Note

Go to the end to download the full example code.

Compute grouped indices for the Ishigami function

In this example, we compute grouped Sobol’ indices for the Ishigami function.

from openturns.usecases import ishigami_function
import openturns as ot

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

We load the Ishigami test function from usecases module:

im = ishigami_function.IshigamiModel()

The IshigamiModel data class contains the input distribution of the random vector \vect{X}=\Tr{(X_1, X_2, X_3)} in im.inputDistribution and the Ishigami function in im.model. We also have access to the input variable names with:

input_names = im.inputDistribution.getDescription()

Create a training sample.

N = 100
inputTrain = im.inputDistribution.getSample(N)
outputTrain = im.model(inputTrain)

Create the chaos.

multivariateBasis = ot.OrthogonalProductPolynomialFactory([im.X1, im.X2, im.X3])
selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
projectionStrategy = ot.LeastSquaresStrategy(
    inputTrain, outputTrain, selectionAlgorithm
)
totalDegree = 8
enumfunc = multivariateBasis.getEnumerateFunction()
P = enumfunc.getStrataCumulatedCardinal(totalDegree)
adaptiveStrategy = ot.FixedStrategy(multivariateBasis, P)
chaosalgo = ot.FunctionalChaosAlgorithm(
    inputTrain, outputTrain, im.inputDistribution, adaptiveStrategy, projectionStrategy
)

chaosalgo.run()
result = chaosalgo.getResult()
metamodel = result.getMetaModel()

Print Sobol’ indices.

chaosSI = ot.FunctionalChaosSobolIndices(result)
chaosSI
FunctionalChaosSobolIndices
  • input dimension: 3
  • output dimension: 1
  • basis size: 16
  • mean: [3.48273]
  • std-dev: [3.70612]
Input Variable Sobol' index Total index
0 X1 0.316757 0.557175
1 X2 0.442820 0.442853
2 X3 0.000000 0.240424
Index Multi-index Part of variance
5 [0,4,0] 0.278531
1 [1,0,0] 0.193813
4 [1,0,2] 0.131990
10 [0,6,0] 0.131954
3 [3,0,0] 0.119909
7 [3,0,2] 0.087691
2 [0,2,0] 0.024288
8 [1,0,4] 0.010603


We compute the first order indice of the group [0,1] .

chaosSI.getSobolGroupedIndex([0, 1])

0.7595764133948941

This group collects all the multi-indices containing variables only in this group, including interactions within the group (by decreasing order of significance):

  • [0,4,0] : 0.279938

  • [1,0,0] : 0.190322

  • [0,6,0] : 0.130033

  • [3,0,0] : 0.12058

  • [0,2,0] : 0.0250262

0.279938 + 0.190322 + 0.130033 + 0.12058 + 0.0250262

0.7458992

The difference between the previous sum and the output of getSobolGroupedIndex is lower than 0.01, which is the threshold used by the __str__ method.

We compute the total order indice of the group [1,2].

chaosSI.getSobolGroupedTotalIndex([1, 2])

0.6832431353476381

This group collects all the multi-indices containing variables in this group, including interactions with variables outside the group:

  • [0,4,0] : 0.279938

  • [1,0,2] : 0.136823

  • [0,6,0] : 0.130033

  • [3,0,2] : 0.0837457

  • [0,2,0] : 0.0250262

  • [1,0,4] : 0.0111867

0.279938 + 0.136823 + 0.130033 + 0.0837457 + 0.0250262 + 0.0111867

0.6667526