaN	0.0976
	160	GT0522	SS_D6.2_a0.6_th045_M103	1	0.4243	0	0.4243	-0.4243	0	0.4243	0	0	0.8051	NaN	NaN	0.09812
	161	GT0523	SS_D6.2_a0.6_th060_M103	1	0.5196	0	0.3	-0.5196	0	0.3	0	0	0.8174	NaN	NaN	0.09891
	162	GT0524	SS_D6.2_a0.6_th075_M103	1	0.5796	0	0.1553	-0.5796	0	0.1553	0	0	0.8318	NaN	NaN	0.09995
	163	GT0525	SS_D6.2_a0.6_th090_M103	1	0.6	0	3.674E-17	-0.6	0	3.674E-17	0	0	0.8473	NaN	NaN	0.1012
	164	GT0526	SS_D6.2_a0.6_th105_M103	1	0.5796	0	-0.1553	-0.5796	0	-0.1553	0	0	0.8631	NaN	NaN	0.1026
	165	GT0527	SS_D6.2_a0.6_th120_M103	1	0.5196	0	-0.3	-0.5196	0	-0.3	0	0	0.8778	NaN	NaN	0.104
	166	GT0528	SS_D6.2_a0.6_th135_M103	1	0.4243	0	-0.4243	-0.4243	0	-0.4243	0	0	0.8906	0.9614	0.5504	0.1053
	167	GT0529	SS_D6.2_a0.6_th150_M103	1	0.3	0	-0.5196	-0.3	0	-0.5196	0	0	0.9005	0.9631	0.5182	0.1063
	168	GT0530	SS_D6.2_a0.6_th165_M103	1	0.1553	0	-0.5796	-0.1553	0	-0.5796	0	0	0.9068	0.9641	0.4977	0.107
	169	GT0532	SS_D6.2_a0.6_th195_M103	1	-0.1553	0	-0.5796	0.1553	0	-0.5796	0	0	0.9068	0.9641	0.4977	0.107
	170	GT0533	SS_D6.2_a0.6_th210_M103	1	-0.3	0	-0.5196	0.3	0	-0.5196	0	0	0.9005	0.9631	0.5182	0.1063
	171	GT0534	SS_D6.2_a0.6_th225_M103	1	-0.4243	0	-0.4243	0.4243	0	-0.4243	0	0	0.8906	0.9614	0.5504	0.1053
	172	GT0535	SS_D6.2_a0.6_th240_M103	1	-0.5196	0	-0.3	0.5196	0	-0.3	0	0	0.8778	NaN	NaN	0.104
	173	GT0536	SS_D6.2_a0.6_th255_M103	1	-0.5796	0	-0.1553	0.5796	0	-0.1553	0	0	0.8631	NaN	NaN	0.1026
	174	GT0537	SS_D6.2_a0.6_th270_M103	1	-0.6	0	-1.102E-16	0.6	0	-1.102E-16	0	0	0.8473	NaN	NaN	0.1012
	175	GT0538	SS_D6.2_a0.6_th285_M103	1	-0.5796	0	0.1553	0.5796	0	0.1553	0	0	0.8318	NaN	NaN	0.09995
	176	GT0539	SS_D6.2_a0.6_th300_M103	1	-0.5196	0	0.3	0.5196	0	0.3	0	0	0.8174	NaN	NaN	0.09891
	177	GT0540	SS_D6.2_a0.6_th315_M103	1	-0.4243	0	0.4243	0.4243	0	0.4243	0	0	0.8051	NaN	NaN	0.09812
	178	GT0541	SS_D6.2_a0.6_th330_M103	1	-0.3	0	0.5196	0.3	0	0.5196	0	0	0.7958	NaN	NaN	0.0976
	179	GT0542	SS_D6.2_a0.6_th345_M103	1	-0.1553	0	0.5796	0.1553	0	0.5796	0	0	0.7899	0.9283	0.8523	0.09727
	152	GT0513	SSaP_D6.2_a0.6_th066_ph000_M77	1	0.5481	0	0.244	-0.5481	0	0.244	0	0	0.8229	NaN	NaN	0.0993
	153	GT0514	SSaP_D6.2_a0.6_th066_ph030_M77	1	0.4747	0.2741	0.244	-0.4747	-0.2741	0.244	0	0	0.8229	NaN	NaN	0.09931
	154	GT0515	SSaP_D6.2_a0.6_th066_ph060_M77	1	0.2741	0.4747	0.244	-0.2741	-0.4747	0.244	0	0	0.8229	NaN	NaN	0.09939
	155	GT0516	SSaP_D6.2_a0.6_th066_ph090_M77	1	3.356E-17	0.5481	0.244	-3.356E-17	-0.5481	0.244	0	0	0.8229	NaN	NaN	0.09945
	156	GT0517	SSaP_D6.2_a0.6_th066_ph120_M77	1	-0.2741	0.4747	0.244	0.2741	-0.4747	0.244	0	0	0.8229	NaN	NaN	0.09944
	157	GT0518	SSaP_D6.2_a0.6_th066_ph150_M77	1	-0.4747	0.2741	0.244	0.4747	-0.2741	0.244	0	0	0.8229	NaN	NaN	0.09937
	9	GT0009	fr_b3.1_a0.2_oth.000_M77	1	0	0	0.2	-0.2	0	0	0	0	0.8362	NaN	NaN	0.1004
	10	GT0010	fr_b3.1_a0.2_oth.045_M77	1	0.1414	0	0.1414	-0.2	0	0	0	0	0.8392	NaN	NaN	0.1006
	11	GT0011	fr_b3.1_a0.2_oth.090_M77	1	0.2	0	1.225E-17	-0.2	0	0	0	0	0.8464	NaN	NaN	0.1012
	12	GT0012	fr_b3.1_a0.2_oth.135_M77	1	-0.2	0	0	0.1414	0	-0.1414	0	0	-0.8535	NaN	NaN	0.1018
	13	GT0013	fr_b3.1_a0.2_oth.180_M77	1	-0.2	0	0	2.45E-17	0	-0.2	0	0	-0.8563	NaN	NaN	0.1021
	14	GT0014	fr_b3.1_a0.2_oth.225_M77	1	-0.2	0	0	-0.1414	0	-0.1414	0	0	-0.8533	NaN	NaN	0.1018
	15	GT0015	fr_b3.1_a0.2_oth.270_M77	1	-0.2	0	-3.674E-17	-0.2	0	0	0	0	0.8461	NaN	NaN	0.1011
	16	GT0016	fr_b3.1_a0.2_oth.315_M77	1	-0.1414	0	0.1414	-0.2	0	0	0	0	0.839	NaN	NaN	0.1006
	1	GT0001	fr_b3.1_a0.4_oth.000_M77	1	0	0	0.3999	-0.4001	0	0	0	0	0.8261	NaN	NaN	0.09957
	2	GT0002	fr_b3.1_a0.4_oth.045_M77	1	0.2828	0	0.2828	-0.4001	0	0	0	0	0.8323	NaN	NaN	0.1001
	3	GT0003	fr_b3.1_a0.4_oth.090_M77	1	0.4	0	2.449E-17	-0.4	0	0	0	0	0.8467	NaN	NaN	0.1012
	4	GT0004	fr_b3.1_a0.4_oth.135_M77	1	-0.3999	0	0	0.2829	0	-0.2829	0	0	-0.8608	NaN	NaN	0.1025
	5	GT0005	fr_b3.1_a0.4_oth.180_M77	1	-0.3999	0	0	4.9E-17	0	-0.4001	0	0	-0.8664	NaN	NaN	0.103
	6	GT0006	fr_b3.1_a0.4_oth.225_M77	1	-0.3999	0	0	-0.2829	0	-0.2829	0	0	-0.8602	NaN	NaN	0.1022
	41	GT0282	fr_b3.1_a0.4_oth.247_M77	1	-0.3682	0	-0.1563	-0.4	0	0	0	0	0.8537	NaN	NaN	0.1016
	42	GT0283	fr_b3.1_a0.4_oth.255_M77	1	-0.3864	0	-0.1035	-0.4	0	0	0	0	0.851	NaN	NaN	0.1013
	7	GT0007	fr_b3.1_a0.4_oth.270_M77	1	-0.4	0	-7.348E-17	-0.4	0	0	0	0	0.8458	NaN	NaN	0.1008
	8	GT0008	fr_b3.1_a0.4_oth.315_M77	1	-0.2828	0	0.2828	-0.4001	0	0	0	0	0.8316	NaN	NaN	0.09983
	26	GT0255	fr_b3.1_a0.6_oth.015_rr_M77	1	0.1553	0	0.5796	-0.6	0	0	0	0	0.8173	NaN	NaN	0.09895
	27	GT0256	fr_b3.1_a0.6_oth.030_rr_M77	1	0.3	0	0.5196	-0.6	0	0	0	0	0.8206	NaN	NaN	0.0992
	28	GT0258	fr_b3.1_a0.6_oth.060_rr_M77	1	0.5196	0	0.3	-0.6	0	0	0	0	0.8321	NaN	NaN	0.1
	29	GT0259	fr_b3.1_a0.6_oth.075_rr_M77	1	0.5796	0	0.1553	-0.6	0	0	0	0	0.8395	NaN	NaN	0.1006
	30	GT0261	fr_b3.1_a0.6_oth.105_rr_M77	1	0.5796	0	-0.1553	-0.6	0	0	0	0	0.8551	NaN	NaN	0.1019
	31	GT0262	fr_b3.1_a0.6_oth.120_rr_M77	1	0.5196	0	-0.3	-0.6	0	0	0	0	0.8623	NaN	NaN	0.1025
	32	GT0264	fr_b3.1_a0.6_oth.150_rr_M77	1	0.3	0	-0.5196	-0.6	0	0	0	0	0.873	NaN	NaN	0.1036
	33	GT0265	fr_b3.1_a0.6_oth.165_rr_M77	1	0.1553	0	-0.5796	-0.6	0	0	0	0	0.8757	NaN	NaN	0.1038
	34	GT0267	fr_b3.1_a0.6_oth.195_rr_M77	1	-0.1553	0	-0.5796	-0.6	0	0	0	0	0.8752	NaN	NaN	0.1036
	46	GT0287	fr_b3.1_a0.6_oth.215_rr_M77	1	-0.3441	0	-0.4915	-0.6	0	0	0	0	0.8704	NaN	NaN	0.1029
	45	GT0286	fr_b3.1_a0.6_oth.220_rr_M77	1	-0.3857	0	-0.4596	-0.6	0	0	0	0	0.8687	NaN	NaN	0.1028
	44	GT0285	fr_b3.1_a0.6_oth.230_rr_M77	1	-0.4596	0	-0.3857	-0.6	0	0	0	0	0.8648	NaN	NaN	0.1023
	43	GT0284	fr_b3.1_a0.6_oth.235_rr_M77	1	-0.4915	0	-0.3441	-0.6	0	0	0	0	0.8627	NaN	NaN	0.1021
	35	GT0272	fr_b3.1_a0.6_oth.250_rr_M77	1	-0.5638	0	-0.2052	-0.6	0	0	0	0	0.8556	NaN	NaN	0.1013
