DoEgen: A Python Library for Optimised Design of Experiment Generation and Evaluation

DoEgen is a Python library aiming to assist in generating optimised Design of Experiments (DoE), evaluating design efficiencies, and analysing experiment results.

In a first step, optimised designs can be automatically generated and efficiencies evaluated for any mixture of factor-levels for numeric and categorical factors. Designs are automatically evaluated as function of number of experiment runs and the most efficient designs are suggested. In particular DoEgen provides computation of a wide range of design efficiencies and allows to import and evaluate externally generated designs as well.

The second part of DoEgen assists in analysing any derived experiment results in terms of factor importance, correlations, and response analysis for best parameter space selection.

Author: Sebastian Haan

Table of Contents

• Definitions
• Functionality
• Installation And Requirements
  • Requirements
  • User Templates
  • Running tests
  • Documentation
• Main Modules and Usage
  • Design Generation
  • Design Efficiencies
  • Design Selection
  • Experiment Result Analysis
• Use Case Study
• Comparison to Other DoE Tools
• Literature
• Attribution and Acknowledgments
• License

Definitions

An Experiment Design is typically defined by:

• Number of Factors: the parameters or variates of the experiment
• Number of Runs: the number of experiments
• Levels: The number of value options for each factor, which can be either numeric values (discrete or continuous) or categorical. Discrete levels for continuous factors can be obtained by providing the minimum and maximum of the factor range and the number of levels. The more levels, the more “fine-grained” the experiment will evaluate this factor, but also more experimental runs are required.

The goal of optimising an experimental design is to provide an efficient design that is near-optimal in terms of, e.g., orthogonality, level balance, and two-way interaction coverage, yet can be performed with a minimum number of experimental runs, which are often costly or time-consuming.

Functionality

If you would like to jumpstart a new experiment and to skip the technical details, you can find a summary of the main usage of DoEgen in Use Case Study.

Currently, the (preliminary) release contains several functions for generating and evaluating designs. Importing and evaluating external designs is supported (e.g. for comparison to other DoE generator tools). DoE also implements several functions for experiment result analysis and visualisation of parameter space.

The main functionalities are (sorted in order of typical experiment process):

• Reading Experiment Setup Table and Settings (Parameter Name, Levels for each factor, Maximum number of runs, Min/Max etc)
• Generating optimised design arrays for a range of runs (given maximum number of runs, and optional computation-time constraints, see settings_design.yaml).
• Evaluation and visualisation of more than ten design efficiencies such as level balance, orthogonality, D-efficiencies etc (see Design Efficiencies for the complete list).
• Automatic suggestion of minimum, optimal, and best designs within a given range of experiment runs.
• Import and evaluation of externally generated design arrays.
• Experiment result analysis: Template table for experiment results, multi-variant RMSE computation, best model/parameter selection, Factor Importance computation, pairwise response surface and correlation computation, factor correlation analysis and Two-way interaction response plots.
• Visualisation of experiment results.

Installation And Requirements

Requirements

• Python >= 3.6
• SWIG >=3.0.12
• OApackage
• xlrd
• XlsxWriter
• openpyxl
• Numpy
• Pandas
• PyYAML
• scikit-learn
• matplotlib
• seaborn

The DoEgen package is currently considered experimental and has been tested with the libraries specified in requirements.txt.

Python Setup Installation

The OApackage requires an installation of SWIG (tested with SWIG 3.0.12), which can be found at https://www.dev2qa.com/how-to-install-swig-on-macos-linux-and-windows/or can be installed via conda

conda install swig

After installing swig and numpy, DoEgen can be installed either with

python setup.py build 
python setup.py install

or using pip

pip install DoEgen

Note that OAPackage can be also installed manually by following installation instructions and documentation for OApackage (tested with OApackage 2.6.6), which can be found at https://pypi.org/project/OApackage/.

Docker Installation

A docker image is provided in the folder docker/ and can be build, e.g., with

docker build -t doegen-app:v1 .

and tested, e.g.,

docker run -it -v <PATH_TO_DOCKER_IMAGE>/project:/project doegen-app:v1 doegen.doegen <SETTINGSFILE_DESIGN>.yaml
docker run -it -v <PATH_TO_DOCKER_IMAGE>/project:/project doegen-app:v1 doegen.doeval <SETTINGSFILE_ANALYSIS>.yaml

replacing <PATH_TO_DOCKER_IMAGE> with the absolute path to the docker image and <SETTINGSFILE_DESIGN> and <SETTINGSFILE_ANALYSIS> with the respective path and filename of the setting file.

