Estimate Sobol’ indices for the beam by simulation algorithm

In this example, we estimate the Sobol’ indices for the cantilever beam by simulation algorithm.

Introduction

In this example we are going to quantify the correlation between the input variables and the output variable of a model thanks to Sobol indices.

Sobol indices are designed to evaluate the importance of a single variable or a specific set of variables. Here the Sobol indices are estimated by sampling from the distributions of the input variables and propagating uncertainty through a function.

In theory, Sobol indices range from 0 to 1; the closer an index value is to 1, the better the associated input variable explains the function output.

Let us denote by the input dimension of the model.

Sobol’ indices can be computed at different orders.

• First order indices evaluate the importance of one input variable at a time.

• Total indices give the relative importance of one input variable and all its interactions with other variables. Alternatively, they can be viewed as measuring how much wriggle room remains to the output when all but one input variables are fixed.

• In general, we are only interested in first order and total Sobol’ indices. There are situations, however, where we want to estimate interactions. Second order indices evaluate the importance of every pair of input variables. The number of second order indices is:

In practice, when the number of input variables is not small (say, when ), then the number of second order indices is too large to be easily analyzed. In these situations, we limit the analysis to the first order and total Sobol’ indices.

Define the model

from openturns.usecases import cantilever_beam
import openturns as ot
import openturns.viewer as otv

ot.Log.Show(ot.Log.NONE)

We load the distribution and model from the example:

beam = cantilever_beam.CantileverBeam()
distribution = beam.independentDistribution
model = beam.model

Estimate the Sobol’ indices

We first create the algorithm from SobolSimulationAlgorithm with the Saltelli estimator it will allow one to control the number of evaluations by convergence instead of using a fixed-size experiment

estimator = ot.SaltelliSensitivityAlgorithm()
estimator.setUseAsymptoticDistribution(True)
algo = ot.SobolSimulationAlgorithm(distribution, model, estimator)
algo.setMaximumOuterSampling(50)  # number of iterations
algo.setExperimentSize(1000)  # size of Sobol experiment at each iteration
algo.setBlockSize(4)  # number of points evaluated simultaneously
# algo.setIndexQuantileLevel(0.05)  # alpha
# algo.setIndexQuantileEpsilon(1e-2)  # epsilon
algo.run()

Extract the results

result = algo.getResult()
fo = result.getFirstOrderIndicesEstimate()
to = result.getTotalOrderIndicesEstimate()
foDist = result.getFirstOrderIndicesDistribution()
graph = result.draw()
_ = otv.View(graph)

Using a different estimator

We have used the SaltelliSensitivityAlgorithm class to estimate the indices. Others are available in the library:

• SaltelliSensitivityAlgorithm

• MartinezSensitivityAlgorithm

• JansenSensitivityAlgorithm

• MauntzKucherenkoSensitivityAlgorithm

otv.View.ShowAll()