	36	GT0275	fr_b3.1_a0.6_oth.265_rr_M77	1	-0.5977	0	-0.05229	-0.6	0	0	0	0	0.8478	NaN	NaN	0.1007
	37	GT0277	fr_b3.1_a0.6_oth.285_rr_M77	1	-0.5796	0	0.1553	-0.6	0	0	0	0	0.8374	NaN	NaN	0.09986
	38	GT0278	fr_b3.1_a0.6_oth.300_rr_M77	1	-0.5196	0	0.3	-0.6	0	0	0	0	0.8302	NaN	NaN	0.09938
	39	GT0280	fr_b3.1_a0.6_oth.330_rr_M77	1	-0.3	0	0.5196	-0.6	0	0	0	0	0.8195	NaN	NaN	0.09885
	40	GT0281	fr_b3.1_a0.6_oth.345_rr_M77	1	-0.1553	0	0.5796	-0.6	0	0	0	0	0.8167	NaN	NaN	0.09878
	63	GT0382	fr_b3.1_a0.8_oth.000_M77	1	0	0	0.8	-0.8	0	0	0	0	0.8059	NaN	NaN	0.09818
	64	GT0384	fr_b3.1_a0.8_oth.030_M77	1	0.4	0	0.6928	-0.8	0	0	0	0	0.8123	NaN	NaN	0.09876
	65	GT0385	fr_b3.1_a0.8_oth.060_M77	1	0.6928	0	0.4	-0.8	0	0	0	0	0.8278	NaN	NaN	0.09985
	66	GT0388	fr_b3.1_a0.8_oth.090_M77	1	0.8	0	4.899E-17	-0.8	0	0	0	0	0.8482	NaN	NaN	0.1014
	68	GT0390	fr_b3.1_a0.8_oth.120_M77	1	0.6928	0	-0.4	-0.8	0	0	0	0	0.8681	NaN	NaN	0.1032
	69	GT0391	fr_b3.1_a0.8_oth.150_M77	1	0.4	0	-0.6928	-0.8	0	0	0	0	0.8821	NaN	NaN	0.1046
	71	GT0393	fr_b3.1_a0.8_oth.180_M77	1	9.797E-17	0	-0.8	-0.8	0	0	0	0	0.8866	NaN	NaN	0.1048
	72	GT0395	fr_b3.1_a0.8_oth.210_M77	1	-0.4	0	-0.6928	-0.8	0	0	0	0	0.8802	NaN	NaN	0.1037
	73	GT0396	fr_b3.1_a0.8_oth.240_M77	1	-0.6928	0	-0.4	-0.8	0	0	0	0	0.8647	NaN	NaN	0.1019
	74	GT0397	fr_b3.1_a0.8_oth.255_M77	1	-0.7727	0	-0.2071	-0.8	0	0	0	0	0.8548	NaN	NaN	0.1009
	75	GT0398	fr_b3.1_a0.8_oth.270_M77	1	-0.8	0	-1.47E-16	-0.8	0	0	0	0	0.8443	NaN	NaN	0.1
	76	GT0400	fr_b3.1_a0.8_oth.300_M77	1	-0.6928	0	0.4	-0.8	0	0	0	0	0.8244	NaN	NaN	0.09873
	77	GT0401	fr_b3.1_a0.8_oth.330_M77	1	-0.4	0	0.6928	-0.8	0	0	0	0	0.8103	NaN	NaN	0.09811
	47	GT0304	fr_b5_a0.6_oth.000_M90	1	0	0	0.6	-0.6	0	0	0	0	0.9408	NaN	NaN	0.05464
	17	GT0203	fr_b5_a0.6_oth.045_M77	1	0.4243	0	0.4243	-0.6	0	0	0	0	0.9467	NaN	NaN	0.05488
	18	GT0204	fr_b5_a0.6_oth.090_M77	1	0.6	0	3.674E-17	-0.6	0	0	0	0	0.9601	NaN	NaN	0.05553
	19	GT0205	fr_b5_a0.6_oth.135_M77	1	0.4243	0	-0.4243	-0.6	0	0	0	0	0.9732	NaN	NaN	0.05651
	20	GT0206	fr_b5_a0.6_oth.180_M77	1	7.348E-17	0	-0.6	-0.6	0	0	0	0	0.9783	NaN	NaN	0.05682
	48	GT0308	fr_b5_a0.6_oth.210_M77	1	-0.3	0	-0.5196	-0.6	0	0	0	0	0.9755	NaN	NaN	0.05648
	21	GT0207	fr_b5_a0.6_oth.225_M77	1	-0.4243	0	-0.4243	-0.6	0	0	0	0	0.9724	NaN	NaN	0.05618
	24	GT0227	fr_b5_a0.6_oth.240_M77	1	-0.5196	0	-0.3	-0.6	0	0	0	0	0.9685	NaN	NaN	0.05581
	49	GT0311	fr_b5_a0.6_oth.245_M77	1	-0.5438	0	-0.2536	-0.6	0	0	0	0	0.967	NaN	NaN	0.05567
	25	GT0229	fr_b5_a0.6_oth.255_M77	1	-0.5796	0	-0.1553	-0.6	0	0	0	0	0.9639	NaN	NaN	0.05544
	50	GT0314	fr_b5_a0.6_oth.260_M77	1	-0.5909	0	-0.1042	-0.6	0	0	0	0	0.9623	NaN	NaN	0.05532
	22	GT0208	fr_b5_a0.6_oth.270_M77	1	-0.6	0	-1.102E-16	-0.6	0	0	0	0	0.959	NaN	NaN	0.0551
	23	GT0209	fr_b5_a0.6_oth.315_M77	1	-0.4243	0	0.4243	-0.6	0	0	0	0	0.9459	NaN	NaN	0.05457
	439	GT0885	q1_LL_D9_a0.6_th1_45_th2_225_m140	1	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.9294	NaN	NaN	0.06485
	380	GT0803	ro_b5_a0.6_oth.000_M103	1	0.4243	0	0.4243	0	0	0.6	0	0	0.9285	NaN	NaN	0.05403
	381	GT0805	ro_b5_a0.6_oth.120_M103	1	-0.2121	0.3674	0.4243	0	0	0.6	0	0	0.9285	NaN	NaN	0.0542
	382	GT0807	ro_b5_a0.6_oth.240_M103	1	-0.2121	-0.3674	0.4243	0	0	0.6	0	0	0.9285	NaN	NaN	0.05417
	317	GT0717	D9_q1.1_a0.0_m160	1.1	0	0	0	0	0	0	0	-0	0.9283	0.9518	0.6851	0.06643
	55	GT0370	D10_a0.0_q1.15_m103_Qs	1.15	0	0	0	0	0	0	0	-0	0.9565	0.952	0.6836	0.05663
	451	GT0898	GW15_D12_q1.2_a0.0_m140	1.2	0	0	0	0	0	0	0	-0	1.012	0.9523	0.6817	0.04343
	452	GT0900	q1.2_a0.4x_0.4x_GW15_D11_m140	1.2	0.4	0	0	0.4	0	0	0	0	0.9812	0.9506	0.7034	0.04838
	78	GT0405	D10_q1.25_a0.2_m100	1.25	0	0	0.2	0	0	0.2	0	-0	0.9372	0.9463	0.7403	0.05586
	79	GT0406	D10_q1.25_a0.4_m100	1.25	0	0	0.4	0	0	0.4	0	-0	0.9251	0.9385	0.7999	0.05514
	80	GT0407	D10_q1.25_a0.6_m100	1.25	0	0	0.6	0	0	0.6	0	-0	0.9133	0.9281	0.8555	0.05451
	334	GT0738	D11_q1.25_a0.0_m200	1.25	0	0	0	0	0	0	0	-0	0.979	0.9527	0.6794	0.04913
	430	GT0876	q1.25_LL_D9_a0.6_th1_45_th2_225_m140	1.25	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.9142	NaN	NaN	0.06444
	56	GT0371	D10_a0.0_q1.30_m103_Qs	1.3	0	0	0	0	0	0	0	-0	0.945	0.9531	0.6768	0.0566
	329	GT0730	D9_q1.4_a0.0_m160	1.4	0	0	0	0	0	0	0	-0	0.9052	0.9541	0.6709	0.06631
	57	GT0372	D10_a0.0_q1.45_m103_Qs	1.45	0	0	0	0	0	0	0	-0	0.9292	0.9546	0.6675	0.05655
	81	GT0411	D10_q1.50_a0.6_m100	1.5	0	0	0.6	0	0	0.6	0	-0	0.8879	0.9313	0.851	0.05456
	396	GT0839	D11_q1.50_a-0.2_m200	1.5	0	0	-0.2	0	0	-0.2	0	-0	0.9633	0.9598	0.5965	0.0497
	393	GT0836	D11_q1.50_a-0.4_m200	1.5	0	0	-0.4	0	0	-0.4	0	-0	0.9748	0.9636	0.5272	0.05036
	392	GT0835	D11_q1.50_a-0.6_m200	1.5	0	0	-0.6	0	0	-0.6	0	-0	0.9864	0.9665	0.4564	0.05107
	124	GT0477	D11_q1.50_a0.0_m200	1.5	0	0	0	0	0	0	0	-0	0.9519	0.9552	0.6639	0.0491
	126	GT0479	D11_q1.50_a0.2_m200	1.5	0	0	0.2	0	0	0.2	0	-0	0.9408	0.9494	0.729	0.04855
	193	GT0558	D11_q1.50_a0.4_m200	1.5	0	0	0.4	0	0	0.4	0	-0	0.9298	0.9419	0.7917	0.04807
	198	GT0567	D6.2_q1.50_S0_m120	1.5	0.2778	0	0	-0.625	0	0	0	0	0.8139	0.9551	0.6644	0.1014
	180	GT0543	D9_q1.50_a0.1_m200	1.5	0	0	0.1	0	0	0.1	0	-0	0.8876	0.9525	0.6968	0.06551
	105	GT0456	RO3_D10_q1.50_a0.40_oth.060_M120	1.5	0.3464	0	0.2	0	0	0.4	0	0	0.9053	0.9465	0.7535	0.05458
	104	GT0455	RO3_D10_q1.50_a0.60_oth.045_M120	1.5	0.4243	0	0.4243	0	0	0.6	0	0	0.8928	0.937	0.8223	0.05413
	106	GT0457	RO3_D10_q1.50_a0.60_oth.060_M120	1.5	0.5196	0	0.3	0	0	0.6	0	0	0.8973	NaN	NaN	0.05428
	356	GT0764	RO3_D10_q1.50_a0.60_oth.090_M120	1.5	0.6	0	3.674E-17	0	0	0.6	0	0	0.908	NaN	NaN	0.05476
	404	GT0847	SO_D9_q1.5_th2_135_ph1_0_m120-all	1.5	0.6	0	0	-0.4243	0	0.4243	0	0	0.8826	NaN	NaN	0.06353