User Templates

1. The factor (parameter) settings of experiment are defined in an experiment setup table (see Experiment_results_template.xlsx). A new excel setup template table can be also created with create_setupfile.py. Each factor is on a new row and specified by Parameter Name, Parameter Type , Level Number, Minimum, Maximum, Include (Y/N) (optional, by default all will be included), Levels (optional) If Levels are provided pleae seperate each level by a comma; Levels can be a mix of numerical and string entries (NUmbre of entries should match Level Number)

2. After the experiment is run, the results have to be filled in an experiment result table (see Experiment_results_template.xlsx). A new excel result template table can be also created with create_resultfile.py The result table allows to fill in multiple output properties (Y_label: output target to be predicted) and experiment positions. The results have to be provided in the table with the following columns:

• Nexp: Run# of experiment, need to match Run# in Experiment setup and design.
• PID: Identifier# of label of location (point) in experiment (e.g. if experiment is run at different locations simultaneously).
• Y Label: Identifier# or label of Y-Variate (target property that has to be predicted or evaluated, e.g. Rain and Temperature). This allows to include multi-output models with distinct target properties. Note that currently each Y variate is evaluated separately.
• Y Exp The experiment result for Y
• Y Truth (optional) if the true value available is available for Y. This is required to calculate the RMSE and to select best parameter space.
• Not currently considered (yet) in result stats computation: Std Y Exp, Std Y Truth, Weight PID

Experiment Setup Table Header.

Experiment Result Table Header.

Running Tests

To verify that DoEgen works, you can run the example experiment

$ python -m doegen.init_tests
$ python -m doegen.doegen test/settings_design_test.yaml
$ python -m doegen.doeval test/settings_expresults_test.yaml

Documentation

Please do not modify README.md. Instead make any changes in the master documentation file MANUAL.md (uses pandoc markdown syntax) and then convert to the inferior Github markdown flavor (note that the new github-flavored markdown format gfm option does not correctly solve figure caption and resize options):

pandoc -f markdown -t markdown_github MANUAL.md -o README.md

and to pdf:

pandoc -V geometry:margin=1.2in MANUAL.md -o docs/MANUAL.pdf

or as standalone html:

pandoc MANUAL.md -o MANUAL.html

Main Modules and Usage

Design Generation

Design generation with doegen.py: Main model for generating optimised designs and computation of efficiencies. Settings are specified in settings yaml file settings_design.yaml. If the yaml and .xlsx template files are not yet in your working directory (e.g. after first DoEgen installation), you can create in the the yaml and excel template files with

$ python -m doegen.init_config

Before running doegen.py,two things have to be the done:

1. fill in experiment setup table (see template provided Experiment_setup_template.xlsx or example in test/ folder)
2. provide settings in settings file (see settings_design.yaml)

Now you are ready to run the design generation

$ python -m doegen.doegen settings_design.yaml

This will produce a number of files for different experiment run length (see folder test/results/DesignArray_Nrun...):

• The optimised design array EDarray_[factor_levelels]_Nrun.csv.
• A table of design efficiencies Efficiencies_[factor_levelels]_Nrun.csv
• Table of two-way Interaction balance Table_Interaction_Balance.txt
• Table of Pearson correlation coefficients between all factor pairs Table_Pearson_Correlation.csv
• Plot of pairwise correlation including regression fit pairwise_correlation.png (see example plot below)

Besides the default optimisation (based on function doegen.deogen.optimize_design), DoEgen also allows the to construct full orthogonal designs using the function doegen.doegen.gen_highD, which is based on OApackage orthogonal arrays and extensions. However, this works only for special cases with limited number of factors and design levels. Thus, it is currently not fully automated but might assist advanced users to construct optimal designs.

