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 in im.distributionX and the Ishigami function in im.model. We also have access to the input variable names with:

input_names = im.distributionX.getDescription()

Create a training sample.

N = 100
inputTrain = im.distributionX.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.distributionX, adaptiveStrategy, projectionStrategy
)

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

Print Sobol’ indices.

chaosSI = ot.FunctionalChaosSobolIndices(result)
print(chaosSI)

 input dimension: 3
 output dimension: 1
 basis size: 26
 mean: [3.50739]
 std-dev: [3.70413]
------------------------------------------------------------
Index   | Multi-indice                  | Part of variance
------------------------------------------------------------
      7 | [0,4,0]                       | 0.274425
      1 | [1,0,0]                       | 0.191936
      6 | [1,0,2]                       | 0.135811
     13 | [0,6,0]                       | 0.134001
      5 | [3,0,0]                       | 0.122952
     10 | [3,0,2]                       | 0.0856397
      3 | [0,2,0]                       | 0.0237185
     11 | [1,0,4]                       | 0.0112027
------------------------------------------------------------


------------------------------------------------------------
Component | Sobol index            | Sobol total index
------------------------------------------------------------
        0 | 0.31752                | 0.559269
        1 | 0.440685               | 0.440794
        2 | 1.87833e-05            | 0.241742
------------------------------------------------------------

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

chaosSI.getSobolGroupedIndex([0, 1])

0.7582578489711685

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])

0.6824803087795113

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