	412	GT0858	SO_D9_q1.5_th2_135_ph1_135_m140-all	1.5	-0.4243	0.4243	0	-0.4243	0	0.4243	0	0	0.8819	0.9482	0.7343	0.06341
	425	GT0871	SO_D9_q1.5_th2_135_ph1_180_m140	1.5	-0.6	0	0	-0.4243	0	0.4243	0	0	0.8818	0.948	0.7405	0.063
	405	GT0848	SO_D9_q1.5_th2_135_ph1_45_m140-all	1.5	0.4243	0.4243	0	-0.4243	0	0.4243	0	0	0.8824	NaN	NaN	0.06359
	424	GT0870	SO_D9_q1.5_th2_135_ph1_90_m140	1.5	0	0.6	0	-0.4243	0	0.4243	0	0	0.8822	0.9487	0.7238	0.06373
	427	GT0873	q1.5_LL_D9_a0.6_th1_45_th2_225_m140	1.5	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.8859	NaN	NaN	0.0641
	58	GT0374	D10_a0.0_q1.60_m103_Qs	1.6	0	0	0	0	0	0	0	-0	0.9107	0.9563	0.6565	0.0565
	230	GT0615	D9_q1.7_a0.0_m160	1.7	0	0	0	0	0	0	0	-0	0.8693	0.9576	0.6486	0.06608
	82	GT0413	D10_q1.75_a0.2_m100	1.75	0	0	0.2	0	0	0.2	0	-0	0.8791	0.9523	0.7139	0.05566
	83	GT0414	D10_q1.75_a0.4_m100	1.75	0	0	0.4	0	0	0.4	0	-0	0.8676	0.945	0.781	0.05485
	84	GT0415	D10_q1.75_a0.6_m100	1.75	0	0	0.6	0	0	0.6	0	-0	0.8563	0.9352	0.8449	0.05429
	326	GT0727	D11_q1.75_a0.0_m200	1.75	0	0	0	0	0	0	0	-0	0.9183	0.9582	0.6444	0.04905
	431	GT0877	q1.75_LL_D9_a0.6_th1_45_th2_225_m140	1.75	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.8523	NaN	NaN	0.0637
	232	GT0618	D9_q1.8_a0.0_m160	1.8	0	0	0	0	0	0	0	-0	0.8562	0.9588	0.6403	0.06601
	59	GT0376	D10_a0.0_q1.90_m103_Qs	1.9	0	0	0	0	0	0	0	-0	0.87	0.96	0.6318	0.05639
	401	GT0844	D11_q2.00_a-0.2_m200	2	0	0	-0.2	0	0	-0.2	0	-0	0.893	0.9653	0.5466	0.04961
	395	GT0838	D11_q2.00_a-0.4_m200	2	0	0	-0.4	0	0	-0.4	0	-0	0.9038	0.9686	0.4681	0.05032
	394	GT0837	D11_q2.00_a-0.6_m200	2	0	0	-0.6	0	0	-0.6	0	-0	0.9148	0.9712	0.3876	0.05099
	100	GT0446	D11_q2.00_a0.0_m200	2	0	0	0	0	0	0	0	-0	0.8822	0.9612	0.6233	0.049
	120	GT0471	D11_q2.00_a0.2_m160	2	0	0	0.2	0	0	0.2	0	-0	0.8717	0.956	0.6978	0.04843
	121	GT0472	D11_q2.00_a0.4_m160	2	0	0	0.4	0	0	0.4	0	-0	0.8614	0.9491	0.7697	0.0479
	211	GT0588	D11_q2.00_a0.60_m200	2	0	0	0.6	0	0	0.6	0	-0	0.8512	0.9396	0.8387	0.04782
	207	GT0582	D12_q2.00_a0.15_-0.60_m200	2	0	0	0.15	0	0	-0.6	0	-0	0.9117	0.9616	0.6353	0.04326
	210	GT0587	D12_q2.00_a0.30_0.00_m200	2	0	0	0.3	0	0	0	0	-0	0.8978	0.9549	0.7175	0.0427
	345	GT0749	D8_q2.00_a0.7_0.0_th045_ph000_m120	2	0.4116	0.281	0.4915	0	0	0	-0	0	0.7774	0.9477	0.8026	0.07652
	346	GT0750	D8_q2.00_a0.7_0.0_th045_ph120_m120	2	-0.4461	-0.2204	0.4923	0	0	0	0	-0	0.7772	0.9475	0.8024	0.07652
	348	GT0756	D8_q2.00_a0.7_0.0_th045_ph240_m120	2	0.4087	-0.2852	0.4915	0	0	0	0	-0	0.7774	0.9469	0.8001	0.0762
	347	GT0754	D8_q2.00_a0.7_0.0_th060_ph045_m120	2	0.194	0.5803	0.34	0	0	0	-0	0	0.7847	0.9507	0.7722	0.07782
	354	GT0762	D8_q2.00_a0.7_0.0_th060_ph135_m120	2	-0.5905	0.1445	0.3471	0	0	0	0	-0	0.7824	0.9509	0.7724	0.07776
	355	GT0763	D8_q2.00_a0.7_0.0_th060_ph225_m120	2	-0.194	-0.5803	0.34	0	0	0	0	-0	0.7847	0.9507	0.7721	0.07782
	359	GT0767	D8_q2.00_a0.7_0.0_th060_ph315_m120	2	0.5905	-0.1445	0.3471	0	0	0	0	-0	0.7824	0.9509	0.7724	0.07776
	358	GT0766	D8_q2.00_a0.7_0.0_th090_ph000_m120	2	0.5102	-0.4792	-0.008191	0	0	0	-0	0	0.7992	0.9579	0.6849	0.08107
	369	GT0787	D8_q2.00_a0.7_0.0_th090_ph060_m120	2	-0.3225	0.6212	-0.011	0	0	0	0	-0	0.8003	0.9576	0.6832	0.08107
	368	GT0781	D8_q2.00_a0.7_0.0_th090_ph120_m120	2	-0.6989	-0.03924	-0.003803	0	0	0	0	-0	0.7975	0.9591	0.6893	0.08167
	371	GT0792	D8_q2.00_a0.7_0.0_th090_ph180_m120	2	-0.5102	0.4792	-0.008191	0	0	0	0	-0	0.7992	0.9579	0.685	0.08107
	374	GT0796	D8_q2.00_a0.7_0.0_th090_ph240_m120	2	0.3225	-0.6212	-0.011	0	0	0	0	-0	0.8003	0.9576	0.6832	0.08107
	372	GT0794	D8_q2.00_a0.7_0.0_th090_ph300_m120	2	0.6989	0.03924	-0.003803	0	0	0	-0	0	0.7975	0.9591	0.6893	0.08167
	377	GT0799	D8_q2.00_a0.7_0.0_th120_ph045_m120	2	0.5996	0.07481	-0.3533	0	0	0	0	-0	0.8157	0.9647	0.5674	0.08579
	378	GT0800	D8_q2.00_a0.7_0.0_th120_ph135_m120	2	-0.0252	0.6006	-0.3586	0	0	0	-0	0	0.8183	0.9649	0.5684	0.08544
	383	GT0808	D8_q2.00_a0.7_0.0_th120_ph225_m120	2	-0.5996	-0.07481	-0.3533	0	0	0	-0	0	0.8157	0.9647	0.5674	0.08579
	379	GT0801	D8_q2.00_a0.7_0.0_th120_ph315_m120	2	0.0252	-0.6006	-0.3586	0	0	0	0	-0	0.8183	0.9649	0.5684	0.08544
	385	GT0810	D8_q2.00_a0.7_0.0_th135_ph000_m120	2	-0.4901	-0.04985	-0.4973	0	0	0	-0	0	0.8241	0.9672	0.5069	0.07531
	384	GT0809	D8_q2.00_a0.7_0.0_th135_ph120_m120	2	0.4922	-0.0231	-0.4972	0	0	0	-0	0	0.8241	0.9672	0.5066	0.07529
	388	GT0825	D8_q2.00_a0.7_0.0_th135_ph240_m120	2	-0.2067	0.4443	-0.4999	0	0	0	0	-0	0.8255	0.9669	0.5041	0.0753
	444	GT0890	D9_q2.0_a0.6_0.6_th0_r100_res140_CE	2	0	0	0.6	0	0	-0.6	0	-0	0.814	0.9511	0.7753	0.06432
	445	GT0891	D9_q2.0_a0.6_0.6_th135_r100_res140_CE	2	0.3282	-0.2703	-0.4234	-0.425	-0.07586	0.4168	0	-0	0.8402	0.9651	0.5362	0.06741
	446	GT0893	D9_q2.0_a0.6_0.6_th225_r100_res140_CE	2	-0.3282	0.2703	-0.4234	0.425	0.07586	0.4168	0	-0	0.8402	0.9651	0.5362	0.06741
	447	GT0894	D9_q2.0_a0.6_0.6_th270_r100_res140_CE	2	-0.5001	0.3241	-0.06956	0.1881	0.5565	0.1222	0	-0	0.8277	0.9593	0.6662	0.06644
	448	GT0895	D9_q2.0_a0.6_0.6_th315_r100_res140_CE	2	0.5115	-0.00915	0.3136	0.5653	0.02502	-0.1997	0	-0	0.814	0.9531	0.7541	0.0657
	449	GT0896	D9_q2.0_a0.6_0.6_th45_r100_res140_CE	2	-0.5115	0.00915	0.3136	-0.5653	-0.02502	-0.1997	0	-0	0.814	0.9531	0.7541	0.0657
	450	GT0897	D9_q2.0_a0.6_0.6_th90_r100_res140_CE	2	0.5001	-0.3241	-0.06956	-0.1881	-0.5565	0.1222	0	-0	0.8277	0.9593	0.6662	0.06644
	107	GT0458	RO3_D10_q2.00_a0.40_oth.060_M120	2	0.3464	0	0.2	0	0	0.4	0	0	0.8397	0.9539	0.7223	0.05459
	109	GT0460	RO3_D10_q2.00_a0.60_oth.045_M120	2	0.4243	0	0.4243	0	0	0.6	0	0	0.8277	0.9451	0.8045	0.05393
	110	GT0461	RO3_D10_q2.00_a0.60_oth.060_M120	2	0.5196	0	0.3	0	0	0.6	0	0	0.8325	0.9481	0.7766	0.05412
