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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 X=(X_1, X_2, X_3) 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)

FunctionalChaosSobolIndices
- input dimension=3
- output dimension=1
- basis size=26
- mean=[3.50739]
- std-dev=[3.70413]

| Index | Multi-index   | Variance part |
|-------|---------------|---------------|
|     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     |

| Input | Name          | Sobol' index  | Total index   |
|-------|---------------|---------------|---------------|
|     0 | X1            | 0.31752       | 0.559269      |
|     1 | X2            | 0.440685      | 0.440794      |
|     2 | X3            | 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