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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 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