	112	GT0463	RO3_D10_q2.00_a0.60_oth.090_M120	2	0.6	0	3.674E-17	0	0	0.6	0	0	0.8439	0.9547	0.6978	0.05476
	187	GT0550	RO3_D10_q2.00_a0.60_oth.135_M120	2	0.4243	0	-0.4243	0	0	0.6	0	0	0.8601	0.9639	0.5495	0.0562
	182	GT0545	RO3_D10_q2.00_a0.60_oth.180_M120	2	7.348E-17	0	-0.6	0	0	0.6	0	0	0.8668	0.9671	0.4646	0.0571
	191	GT0556	RO3_D10_q2.00_a0.60_oth.270_M120	2	-0.6	0	-1.102E-16	0	0	0.6	0	0	0.8439	0.9547	0.6978	0.05476
	88	GT0428	Sq2_d6.2_a0.6_oth.000_rr_M140	2	0	0	0.6	-0.6	0	0	0	0	0.7167	0.9458	0.8095	0.09791
	90	GT0431	Sq2_d6.2_a0.6_oth.030_rr_M140	2	0.3	0	0.5196	-0.6	0	0	0	0	0.7223	NaN	NaN	0.09845
	93	GT0434	Sq2_d6.2_a0.6_oth.090_rr_M140	2	0.6	0	3.674E-17	-0.6	0	0	0	0	0.7558	0.9596	0.6523	0.1015
	95	GT0437	Sq2_d6.2_a0.6_oth.150_rr_M140	2	0.3	0	-0.5196	-0.6	0	0	0	0	0.7885	0.9688	0.4574	0.1056
	97	GT0440	Sq2_d6.2_a0.6_oth.180_rr_M140	2	7.348E-17	0	-0.6	-0.6	0	0	0	0	0.7932	0.9697	0.4287	0.1062
	99	GT0443	Sq2_d6.2_a0.6_oth.270_rr_M140	2	-0.6	0	-1.102E-16	-0.6	0	0	0	0	0.7541	NaN	NaN	0.1008
	426	GT0872	q2_LL_D9_a0.6_th1_45_th2_225_m140	2	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.817	NaN	NaN	0.06335
	413	GT0859	q2_Seff_D9_a_-0.15_0.6_th1_225_m140	2	-0.1061	0	-0.1061	0.4243	0	0.4243	0	0	0.8242	0.9609	0.615	0.06447
	410	GT0856	q2_Seff_D9_a_0.15_-0.6_th1_45_m140	2	0.1061	0	0.1061	-0.4243	0	-0.4243	0	0	0.8312	0.9615	0.6322	0.06466
	406	GT0851	th_0_SSeq_D9_q2_m140	2	0.6	0	0	0.6	0	0	0	0	0.8273	0.9583	0.6907	0.06355
	423	GT0869	th_135_SSeq_D9_q2_m140	2	-0.4243	0	0.4243	-0.4243	0	0.4243	0	0	0.8012	0.9459	0.803	0.06211
	407	GT0852	th_180_SSeq_D9_q2_m140	2	-0.6	0	0	-0.6	0	0	0	0	0.8273	0.9583	0.6907	0.06355
	408	GT0853	th_225_SSeq_D9_q2_m140	2	-0.4243	0	-0.4243	-0.4243	0	-0.4243	0	0	0.8548	0.968	0.5091	0.06699
	409	GT0854	th_315_SSeq_D9_q2_m140	2	0.4243	0	-0.4243	0.4243	0	-0.4243	0	0	0.8548	0.968	0.5091	0.06699
	422	GT0868	th_45_SSeq_D9_q2_m140	2	0.4243	0	0.4243	0.4243	0	0.4243	0	0	0.8012	0.9459	0.803	0.06211
	60	GT0378	D10_a0.0_q2.05_m103_Qs	2.05	0	0	0	0	0	0	0	-0	0.849	0.9618	0.619	0.05632
	231	GT0617	D9_q2.1_a0.0_m160	2.1	0	0	0	0	0	0	0	-0	0.8158	0.9624	0.6147	0.06578
	61	GT0379	D10_a0.0_q2.20_m103_Qs	2.2	0	0	0	0	0	0	0	-0	0.828	0.9635	0.606	0.05626
	349	GT0757	D11_q2.25_a0.0_m200	2.25	0	0	0	0	0	0	0	-0	0.8461	0.9641	0.6018	0.04894
	432	GT0878	q2.25_LL_D9_a0.6_th1_45_th2_225_m140	2.25	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.7821	0.9556	0.7477	0.06328
	233	GT0621	D9_q2.3_a0.0_m160	2.3	0	0	0	0	0	0	0	-0	0.7891	0.9646	0.5975	0.06561
	62	GT0380	D10_a0.0_q2.35_m103_Qs	2.35	0	0	0	0	0	0	0	-0	0.8073	0.9651	0.5932	0.0562
	234	GT0622	D9_q2.4_a0.0_m160	2.4	0	0	0	0	0	0	0	-0	0.776	0.9657	0.589	0.06554
	400	GT0843	D11_q2.50_a-0.2_m200	2.5	0	0	-0.2	0	0	-0.2	0	-0	0.821	0.9703	0.4943	0.04952
	397	GT0840	D11_q2.50_a-0.4_m200	2.5	0	0	-0.4	0	0	-0.4	0	-0	0.8312	0.9731	0.4062	0.05024
	402	GT0845	D11_q2.50_a-0.6_m200	2.5	0	0	-0.6	0	0	-0.6	0	-0	0.8415	0.9754	0.3162	0.05119
	196	GT0565	D11_q2.50_a0.0_m200	2.5	0	0	0	0	0	0	0	-0	0.811	0.9667	0.5806	0.0489
	122	GT0475	D11_q2.50_a0.2_0.00_m160	2.5	0	0	0.2	0	0	0	0	-0	0.8036	0.9629	0.6567	0.04846
	127	GT0480	D11_q2.50_a0.4_0.00_m160	2.5	0	0	0.4	0	0	0	0	-0	0.7962	0.9581	0.7312	0.04802
	237	GT0629	Eq_D7_q2.50_a0.6_ph000_m140	2.5	0.6	0	0	-0.6	-0	0	0	0	0.7137	0.9647	0.6317	0.08572
	239	GT0632	Eq_D7_q2.50_a0.6_ph030_m140	2.5	0.5196	0.3	0	-0.5196	-0.3	0	0	0	0.7137	0.9646	0.6321	0.08559
	240	GT0633	Eq_D7_q2.50_a0.6_ph060_m140	2.5	0.3	0.5196	0	-0.3	-0.5196	0	0	0	0.7137	0.9655	0.636	0.08556
	241	GT0634	Eq_D7_q2.50_a0.6_ph090_m140	2.5	0	0.6	0	-0	-0.6	0	0	0	0.7137	0.966	0.6373	0.08567
	248	GT0641	Eq_D7_q2.50_a0.6_ph120_m140	2.5	-0.3	0.5196	0	0.3	-0.5196	0	0	0	0.7137	0.9657	0.6358	0.08579
	250	GT0643	Eq_D7_q2.50_a0.6_ph150_m140	2.5	-0.5196	0.3	0	0.5196	-0.3	0	0	0	0.7137	0.9653	0.6338	0.08581
	247	GT0640	Eq_D7_q2.50_a0.6_ph180_m140	2.5	-0.6	0	0	0.6	-0	0	0	0	0.7137	0.9647	0.6317	0.08572
	246	GT0639	Eq_D7_q2.50_a0.6_ph210_m140	2.5	-0.5196	-0.3	0	0.5196	0.3	0	0	0	0.7137	0.9646	0.6322	0.08559
	245	GT0638	Eq_D7_q2.50_a0.6_ph240_m140	2.5	-0.3	-0.5196	0	0.3	0.5196	0	0	0	0.7137	0.9655	0.636	0.08556
	244	GT0637	Eq_D7_q2.50_a0.6_ph270_m140	2.5	-0	-0.6	0	0	0.6	0	0	0	0.7137	0.966	0.6373	0.08567
	270	GT0665	Eq_D7_q2.50_a0.6_ph300_m140	2.5	0.3	-0.5196	0	-0.3	0.5196	0	0	0	0.7137	0.9657	0.6358	0.08579
	269	GT0664	Eq_D7_q2.50_a0.6_ph330_m140	2.5	0.5196	-0.3	0	-0.5196	0.3	0	0	0	0.7137	0.9653	0.6338	0.08581
	280	GT0676	Lq_D6.2_q2.50_a0.6_th000_m140	2.5	0	0	0.6	0	0	0.6	0	0	0.6453	0.9472	0.8255	0.1024
	281	GT0677	Lq_D6.2_q2.50_a0.6_th015_m140	2.5	0.1553	0	0.5796	-0.1553	-0	0.5796	0	0	0.6469	0.9481	0.8203	0.1025
	283	GT0679	Lq_D6.2_q2.50_a0.6_th030_m140	2.5	0.3	0	0.5196	-0.3	-0	0.5196	0	0	0.6519	0.9506	0.8038	0.1029
	274	GT0669	Lq_D6.2_q2.50_a0.6_th045_m140	2.5	0.4243	0	0.4243	-0.4243	-0	0.4243	0	0	0.6599	0.9535	0.7753	0.104
	276	GT0671	Lq_D6.2_q2.50_a0.6_th060_m140	2.5	0.5196	0	0.3	-0.5196	-0	0.3	0	0	0.6703	0.9574	0.737	0.09913
	277	GT0672	Lq_D6.2_q2.50_a0.6_th075_m140	2.5	0.5796	0	0.1553	-0.5796	-0	0.1553	0	0	0.6825	0.9622	0.6929	0.1003
	279	GT0674	Lq_D6.2_q2.50_a0.6_th120_m140	2.5	0.5196	0	-0.3	-0.5196	-0	-0.3	0	0	0.7215	0.9709	0.4991	0.1048
	266	GT0661	Lq_D6.2_q2.50_a0.6_th150_m140	2.5	0.3	0	-0.5196	-0.3	-0	-0.5196	0	0	0.7406	0.9749	0.3726	0.1075
	275	GT0670	Lq_D6.2_q2.50_a0.6_th210_m140	2.5	-0.3	-0	-0.5196	0.3	0	-0.5196	0	0	0.7406	0.9749	0.3726	0.1075
	278	GT0673	Lq_D6.2_q2.50_a0.6_th240_m140	2.5	-0.5196	-0	-0.3	0.5196	0	-0.3	0	0	0.7215	0.9709	0.4991	0.1048
	282	GT0678	Lq_D6.2_q2.50_a0.6_th300_m140	2.5	-0.5196	-0	0.3	0.5196	0	0.3	0	0	0.6703	0.9574	0.737	0.09913