Design Selection

DoEgen will select by default three designs based on the following citeria:

1. minimum Design with the criteria:

• number of runs >= number of factors + 1
• center balance > 95%
• level balance > 95%
• Orthogonal Balance > 90%
• Two Level interaction Balance > 90%
• Two Level Interaction Minimum One = 100%

1. optimal Design with the criteria:

• center balance > 98%
• level balance > 98%
• Orthogonal Balance > 95%
• Two Level interaction Balance > 95%
• Two Level Interaction Minimum One = 100%

1. best design which is based on best score that is sum of efficiencies above and includes a small penalty for runsize relative to maximum runsize

This will deliver (see folder test/results/):

• Overview summary of the three designs and their main efficiencies: Experiment_Design_selection_summary.txt
• Three tables (Designtable_minimum/optimal/best...csv) for the there suggested designs that are converted in the actual level values
• An overview of the efficiencies is plotted as function of exp run and saved in Efficiencies_[factor_levels].png

In case the user wants to select another design for a different run size, one can covert the design array into a design table with the function doegen.deogen.array2valuetable().

Example overview plot of the main efficiencies (from 0=worst to 100=best) as function of number of experiments.

Design Efficiencies

DoEgen computes more than ten efficiencies and saves them as .csv file for each generated design array. All indicators, except for the canonical correlations, have a range from 0 (worst possible) to 1 (optimal):

• Center Balance: 100% [1 - Sum(Center-Deviation)/Array Size], i.e. the average center balance over all factors.
• Level Balance: Defined as 100% [1 - Sum(Imbalance)/Array Size], the average level balance over all factors.
• Orthogonality: Defined as 100% [1 - Orthogonality], i.e. the average orthogonality over all factor pairs.
• Two-way Interaction Balance: Similar to level balance but for pairwise factor balance.
• Two-way Interaction with at least one occurrence: 100% [1 - Sum(Not at least one pairwise factor occurrence)/number of pairwise combinations]; 100% if all factor-level pair combinations occur at least once.
• D-Eff: D-Efficiency (model includes main term and quadratic).
• D1 Eff: only main terms
• D2 Eff: main, quadratic, and interaction terms
• A-Eff: A-efficiency (main term and quadratic)
• A1-Eff: only main terms
• A2-Eff: main, quadratic, and interaction terms

For further inspection, doegen.deogen.evaluate_design2 creates also the following tables and plots:

• Table of Pearson Correlation (same as above if normalised discrete variables)
• Table of Two-way Interaction Balance
• Cornerplot of pairwise factor relation with Y

Pairwise factor correlation plot of an example 8 factor design array with a mix of 3- and 2-level factors. The lines and blue shadows correspond to the linear regression fit and its uncertainty. Two pairs are 100% orthogonal if the linear regression line is horizontal. The diagonal bar charts show the histogram of level values for each factor (perfect level balance if histogram is flat).

Experiment Result Analysis

Experiment Result Analysis with doeval.py: The experiment results have to be provided in a result table with the format as specified in #user-templates, and specifications in the settings_expresults.yaml file. Then run

$ python -m doegen.doeval settings_expresults.yaml

This will create the following stats tables and plots (see folder test/expresults/ as example):

• Valuation of the factors in term of “importance”, which is defined by the maximum change (range) in the average Y between any factor levels. Results are visualized in bar plot (Ybarplot_*.png) and saved as csv (Experiment_Elevation_Factorimportance.csv), including, min, max, std deviation across all levels
• Computes RMSE between experiment result and ground truth; results saved as csv.
• Ranked list of top experiments and their parameters based on RMSE
• Computes average and variance of best parameters weighted with RMSE; saved to csv file
• An overview plot of all the correlation plots between Y and each factor (Expresult_distribution_X-Y_*.png, see function plot_regression)
• Overview plot of the correlations between Y and RMSE (Expresult_distribution_X-RMSE_*.png, see function plot_regression)
• Plot of Y values for each pairwise combination of factors (Y-pairwise-correlation_*.png, see function plot_3dmap), which allows the user to visualise categorical factors
• Plot of RMSE value for each pairwise combination of factors (RMSE-pairwise-correlation_*.png, see function plot_3dmap)

Overview plot of X-Y Correlation for each factor as function of their level values. On top the linear regression coefficient r is shown along the linear regression fit and its uncertainty (line and shadow).

Cornerplot of pairwise factor relation with Y. The color(bar) indicates the value of Y.

Use Case Study

Here we demonstrate a typical use case where we would like to first generate and select an optimal experiment design. Then subsequently after running the experiment we would like to answer the question which is the best parameter space and what parameters are important. Our case study is given by the test example, which consists of 8 factors (parameters) that are specified in the experiment setup table Experiment_setup_test.xlsx.

