Design of experiments¶

Design of experiments (DOE) is a branch of applied statistics to plan, conduct and analyze real or numerical experiments. It consists in selecting input values in a methodical way (sampling) and then performing the experiments to obtain output values (measurement or evaluation).

Note

“DOE” may also refer to the sampling method itself, e.g. Latin hypercube sampling.

A DOE can be used to:

• determine whether an input or an interaction between inputs has an effect on an output (sensitivity analysis),

• model the relationship between inputs and outputs (surrogate modeling),

• optimize an output with respect to inputs while satisfying some constraints (trade-off).

API¶

In GEMSEO, a DOELibrary contains one or several DOE algorithms.

As any DriverLib, a DOELibrary executes an algorithm from an OptimizationProblem and options. Most of the DOE algorithms also need the number of samples when calling execute():

>>> from gemseo.algos.doe.lib_pydoe import PyDOE
>>> pydoe_library = PyDOE()
>>> optimization_result = pydoe_library.execute(problem, "lhs", n_samples=100)


In the presence of an OptimizationProblem, it is advisable to apply DOE algorithms with the function execute_algo() which returns an OptimizationResult:

>>> from gemseo.api import execute_algo
>>> optimization_result = execute_algo(problem, "lhs", algo_type="doe", n_samples=100)


In the presence of an MDODiscipline, it is advisable to create a DOEScenario with the function create_scenario() and pass the DOE algorithm to DOEScenario.execute():

>>> doe_scenario.execute({"algo": "lhs", "n_samples": 100})


Algorithms¶

GEMSEO wraps different kinds of DOE algorithms from the libraries PyDOE and OpenTURNS.

Note

The names of the algorithms coming from OpenTURNS starts with "OT_", e.g. "OT_OPT_LHS". You need to install the full features of GEMSEO in order to use them.

All the DOE algorithms and their settings are listed on this page.

These DOE algorithms can be classified into categories:

• the Monte Carlo sampling generates values in the input space distributed as a multivariate uniform probability distribution with stochastically independent components; the algorithm is "OT_MONTE_CARLO",

• the low-discrepancy sequences are sequences of input values designed to be distributed as uniformly as possible (the deviation from uniform distribution is called discrepancy); the algorithms are "OT_FAURE", "OT_HALTON", "OT_HASELGROVE", "OT_SOBOL" and "OT_REVERSE_HALTON",

• the Latin hypercube sampling (LHS) is an algorithm generating $$N$$ points in the input space based on the generalization of the Latin square: the range of each input is partitioned into $$N$$ equal intervals and, for each interval, one and only one of the points has its corresponding input value inside the interval; the algorithms are "lhs", "OT_LHS" and "OT_LHSC",

• the optimized LHS is an LHS optimized by Monte Carlo replicates or simulated annealing; the algorithm is "OT_OPT_LHS",

• the stratified DOEs makes the inputs, also called factors, vary by level;

• a full factorial DOE considers all the possible combinations of these levels across all the inputs; the algorithms are "ff2n", "fullfact" and "OT_FULLFACT";

• a factorial DOE samples the diagonals of the input space, symmetrically with respect to its center; the algorithm is "OT_FACTORIAL";

• an axial DOE samples the axes of the input space, symmetrically with respect to its center; the algorithm is "OT_AXIAL";

• a central composite DOE combines a factorial and an axial DOEs; the algorithms are "OT_COMPOSITE" and "ccdesign";

• Box–Behnken and Plackett-Burman DOEs for response surface methodology; the algorithms are "bbdesign" and "pbdesign".

GEMSEO also offers a CustomDOE to set its own input values, either as a CSV file or a two-dimensional NumPy array.

Once the functions of the OptimizationProblem have been evaluated, the input samples can be accessed with samples.

Note

GEMSEO applies a DOE algorithm over a unit hypercube of the same dimension as the input space and then project the unit_samples onto the input space using either the probability distributions of the inputs, if the latter are random variables, or their lower and upper bounds.

If we do not want to evaluate the functions but only obtain the input samples, we can use the method compute_doe() which returns the samples as a two-dimensional NumPy array.

The quality of the input samples can be assessed with a DOEQuality computing the $$\varphi_p$$, minimum-distance and discrepancy criteria. The smaller these quality measures, the better, except for the minimum-distance criterion for which the larger it is the better. The qualities can be compared with logical operations, with DOEQuality(doe_1) > DOEQuality(doe_2) meaning that doe_1 is better than doe_2.

Note

When numerical metrics are not sufficient to compare two input samples sets, graphical indicators (e.g. ScatterMatrix) could be considered.

Lastly, a DOELibrary has a seed initialized at 0 and each call to execute() increments it before using it. Thus, two executions generate two distinct set of input-output samples. For the sake of reproducibility, you can pass your own seed to execute() as a DOE option.