Compute grouped indices for the Ishigami function

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

import openturns as ot
from math import pi
import openturns.viewer as otv

Create the Ishigami test function.

ot.RandomGenerator.SetSeed(0)
formula = ['sin(X1) + 7. * sin(X2)^2 + 0.1 * X3^4 * sin(X1)']
input_names = ['X1', 'X2', 'X3']
g = ot.SymbolicFunction(input_names, formula)

Create the probabilistic model

distributionList = [ot.Uniform(-pi, pi)] * 3
distribution = ot.ComposedDistribution(distributionList)

Create a training sample

N = 100
inputTrain = distribution.getSample(N)
outputTrain = g(inputTrain)

Create the chaos.

multivariateBasis = ot.OrthogonalProductPolynomialFactory(distributionList)
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, distribution, adaptiveStrategy, projectionStrategy)

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

Print Sobol’ indices

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

 input dimension: 3
 output dimension: 1
 basis size: 26
 mean: [3.50804]
 std-dev: [3.70043]
------------------------------------------------------------
Index   | Multi-indice                  | Part of variance
------------------------------------------------------------
      7 | [0,4,0]                       | 0.279938
      1 | [1,0,0]                       | 0.190322
      6 | [1,0,2]                       | 0.136823
     15 | [0,6,0]                       | 0.130033
      5 | [3,0,0]                       | 0.12058
     11 | [3,0,2]                       | 0.0837457
      3 | [0,2,0]                       | 0.0250262
     12 | [1,0,4]                       | 0.0111867
------------------------------------------------------------


------------------------------------------------------------
Component | Sobol index            | Sobol total index
------------------------------------------------------------
        0 | 0.313447               | 0.55526
        1 | 0.444616               | 0.444775
        2 | 5.73982e-05            | 0.241886
------------------------------------------------------------

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

chaosSI.getSobolGroupedIndex([0,1])

0.7581141221738299

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 summary method.

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

chaosSI.getSobolGroupedTotalIndex([1,2])

0.6865529054018149

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