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
import openturns.viewer as viewer
from matplotlib import pylab as plt
from math import pi
import openturns.viewer as otv
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.summary())

Out:

 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])

Out:

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

Out:

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])

Out:

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

Out:

0.6667526