	285	GT0681	Lq_D6.2_q2.50_a0.6_th315_m140	2.5	-0.4243	-0	0.4243	0.4243	0	0.4243	0	0	0.6599	0.9535	0.7753	0.0982
	286	GT0682	Lq_D6.2_q2.50_a0.6_th330_m140	2.5	-0.3	-0	0.5196	0.3	0	0.5196	0	0	0.6519	0.9506	0.8038	0.1029
	287	GT0683	Lq_D6.2_q2.50_a0.6_th345_m140	2.5	-0.1553	-0	0.5796	0.1553	0	0.5796	0	0	0.6469	0.9481	0.8204	0.1025
	108	GT0459	RO3_D10_q2.50_a0.40_oth.045_M120	2.5	0.2828	0	0.2828	0	0	0.4	0	0	0.7689	0.9583	0.7151	0.05438
	111	GT0462	RO3_D10_q2.50_a0.40_oth.060_M120	2.5	0.3464	0	0.2	0	0	0.4	0	0	0.7722	0.9604	0.6928	0.05455
	113	GT0464	RO3_D10_q2.50_a0.40_oth.090_M120	2.5	0.4	0	2.449E-17	0	0	0.4	0	0	0.7799	0.964	0.6259	0.05509
	114	GT0465	RO3_D10_q2.50_a0.60_oth.045_M120	2.5	0.4243	0	0.4243	0	0	0.6	0	0	0.761	0.9522	0.7875	0.05403
	115	GT0466	RO3_D10_q2.50_a0.60_oth.060_M120	2.5	0.5196	0	0.3	0	0	0.6	0	0	0.7657	0.955	0.7563	0.05423
	116	GT0467	RO3_D10_q2.50_a0.60_oth.090_M120	2.5	0.6	0	3.674E-17	0	0	0.6	0	0	0.7772	0.9624	0.6707	0.05494
	254	GT0647	Sq_D6.2_q2.50_a0.6_th000_m140	2.5	0	0	0.6	-0.6	0	0	0	0	0.6567	0.9516	0.8047	0.1047
	255	GT0648	Sq_D6.2_q2.50_a0.6_th015_m140	2.5	0.1553	0	0.5796	-0.6	0	0	0	0	0.6582	0.9521	0.797	0.1052
	256	GT0649	Sq_D6.2_q2.50_a0.6_th030_m140	2.5	0.3	0	0.5196	-0.6	0	0	0	0	0.6622	0.9545	0.7833	0.09828
	253	GT0646	Sq_D6.2_q2.50_a0.6_th045_m140	2.5	0.4243	0	0.4243	-0.6	0	0	0	0	0.6684	0.9566	0.7585	0.09884
	252	GT0645	Sq_D6.2_q2.50_a0.6_th060_m140	2.5	0.5196	0	0.3	-0.6	0	0	0	0	0.6765	0.9584	0.7241	0.09962
	251	GT0644	Sq_D6.2_q2.50_a0.6_th075_m140	2.5	0.5796	0	0.1553	-0.6	0	0	0	0	0.6858	0.9629	0.6865	0.1006
	249	GT0642	Sq_D6.2_q2.50_a0.6_th120_m140	2.5	0.5196	0	-0.3	-0.6	0	0	0	0	0.7148	0.9704	0.5102	0.1041
	264	GT0659	Sq_D6.2_q2.50_a0.6_th150_m140	2.5	0.3	0	-0.5196	-0.6	0	0	0	0	0.7285	0.9737	0.3894	0.1062
	265	GT0660	Sq_D6.2_q2.50_a0.6_th180_m140	2.5	0	0	-0.6	-0.6	0	0	0	0	0.7333	0.9745	0.3439	0.107
	272	GT0667	Sq_D6.2_q2.50_a0.6_th210_m140	2.5	-0.3	-0	-0.5196	-0.6	0	0	0	0	0.7278	0.9731	0.421	0.1058
	268	GT0663	Sq_D6.2_q2.50_a0.6_th240_m140	2.5	-0.5196	-0	-0.3	-0.6	0	0	0	0	0.7135	0.9696	0.5521	0.1035
	267	GT0662	Sq_D6.2_q2.50_a0.6_th270_m140	2.5	-0.6	-0	-0	-0.6	0	0	0	0	0.6943	0.9638	0.6684	0.101
	236	GT0628	Sq_D6.2_q2.50_a0.6_th300_m140	2.5	-0.5196	-0	0.3	-0.6	0	0	0	0	0.6752	0.9576	0.7499	0.09906
	238	GT0630	Sq_D6.2_q2.50_a0.6_th315_m140	2.5	-0.4243	-0	0.4243	-0.6	0	0	0	0	0.6674	0.9548	0.7762	0.09841
	271	GT0666	Sq_D6.2_q2.50_a0.6_th330_m140	2.5	-0.3	-0	0.5196	-0.6	0	0	0	0	0.6615	0.9531	0.7947	0.09798
	273	GT0668	Sq_D6.2_q2.50_a0.6_th345_m140	2.5	-0.1553	-0	0.5796	-0.6	0	0	0	0	0.6578	0.9521	0.8043	0.1041
	313	GT0712	Sq_D9.0_q2.50_a0.6_th000_m140	2.5	0.1096	0.006908	0.5899	0.5974	-0.04858	0.02815	-0	0.01952	0.7358	0.9515	0.8064	0.06295
	318	GT0718	Sq_D9.0_q2.50_a0.6_th030_m140	2.5	-0.1866	0.2206	0.5259	0.09963	0.5412	0.2391	0	-0.1596	0.7225	0.9537	0.7877	0.06283
	324	GT0725	Sq_D9.0_q2.50_a0.6_th045_m140	2.5	-0.2735	0.2526	0.4706	0.5203	0.2865	0.08488	0	-0.1529	0.7282	0.9557	0.7663	0.0629
	316	GT0715	Sq_D9.0_q2.50_a0.6_th060_m140	2.5	-0.1921	0.3999	0.4039	0.5358	0.225	0.1492	0	-0.2291	0.7134	0.9584	0.7333	0.06245
	308	GT0706	Sq_D9.0_q2.50_a0.6_th090_m140	2.5	-0.4028	0.4444	0.01711	-0.5513	-0.2211	0.08441	0	-0.1148	0.7511	0.9641	0.6393	0.06617
	435	GT0881	q2.5_LL_D9_a0.6_th1_45_th2_225_m140	2.5	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.7485	0.9594	0.7508	0.06303
	258	GT0651	D9_q2.6_a0.0_m160	2.6	0	0	0	0	0	0	0	-0	0.7504	0.9677	0.5723	0.06539
	257	GT0650	D9_q2.7_a0.0_m160	2.7	0	0	0	0	0	0	0	-0	0.738	0.9686	0.5641	0.0653
	436	GT0882	q2.75_LL_D9_a0.6_th1_45_th2_225_m140	2.75	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.7165	NaN	NaN	0.06296
	260	GT0653	D9_q2.8_a0.0_m160	2.8	0	0	0	0	0	0	0	-0	0.7258	0.9695	0.5561	0.06522
	261	GT0654	D9_q2.9_a0.0_m160	2.9	0	0	0	0	0	0	0	-0	0.7139	0.9704	0.5482	0.06517
	213	GT0596	D10_q3.00_a0.4_0.0_m240	3	0	0	0.4	0	0	0	0	-0	0.7083	0.9631	0.7126	0.05417
	216	GT0600	D10_q3.00_a0.6_0.0_m240	3	0	0	0.6	0	0	0	0	-0	0.7005	0.9567	0.7957	0.05355
	399	GT0842	D11_q3.00_a-0.2_m200	3	0	0	-0.2	0	0	-0.2	0	-0	0.7551	0.9744	0.4452	0.04943
	398	GT0841	D11_q3.00_a-0.4_m200	3	0	0	-0.4	0	0	-0.4	0	-0	0.7646	0.9769	0.3482	0.05018
	403	GT0846	D11_q3.00_a-0.6_m200	3	0	0	-0.6	0	0	-0.6	0	-0	0.7743	0.9789	0.2494	0.05101
x	102	GT0453	D11_q3.00_a0.0_m200	3	0	0	0	0	0	0	0	-0	0.7458	0.9713	0.5405	0.04888
	214	GT0598	D11_q3.00_a0.2_0.00_m200	3	0	0	0.2	0	0	0	0	-0	0.7385	0.9678	0.6277	0.04845
	212	GT0590	D11_q3.00_a0.8_0.00_m200	3	0	0	0.8	0	0	0	0	-0	0.7171	NaN	NaN	0.04753
	219	GT0603	D8_q3.00_a0.5_0.0_th000_ph000_m120	3	0	0	0.5	0	0	0	0	-0	0.6573	0.9603	0.7552	0.07343
	223	GT0607	D8_q3.00_a0.5_0.0_th045_ph000_m120	3	-0.276	0.1482	0.3897	0	0	0	0	-0.1353	0.6491	0.9633	0.7176	0.0741
	229	GT0613	D8_q3.00_a0.5_0.0_th045_ph120_m120	3	0.1755	-0.2213	0.4126	0	0	0	-0	0.1531	0.6455	0.9634	0.7178	0.07364
	235	GT0626	D8_q3.00_a0.5_0.0_th045_ph240_m120	3	0.179	0.254	0.3918	0	0	0	0	-0.09712	0.6563	0.9637	0.7189	0.07444
	222	GT0606	D8_q3.00_a0.5_0.0_th060_ph045_m120	3	-0.411	0.1491	0.2425	0	0	0	-0	0.02363	0.6666	0.9654	0.6885	0.07611
	335	GT0739	D8_q3.00_a0.5_0.0_th060_ph135_m120	3	-0.1616	-0.3256	0.3433	0	0	0	-0	0.1768	0.6443	0.9654	0.6884	0.07389
	340	GT0744	D8_q3.00_a0.5_0.0_th060_ph225_m120	3	0.411	-0.1491	0.2425	0	0	0	0	-0.02363	0.6666	0.9654	0.6886	0.0761
	337	GT0741	D8_q3.00_a0.5_0.0_th060_ph315_m120	3	0.1616	0.3256	0.3433	0	0	0	0	-0.1768	0.6443	0.9654	0.6885	0.07389
	221	GT0605	D8_q3.00_a0.5_0.0_th090_ph000_m120	3	0.4636	-0.1815	0.04629	0	0	0	-0	0.1796	0.6541	0.9699	0.6039	0.07729