Test Experiment Setup Table with 6 discrete and 2 categorical factors. Each factor can have a certain number of levels (values), which are here either 3 or 2

The first goal is to generate an efficient design with only a fraction of the entire parameter combination (in our case the full factorial would be 3⁶ × 2² = 2916). The maximum number of experiments (in this case we choose 150) is set in the file settings_design_test.yaml, which also specifies input and output directory names, as well as the maximum time for optimising one run (in this case 100 seconds per design optimisation). This configuration will generate and optimize a range of experiments with different design run sizes from 12 to 150, in steps of 6 runsizes (since the lowest common multiple of our mix of 2 and 3 factor levels is 6). Note that the user can also choose a different stepsize, which can done by setting the value in the setting parameter delta_nrun. Now we are all setup to start the experiment design generation and optimisation script, which we do by running the script doegen.py with the settings file as argument:

$ cd DoEgen
$ python -m doegen.doegen test/settings_design_test.yaml

This will generate for each runsize an optimised design array and a list of efficiencies and diagnostic tables and plots (see Design Generation for more details). To simplify the selection of the generated experiment designs, DoEgen suggests automatically three designs: 1) one minimum design (lowest number of runs at given efficiency threshold), 2) one optimal design, and 3) one best design (either equal or has larger experiment run number than optimal design). In our case the three design are selected for run numbers 30 (minimum), 72 (optimal), 90 (best). Since the optimal design has basically almost the same efficiencies as the best design (see figure below) but at a lower cost of experiment runs, we choose for our experiment the optimal design, which is given in the table Designtable_optimal_Nrun72.csv.

Result Overview of Experiment Design Generation and the three suggested choices. The most important criteria for a good design are orthogonality (100% means that all factor pairs are 100% orthogonal to each other), level/center balance (100% is best) and two-way interaction balance (100% is best). We also want to make sure that at each pairwise interaction occurs at least one (100% Two-Level Min Efficiency). D-efficiency maximises the determinant of the information matrix |X^TX|, which corresponds to minimizing the generalized variance of the parameter estimates for a pre-specified model X. Here, D1-efficiency defines the model with only the main effects, while D-efficiency includes also all quadratic terms in the model X. Typically D1-efficiency should be larger than 60%, while D-efficiency only increases if number of experiments is much larger than the number of model terms. In this case study we consider only D1-efficiency given that we want to minimize the number of experiments.

Header with first 5 rows of the optimal design with 72 experiments

Now it is time to run the experiment. In our example, we produce just some random data for the 72 experiments with 10 sensor locations (PID 1 to 10) and one output variable Y (e.g. temperature). To analyse the experiment, the results have to written in a structured table with the format as given in experiment_results_Nrun72.xlsx (see description in figure below).

Header with first rows of the experiment result table for 72 experiments. Note that the Nexp number has to match the experiment design table Nexp. Each experiment (label Nexp) can have multiple locations or points (identifier# PID), e.g., if experiment is run at different locations simultaneously. In addition, it is possible that one has multiple output Y-variates, labeled with identifier Y :abel (target property that has to be predicted or evaluated, e.g. Rain and Temperature). The column Y Exp holds the experiment result for Y while the column Y Truthholds the ground truth value, which is required to calculate the RMSE and to select best parameter space.

To run the experiment analysis script, settings such as for input output directory names are given in the settings file settings_expresults_test.yaml, and we can now run the analysis script with

$ python -m doegen.doeval test/settings_expresults_test.yaml

This analysis produces a range of diagnostic tables and result plots for each output variable Y (in our case we have only one Y). One of the question of this example use case is to identify what factors are important, which is given in the figure Ybarplot.png. The “importance” basically indicates how much a factor changes Y (defined by the maximum average change in Y between any levels). This has the advantage to identify also important factors that have either a low linear regression coefficients with Y (see r values in plot Expresult_correlation_X.png) or are categorical. Such insight can be valuable to determine, e.g., which factors should be investigated in more detail in a subsequent experiment or to estiamate which factors have no effect on Y.