	263	GT0657	D8_q3.00_a0.5_0.0_th090_ph060_m120	3	-0.2006	0.4052	0.2134	0	0	0	0	-0.3247	0.5971	0.9696	0.602	0.07147
	243	GT0636	D8_q3.00_a0.5_0.0_th090_ph120_m120	3	-0.4996	0.01877	-0.001172	0	0	0	0	-0.05987	0.6754	0.9701	0.6047	0.07898
	323	GT0724	D8_q3.00_a0.5_0.0_th090_ph180_m120	3	-0.4636	0.1815	0.04629	0	0	0	0	-0.1796	0.6541	0.9699	0.6039	0.07729
	242	GT0635	D8_q3.00_a0.5_0.0_th090_ph240_m120	3	0.2006	-0.4052	0.2134	0	0	0	-0	0.3247	0.5971	0.9696	0.6019	0.07147
	284	GT0680	D8_q3.00_a0.5_0.0_th090_ph300_m120	3	0.4996	-0.01877	-0.001172	0	0	0	-0	0.05987	0.6754	0.9701	0.6047	0.07898
	225	GT0609	D8_q3.00_a0.5_0.0_th120_ph045_m120	3	-0.3352	-0.3173	-0.1921	0	0	0	-0	0.142	0.6767	0.9736	0.4883	0.08041
	339	GT0743	D8_q3.00_a0.5_0.0_th120_ph135_m120	3	0.2674	-0.4119	-0.09392	0	0	0	-0	0.282	0.6317	0.974	0.4904	0.07674
	343	GT0747	D8_q3.00_a0.5_0.0_th120_ph225_m120	3	0.3352	0.3173	-0.1921	0	0	0	0	-0.142	0.6767	0.9736	0.4883	0.08041
	342	GT0746	D8_q3.00_a0.5_0.0_th120_ph315_m120	3	-0.2674	0.4119	-0.09392	0	0	0	0	-0.282	0.6317	0.974	0.4904	0.07674
	288	GT0684	D8_q3.00_a0.5_0.0_th135_ph000_m120	3	0.2891	-0.2813	-0.2955	0	0	0	-0	0.1714	0.6756	0.9751	0.4274	0.08157
	325	GT0726	D8_q3.00_a0.5_0.0_th135_ph120_m120	3	-0.1969	0.3849	-0.2512	0	0	0	0	-0.2114	0.6646	0.9751	0.4272	0.08046
	333	GT0735	D8_q3.00_a0.5_0.0_th135_ph240_m120	3	-0.1554	-0.3943	-0.2653	0	0	0	-0	0.1759	0.6749	0.9753	0.4284	0.08103
	307	GT0705	Eq_D7_q3.00_a0.6_ph030_m140	3	0.5196	0.3	0	-0.5196	-0.3	0	0	0	0.6568	0.9693	0.6164	0.08577
	298	GT0695	Eq_D7_q3.00_a0.6_ph060_m140	3	0.3	0.5196	0	-0.3	-0.5196	0	0	0	0.6568	0.9693	0.6165	0.08576
	299	GT0696	Eq_D7_q3.00_a0.6_ph090_m140	3	0	0.6	0	-0	-0.6	0	0	0	0.6568	0.97	0.6194	0.08591
	309	GT0707	Eq_D7_q3.00_a0.6_ph120_m140	3	-0.3	0.5196	0	0.3	-0.5196	0	0	0	0.6568	0.9703	0.6202	0.08605
	303	GT0701	Eq_D7_q3.00_a0.6_ph150_m140	3	-0.5196	0.3	0	0.5196	-0.3	0	0	0	0.6568	0.9703	0.6197	0.08604
	300	GT0697	Eq_D7_q3.00_a0.6_ph180_m140	3	-0.6	0	0	0.6	-0	0	0	0	0.6568	0.97	0.6185	0.08591
	310	GT0708	Eq_D7_q3.00_a0.6_ph210_m140	3	-0.5196	-0.3	0	0.5196	0.3	0	0	0	0.6568	0.9693	0.6164	0.08577
	304	GT0702	Eq_D7_q3.00_a0.6_ph240_m140	3	-0.3	-0.5196	0	0.3	0.5196	0	0	0	0.6568	0.9693	0.6165	0.08576
	302	GT0699	Eq_D7_q3.00_a0.6_ph270_m140	3	-0	-0.6	0	0	0.6	0	0	0	0.6568	0.97	0.6194	0.08591
	305	GT0703	Eq_D7_q3.00_a0.6_ph300_m140	3	0.3	-0.5196	0	-0.3	0.5196	0	0	0	0.6568	0.9703	0.6202	0.08605
	306	GT0704	Eq_D7_q3.00_a0.6_ph330_m140	3	0.5196	-0.3	0	-0.5196	0.3	0	0	0	0.6568	0.9703	0.6197	0.08604
	330	GT0732	Lq_D6.2_q3.00_a0.6_th030_m140	3	0.3	0	0.5196	-0.3	-0	0.5196	0	0	0.5997	0.9566	0.7915	0.1019
	331	GT0733	Lq_D6.2_q3.00_a0.6_th045_m140	3	0.4243	0	0.4243	-0.4243	-0	0.4243	0	0	0.6071	0.9593	0.7629	0.1028
	332	GT0734	Lq_D6.2_q3.00_a0.6_th060_m140	3	0.5196	0	0.3	-0.5196	-0	0.3	0	0	0.6169	0.9635	0.727	0.09917
	327	GT0728	Lq_D6.2_q3.00_a0.6_th210_m140	3	-0.3	-0	-0.5196	0.3	0	-0.5196	0	0	0.6822	0.9782	0.3235	0.1077
	328	GT0729	Lq_D6.2_q3.00_a0.6_th240_m140	3	-0.5196	-0	-0.3	0.5196	0	-0.3	0	0	0.6645	0.9749	0.4718	0.1049
	297	GT0694	Sq_D6.2_q3.00_a0.6_th000_m140	3	0	0	0.6	-0.6	0	0	0	0	0.6025	0.9572	0.798	0.1032
	301	GT0698	Sq_D6.2_q3.00_a0.6_th015_m140	3	0.1553	0	0.5796	-0.6	0	0	0	0	0.6039	0.9579	0.7912	0.1036
	292	GT0689	Sq_D6.2_q3.00_a0.6_th030_m140	3	0.3	0	0.5196	-0.6	0	0	0	0	0.6078	0.9587	0.7738	0.104
	293	GT0690	Sq_D6.2_q3.00_a0.6_th045_m140	3	0.4243	0	0.4243	-0.6	0	0	0	0	0.6139	0.9618	0.7516	0.09878
	294	GT0691	Sq_D6.2_q3.00_a0.6_th060_m140	3	0.5196	0	0.3	-0.6	0	0	0	0	0.6217	0.9637	0.715	0.09957
	295	GT0692	Sq_D6.2_q3.00_a0.6_th075_m140	3	0.5796	0	0.1553	-0.6	0	0	0	0	0.6308	0.9673	0.6739	0.1006
	296	GT0693	Sq_D6.2_q3.00_a0.6_th090_m140	3	0.6	0	0	-0.6	0	0	0	0	0.6405	0.9693	0.6153	0.1018
	289	GT0685	Sq_D6.2_q3.00_a0.6_th120_m140	3	0.5196	0	-0.3	-0.6	0	0	0	0	0.6591	0.9746	0.4793	0.1045
	290	GT0686	Sq_D6.2_q3.00_a0.6_th150_m140	3	0.3	0	-0.5196	-0.6	0	0	0	0	0.6726	0.9775	0.333	0.1068
	312	GT0710	Sq_D6.2_q3.00_a0.6_th180_m140	3	0	0	-0.6	-0.6	0	0	0	0	0.6773	0.9783	0.269	0.1076
	311	GT0709	Sq_D6.2_q3.00_a0.6_th210_m140	3	-0.3	-0	-0.5196	-0.6	0	0	0	0	0.672	0.9769	0.3642	0.1064
	314	GT0713	Sq_D6.2_q3.00_a0.6_th240_m140	3	-0.5196	-0	-0.3	-0.6	0	0	0	0	0.6581	0.9739	0.5173	0.1039
	315	GT0714	Sq_D6.2_q3.00_a0.6_th270_m140	3	-0.6	-0	-0	-0.6	0	0	0	0	0.6393	0.9688	0.648	0.1012
	322	GT0723	Sq_D6.2_q3.00_a0.6_th300_m140	3	-0.5196	-0	0.3	-0.6	0	0	0	0	0.6207	0.9629	0.7364	0.09907
	319	GT0720	Sq_D6.2_q3.00_a0.6_th315_m140	3	-0.4243	-0	0.4243	-0.6	0	0	0	0	0.613	0.9603	0.7647	0.102
	320	GT0721	Sq_D6.2_q3.00_a0.6_th330_m140	3	-0.3	-0	0.5196	-0.6	0	0	0	0	0.6072	0.9587	0.7855	0.1022
	321	GT0722	Sq_D6.2_q3.00_a0.6_th345_m140	3	-0.1553	-0	0.5796	-0.6	0	0	0	0	0.6036	0.9573	0.7959	0.1027
	428	GT0874	q3_LL_D9_a0.6_th1_45_th2_225_m140	3	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.6865	0.9632	0.7449	0.06296
	411	GT0857	q3_Seff_D9_a_0.066667_-0.6_th1_45_m140	3	0.04714	0	0.04714	-0.4243	0	-0.4243	0	0	0.7028	0.9716	0.5484	0.06494
	350	GT0758	D9_q3.1_a0.0_m160	3.1	0	0	0	0	0	0	0	-0	0.691	0.972	0.5329	0.06503
	351	GT0759	D9_q3.2_a0.0_m160	3.2	0	0	0	0	0	0	0	-0	0.6799	0.9728	0.5255	0.06493
	437	GT0883	q3.25_LL_D9_a0.6_th1_45_th2_225_m140	3.25	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.6583	NaN	NaN	0.06287
	352	GT0760	D9_q3.3_a0.0_m160	3.3	0	0	0	0	0	0	0	-0	0.6691	0.9735	0.5182	0.06488
	353	GT0761	D9_q3.4_a0.0_m160	3.4	0	0	0	0	0	0	0	-0	0.6586	0.9743	0.5111	0.0648
	259	GT0652	D9_q3.5_a0.0_m160	3.5	0	0	0	0	0	0	0	-0	0.6484	0.9749	0.5041	0.0649
	433	GT0879	q3.5_LL_D9_a0.6_th1_45_th2_225_m140	3.5	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.632	0.9668	0.7369	0.0626
	344	GT0748	D9_q3.6_a0.0_m160	3.6	0	0	0	0	0	0	0	-0	0.6384	0.9756	0.4973	0.06481
	365	GT0778	D9_q3.7_a0.0_m160	3.7	0	0	0	0	0	0	0	-0	0.6287	0.9762	0.4907	0.06475
	438	GT0884	q3.75_LL_D9_a0.6_th1_45_th2_225_m140	3.75	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.6075	NaN	NaN	0.06245
	367	GT0780	D9_q3.8_a0.0_m160	3.8	0	0	0	0	0	0	0	-0	0.6192	0.9768	0.4842	0.06469
	364	GT0777	D9_q3.9_a0.0_m160	3.9	0	0	0	0	0	0	0	-0	0.61	0.9774	0.4779	0.06473
	200	GT0570	D10_q4.00_a0.2_0.0_m240	4	0	0	0.2	0	0	0	0	-0	0.6117	0.9749	0.5767	0.05425
	201	GT0573	D10_q4.00_a0.4_0.0_m240	4	0	0	0.4	0	0	0	0	-0	0.6044	0.971	0.6784	0.05352
	204	GT0578	D10_q4.00_a0.6_0.0_m240	4	0	0	0.6	0	0	0	0	-0	0.5972	0.9653	0.7784	0.05285
	390	GT0833	D10_q4.00_a0.8_th174_Q20	4	-0.08282	-0.0009879	-0.7957	0	0	0	-0	0	0.65	0.984	0.06711	0.05957
	181	GT0544	D10_q4_a0.6_th90_m120	4	-0.5624	-0.209	-0.008836	0.1004	-0.0269	0.5909	0	-0.01942	0.6127	0.9747	0.6058	0.05579
	197	GT0566	D11_q4.00_a-0.8_0.0_m160	4	0	0	-0.8	0	0	0	0	-0	0.6657	0.9841	0.03911	0.05106
	103	GT0454	D11_q4.00_a0.0_m200	4	0	0	0	0	0	0	0	-0	0.6374	0.9779	0.4716	0.04876
	125	GT0478	RO3_D10_q4.00_a0.40_oth.060_M120	4	0.3464	0	0.2	0	0	0.4	0	0	0.6071	0.9737	0.6215	0.05464
	117	GT0468	RO3_D10_q4.00_a0.60_oth.045_M120	4	0.4243	0	0.4243	0	0	0.6	0	0	0.5978	0.9675	0.7485	0.05389
	118	GT0469	RO3_D10_q4.00_a0.60_oth.060_M120	4	0.5196	0	0.3	0	0	0.6	0	0	0.6022	0.9702	0.7148	0.05406
	119	GT0470	RO3_D10_q4.00_a0.60_oth.090_M120	4	0.6	0	3.674E-17	0	0	0.6	0	0	0.6127	0.9749	0.6095	0.05484
	91	GT0432	Sq4_d6.2_a0.6_oth.030_rr_M140	4	0.3	0	0.5196	-0.6	0	0	0	0	0.5182	0.9673	0.7585	0.1024
	192	GT0557	Sq4_d9_a0.0_0.6_oth.090_rr_M140	4	0.6	0	3.674E-17	0	0	0	0	0	0.5983	0.9756	0.6015	0.06372
	89	GT0430	Sq4_d9_a0.6_oth.000_rr_M140	4	0	0	0.6	-0.6	0	0	0	0	0.5744	0.9654	0.7794	0.0618
	92	GT0433	Sq4_d9_a0.6_oth.090_rr_M140	4	0.6	0	3.674E-17	-0.6	0	0	0	0	0.5985	0.976	0.5926	0.06376
	94	GT0435	Sq4_d9_a0.6_oth.150_rr_M140	4	0.3	0	-0.5196	-0.6	0	0	0	0	0.6191	0.9826	0.258	0.06851
	96	GT0438	Sq4_d9_a0.6_oth.180_rr_M140	4	7.348E-17	0	-0.6	-0.6	0	0	0	0	0.6222	0.9833	0.1498	0.06945
	98	GT0441	Sq4_d9_a0.6_oth.210_rr_M140	4	-0.3	0	-0.5196	-0.6	0	0	0	0	0.6188	0.9823	0.2849	0.06817
	194	GT0560	Sq4_d9_a0.6_oth.270_rr_M180	4	-0.6	0	-1.102E-16	-0.6	0	0	0	0	0.598	0.9752	0.6128	0.06325
	429	GT0875	q4_LL_D9_a0.6_th1_45_th2_225_m166	4	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.5846	0.9697	0.7293	0.06246
	375	GT0797	D9_q4.1_a0.0_m160	4.1	0	0	0	0	0	0	0	-0	0.5923	0.9784	0.4656	0.0646
	376	GT0798	D9_q4.2_a0.0_m160	4.2	0	0	0	0	0	0	0	-0	0.5838	0.9789	0.4597	0.06454
	370	GT0791	D9_q4.3_a0.0_m160	4.3	0	0	0	0	0	0	0	-0	0.5755	0.9794	0.4538	0.0645
	373	GT0795	D9_q4.4_a0.0_m160	4.4	0	0	0	0	0	0	0	-0	0.5674	0.9799	0.4482	0.06438
	262	GT0656	D9_q4.5_a0.0_m160	4.5	0	0	0	0	0	0	0	-0	0.5595	0.9804	0.4426	0.06431
	434	GT0880	q4.5_LL_D9_a0.6_th1_45_th2_225_m186	4.5	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.5431	0.9726	0.7212	0.0621
	363	GT0771	D10_q5.00_a-0.68_0.00_m240	5	0	0	-0.68	0	0	0	0	-0	0.5626	0.987	0.007675	0.05877
	366	GT0779	D10_q5.00_a-0.69273917_0.00_m280	5	0	0	-0.6927	0	0	0	0	-0	0.563	0.9871	0.000696	0.05886
	362	GT0770	D10_q5.00_a-0.69_0.00_m240	5	0	0	-0.69	0	0	0	0	-0	0.5629	0.987	0.001662	0.05884
	360	GT0768	D10_q5.00_a-0.71_0.00_m240	5	0	0	-0.71	0	0	0	0	-0	0.5637	0.987	0.01046	0.05897
	361	GT0769	D10_q5.00_a-0.72_0.00_m240	5	0	0	-0.72	0	0	0	0	-0	0.564	0.9871	0.01637	0.05904
	357	GT0765	D10_q5.00_a-0.73_0.00_m240	5	0	0	-0.73	0	0	0	0	-0	0.5644	0.9871	0.02218	0.05911
	203	GT0577	D10_q5.00_a0.0_0.0_m240	5	0	0	0	0	0	0	0	-0	0.5383	0.9824	0.4175	0.05463
	215	GT0599	D10_q5.00_a0.2_0.0_m240	5	0	0	0.2	0	0	0	0	-0	0.5314	0.9799	0.5351	0.05371
	218	GT0602	D10_q5.00_a0.4_0.0_m240	5	0	0	0.4	0	0	0	0	-0	0.5247	0.9766	0.6496	0.05302
	227	GT0611	D10_q5.00_a0.6_0.0_m240	5	0	0	0.6	0	0	0	0	-0	0.5181	0.9716	0.762	0.05238
	441	GT0887	q5_LL_D9_a0.6_th1_45_th2_225_m320	5	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.5068	NaN	NaN	0.06126
	387	GT0824	D9_q5.5_a0.0_Q20	5.5	0	0	0	0	0	0	0	-0	0.4906	0.9839	0.3932	0.06221
	220	GT0604	D10_q6.00_a0.00_0.00_m280	6	0	0	0	0	0	0	0	-0	0.4752	0.9855	0.3732	0.05417
	224	GT0608	D10_q6.00_a0.20_0.00_m280	6	0	0	0.2	0	0	0	0	-0	0.4689	0.9834	0.5007	0.05322
	226	GT0610	D10_q6.00_a0.40_0.00_m280	6	0	0	0.4	0	0	0	0	-0	0.4627	0.9805	0.6257	0.05254
	228	GT0612	D10_q6.00_a0.60_0.00_m280	6	0	0	0.6	0	0	0	0	-0	0.4566	0.9762	0.748	0.05189
	341	GT0745	Oq_D9_q6.00_a0.6_th090_m320	6	-0.04612	0.5394	0.2587	0	0	0	0	-0.2052	0.413	0.984	0.5707	0.05757
	443	GT0889	q6_LL_D9_a0.6_th1_45_th2_225_m320	6	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.4465	0.9794	0.7038	0.06124
	291	GT0688	D10_q7.00_a0.20_0.00_m320	7	0	0	0.2	0	0	0	0	-0	0.4191	0.9859	0.4724	0.05284
	336	GT0740	D10_q7.00_a0.40_0.00_m320	7	0	0	0.4	0	0	0	0	-0	0.4134	0.9834	0.6056	0.05216
	391	GT0834	D10_q7.00_a0.8_th168_Q20	7	-0.1459	0.07588	-0.7829	0	0	0	0	-0	0.4485	0.9908	0.2388	0.05941
	386	GT0818	EccQ_q7_c0.01_m320	7	0	0	0	0	0	0	0	0	0.4186	0.9877	0.3367	0.05746
	338	GT0742	Oq_D9_q7.00_a0.6_th090_m320	7	-0.02811	0.5993	-0.008343	0	0	0	0	-0	0.4125	0.9866	0.5657	0.06279
	442	GT0888	q7_LL_D9_a0.6_th1_45_th2_225_m320	7	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.3986	0.9825	0.6952	0.06091
	440	GT0886	q8_LL_D9_a0.6_th1_45_th2_225_m400	8	0.4243	0	0.4243	-0.4243	0	-0.4243	0	0	0.3598	NaN	NaN	0.06115
	199	GT0568	D8.4_q10.00_a0.0_m400	9.996	0	0	0	0	0	0	0	-0	0.3054	0.9918	0.2588	0.06976
	217	GT0601	D7.5_q15.00_a0.0_CHgEEB_m800	15.04	0	0	0	0	0	0	0	-0	0.2095	0.9948	0.1884	0.08475