Factor Importance ranked from maximum to lowest change (range) in Y

Another important question is what are the best parameter values based on the obtained experiment results so far? This question can be answered by computing the Root-Mean-Square-Error between experiment results and ground truth (or alternatively the likelihood if the model predictions include also uncertainties). Table Experiment_1_RMSE_Top10_sorted.csv provides an overview of the top 10 experiments sorted as function of their RMSE. Moroever we can calculate the (RMSE-weighted) average of each factor for the top experiments as shown in bar plot below.

Picture of Table Experiment_1_RMSE_Top10_sorted.csv which shows the factor values of the top 10 experiments based on their RSME values.

Factor values of the top 10 experiments based on their RSME values. The bar heights indicate the top factor’s average value and the dark lines their standard deviation. Note that the average and their standard deviation are computed with the weights RMSE^− 2.

Furthermore, multiple other diagnostics plots such as factor-Y correlation and pairwise correlation maps with RMSE are generated (see Experiment Result Analysis for more details).

Comparison to Other DoE Tools

The aim of DoEgen is to provide an open-source tool for researchers to create optimised designs and a framework for transparent evaluation of experiment designs. Moreover, DoEgen aims to assist the result analysis that may allow the researcher a subsequent factor selection, parameter fine-tuning, or model building. The design generation function of DoEgen is build upon the excellent package OApackage and extends it further in terms of design efficiency evaluation, filtering, automation, and experiment analysis. There are multiple other tools available for DoE; the table below provides a brief (preliminary, subjective, and oversimplified) summary of the main advantages and disadvantages for each tool that has been tested. Users are encouraged to test these tools themselves.

Feature	SAS JMP	pyDOE2	OApackage	DoEgen
Open-Source	no (paid)	yes	yes	yes
Design Optimisation Score	very good	limited	good	good
Optimal Runsize Finder	no	no	no	yes
Design Efficiency Eval	yes	no	limited	yes
Exp Result Analysis	yes	no	no	yes
Development Stage	advanced	early	moderate	very early

Literature

OApackage: A Python package for generation and analysis of orthogonal arrays, optimal designs and conference designs, P.T. Eendebak, A.R. Vazquez, Journal of Open Source Software, 2019

pyDOE2: An experimental design package for python

Dean, A., Morris, M., Stufken, J. and Bingham, D. eds., 2015. Handbook of design and analysis of experiments (Vol. 7). CRC Press.

Goos, P. and Jones, B., 2011. Optimal design of experiments: a case study approach. John Wiley & Sons.

Kuhfeld, W.F., 2010. Discrete choice. SAS Technical Papers, 2010, pp.285-663.

Zwerina, K., Huber, J. and Kuhfeld, W.F., 1996. A general method for constructing efficient choice designs. Durham, NC: Fuqua School of Business, Duke University.

Cheong, Y.P. and Gupta, R., 2005. Experimental design and analysis methods for assessing volumetric uncertainties. SPE Journal, 10(03), pp.324-335.

JMP, A. and Proust, M., 2010. Design of experiments guide. Cary, NC: SAS Institute Inc.

Attribution and Acknowledgments

Acknowledgments are an important way for us to demonstrate the value we bring to your research. Your research outcomes are vital for ongoing funding of the Sydney Informatics Hub.

If you make use of this code for your research project, please include the following acknowledgment:

“This research was supported by the Sydney Informatics Hub, a Core Research Facility of the University of Sydney.”

Project Contributors

Key project contributors to the DoEgen project are:

• Sebastian Haan (Sydney Informatics Hub, University of Sydney): Main contributor and software development of DoEgen.
• Christopher Howden (Sydney Informatics Hub, University of Sydney): Statistical consultancy, literature suggestions, and documentation.
• Danial Azam (School of Geophysics, University of Sydney): Testing DoEgen on applications for computational geosciences.
• Joel Nothman (Sydney Informatics Hub, University of Sydney): Code review and improvements with focus on doegen.py.
• Dietmar Muller (School of Geophysics, University of Sydney): Suggesting the need for this project and developing real-world use cases for geoscience research.

DoEgen has benefited from the OApackage library OApackage for the design optimisation code and we would like to thank the researchers who have made their code available as open-source.

License

Copyright 2021 Sebastian Haan, The University of Sydney

DoEgen is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License (LGPL version 3) as published by the Free Software Foundation.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public License along with this program (see LICENSE.md). If not, see https://www.gnu.org/licenses/.