Data model

Predictive model

In order to make a prediction using the Gaussian Process model we require a new input at which the prediction should be made, which is denoted $x^*$. In order to form the predictive distribution we must then calculate the covariance of the new input with the existing training data, which we denote $K_{x, x^*}$, and the autocovariance of the input, $K_{x^*, x^*}$. We then define a new covariance matrix, $K⁺$, which has the block structure

$$\label{eq:blockK-plus-mat} K^+ = \begin{bmatrix} Kx,x & Kx,x^* \ Kx^*,x & Kx^*, x^* \end{bmatrix}$$

for $K_x,x$ the covariance matrix of the training inputs, and $K_x^*,x = K_x,x^*^T$.

The predictive distribution can then be found as

$$\label{eq:predictive-gp} p(y^* | x^*, \mathcal{D}) = \mathcal{N}(y^* | Kx^*,x Kx,x-1 y, Kx^*, x^* - Kx^*,xK-1x,x Kx,x^*).$$

Training the Gaussian Process

Once the Gaussian process model has been constructed it is necessary to train the model. This is done by choosing a set of hyperparameters for the covariance (kernel) function of the model which maximise some figure of merit of the model:

marginal likelihood (evidence): This is the most common method used to train a GPR model, and allows the training step to be conducted in a fully Bayesian framework.
cross validation: By comparing the output of the model to some set of test data we can produce a figure of merit from the deviation of the model predictions from the test set. By minimising this quantity we can find a model which is best capable of reproducing the test data.

Performing this optimisation can be done in a number of ways, either by sampling the hyperparameter space using MCMC or nested sampling, or using a quasi-Newtonian method like BFGS. This approach works well for simple models, but rapidly becomes intractable for larger models, as the hyperparameter space seems to produce an area where there is a plateau in the overall evidence.

This problem has held Heron back for some time, but a solution in the form of nested sampling seems to allow us to find the correct (or at least a sufficiently correct) maximum in the marginal likelihood function for a given model (and in reasonable time: ~30 minutes for a model with 40 waveforms). Even nested sampling suffers from the plateau-ing problem, however, and it seems to be important to find a stopping criterion for the optimisation to prevent the process becoming stuck.

Finished models

Paper Model 1?

This model is composed of 40 waveforms from the fr (S-Series-v2) set from the Georgia tech catalogue, and involves dynamic range in the (t, q, a1x) parameters (check this!)

Check GPR evidence plateau

Is this real, or is this the result of a sign-flip in the code somewhere? It seems odd that the evidence should be the same as the hyperparameters keep getting smaller. Do we just need to penalise for this in the prior?

It might be a good idea to run some tests with a normal distribution around the `best guess’ value from HERON.

Determine the best optimisation method

It looks like ADAM is the most effective method.

Marginalising over the hyperparameters

Assessing the model

In order to assess the global accuracy of our Gaussian Process surrogate to the underlying function it is standing-in for we require some means of quantitatively measuring deviations between the two. We introduce two measures: the root-mean-squared error, and the correlation. Both methods require access to test data; some fraction of the available evaluations of the function which are not used to train the Gaussian process, but which are held aside, and to which predictions from the GP can be compared.

Let $x_*$ and $y_*$ be respectively the test inputs and test outputs for the Gaussian process, then let $\hat{y}$ be the set of model predictions drawn from the Gaussian Process with inputs $\vec{x}_*$.

Mean-squared error

$$\mathrm{RMSE} = \sqrt{ \frac{ ∑i=0n_i (y_*(i) - \hat{y}(i))^2 } { n_t } }$$

for $n_t$ the size of the test set.

Correlation

$$\label{eq:correlation} ρ^2 = \left( \frac{ \cov(y^*, \hat{y})} { \sqrt{ \vary(y) \vary(\hat{y}) } } \right)^2$$

Forrester\cite{forrester2008engineering} suggests that a $ρ^2 \geq 0.8$ provides a surrogate model with good global predictive abilities, which corresponds to an RMSE of around $0.1$.

Matched filtering

We can use the Gaussian Process to generate waveforms with which to perform matched filtering, and perform matched filtering in much the same way as the standard method. However, normal matched filtering assumes the templates to have no uncertainty, whereas the GPR-generated waveforms come with some measure of their error.

The abb:gpr uncertainty can be incorporated into the matched filtering process by considering the covariance (gram) matrix of the Gaussian process, and using this as part of the error term in the matched filter likelihood.

In a situation with Gaussian and stationary noise the likelihood of a match can be defined as

\[ L(s | \vec{λ}) \propto exp(-0.5 \inp{s - h(\vec{λ})} ) \]

With the noise-weighted inner product is defined as

\[ \inp{x}{y} = 4 \re{ ∫₀^∞ \dd{f} \frac{x(f) y(f)}{Sₙ(f)} \]

Related Papers

Gaussian Processes

Marginalizing Gaussian Process Hyperparameters using Sequential Monte Carlo

PDF

Gravitational Waves

Optimizing gravitational waveforms using Gaussian Process Regression

Abstract

Models of gravitational waveforms play a principal role in detecting and estimating parameters of grav- itational waves (GWs) from compact binary coalescences. We present a Gaussian-process-regression ( GPR ) method which optimizes three aspects of compact binary coalescence gravitational waveforms. Using only a training set of accurate waveforms, the method (a) regresses waveforms with uncertain- ties to reduce biases in parameter estimates, (b) suggests where new accurate waveforms should be generated to minimize model error, and (c) allows fast waveform evaluation. As a proof of concept, we use a training set of IMRPhenomD waveforms with varied mass ratios and equal, aligned spins to build a GPR model which is compared back to IMRPhenomD. With 1000 training waveforms, a waveform with uncertainties from the GPR model can be evaluated in ∼ 3 ms. We then describe how the GPR uncertainties can be used to select parameter values at which to add new training waveforms. The benefit of such a method is that numerical relativity waveforms can be used as the training set to train a GPR model for use in parameter estimation, and the GPR can then in turn suggest parameter values for new numerical relativity simulations.

Method

This method is comparable to the HERON method except that the training data is generated from IMRPhenom waveforms, which are then reprojected into a ROM basis. The GP is then trained with these waveforms, using a Matérn-5/2 kernel (which they justify because of its differentiation properties, but there are advantages to it for the optimisation of the hyperparameters too). They appear to perform a MAP-based optimisation, however they do provide plots of the hyperposterior for a range of hyperparameter values in the case of non-spinning waveforms.

Placing new training waveforms

They propose using posterior uncertainties to guide the placement of future waveforms; this appears to work by training the GPR with a small number of points near the boundary of the parameter space, and then evaluating the model using a fine grid, and then calculating the waveform mismatch at each point in that grid. They then place a new waveform at the location with the largest value of the mismatch. This is naiively a sensible idea, and works for them in a 2D situation, but is computationally inefficient, and will lead to an extremely large number of samples being evaluated.

Publications

HERON Paper 1: Prototype model

Add new waveform points to the corner plot

Make aligned NR comparison plot

Change the units to code units

Project logbook

Out-of-sample testing issues

During out of sample testing efforts (week of 2019-06-10) a few interesting new issues have arisen with the rebuilt model (which appear to be a problem with the hyperparameter choices for the model). This manifests itself as an incorrectly predicted amplitude for the wave, and doesn’t seem to have any obvious solution.

Make data products for the training data

It would be desirable to be able to split the production of training data away from the actual operation of the model. It should be fairly straight forward to do this, but it would require a bit of alteration of the code.

Bibliography & Reading

Improving gravitational-wave parameter estimation using Gaussian process regression

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Files

heron.org

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