ANCOVA sensitivity indices

In this example we are going to use the ANCOVA decomposition to estimate sensitivity indices from a model with correlated inputs.

ANCOVA allows to estimate the Sobol’ indices, and thanks to a functional decomposition of the model it allows to separate the part of variance explained by a variable itself from the part of variance explained by a correlation which is due to its correlation with the other input parameters.

In theory, ANCOVA indices range is \left[0; 1\right] ; the closer to 1 the index is, the greater the model response sensitivity to the variable is. These indices are a sum of a physical part S_i^U and correlated part S_i^C. The correlation has a weak influence on the contribution of X_i, if |S_i^C| is low and S_i is close to S_i^U. On the contrary, the correlation has a strong influence on the contribution of the input X_i, if |S_i^C| is high and S_i is close to S_i^C.

The ANCOVA indices S_i evaluate the importance of one variable at a time (d indices stored, with d the input dimension of the model). The d uncorrelated parts of variance of the output due to each input S_i^U and the effects of the correlation are represented by the indices resulting from the subtraction of these two previous lists.

To evaluate the indices, the library needs of a functional chaos result approximating the model response with uncorrelated inputs and a sample with correlated inputs used to compute the real values of the output.

[47]:
from __future__ import print_function
import openturns as ot
[48]:
# Create the model (x1,x2) --> (y) = (4.*x1+5.*x2)
model = ot.SymbolicFunction(['x1', 'x2'], ['4.*x1+5.*x2'])
[49]:
# Create the input independent joint distribution
distribution = ot.Normal(2)
distribution.setDescription(['X1', 'X2'])
[50]:
# Create the correlated input distribution
S = ot.CorrelationMatrix(2)
S[1, 0] = 0.3
R = ot.NormalCopula.GetCorrelationFromSpearmanCorrelation(S)
copula = ot.NormalCopula(R)
distribution_corr = ot.ComposedDistribution([ot.Normal()] * 2, copula)
[51]:
# ANCOVA needs a functional decomposition of the model
enumerateFunction = ot.EnumerateFunction(2)
productBasis = ot.OrthogonalProductPolynomialFactory([ot.HermiteFactory()]*2, enumerateFunction)
adaptiveStrategy = ot.FixedStrategy(productBasis, enumerateFunction.getStrataCumulatedCardinal(4))
samplingSize = 250
projectionStrategy = ot.LeastSquaresStrategy(ot.MonteCarloExperiment(samplingSize))
algo = ot.FunctionalChaosAlgorithm(model, distribution, adaptiveStrategy, projectionStrategy)
algo.run()
result = ot.FunctionalChaosResult(algo.getResult())
[52]:
# Create the input sample taking account the correlation
size = 2000
sample = distribution_corr.getSample(size)
[53]:
# Perform the decomposition
ancova = ot.ANCOVA(result, sample)
# Compute the ANCOVA indices (first order and uncorrelated indices are computed together)
indices = ancova.getIndices()
# Retrieve uncorrelated indices
uncorrelatedIndices = ancova.getUncorrelatedIndices()
# Retrieve correlated indices:
correlatedIndices = indices - uncorrelatedIndices
[54]:
# Print Sobol' indices, the physical part, and the correlation part
print("ANCOVA indices ", indices)
print("ANCOVA uncorrelated indices ", uncorrelatedIndices)
print("ANCOVA correlated indices ", correlatedIndices)
ANCOVA indices  [0.415981,0.584019]
ANCOVA uncorrelated indices  [0.30781,0.475848]
ANCOVA correlated indices  [0.108171,0.108171]
[55]:
ot.SobolIndicesAlgorithm.DrawImportanceFactors(indices, distribution.getDescription(), 'ANCOVA indices (Sobol\')')
[55]:
../../_images/examples_reliability_sensitivity_sensitivity_ancova_10_0.svg
[56]:
ot.SobolIndicesAlgorithm.DrawImportanceFactors(uncorrelatedIndices, distribution.getDescription(), 'ANCOVA uncorrelated indices\n(part of physical variance in the model)')
[56]:
../../_images/examples_reliability_sensitivity_sensitivity_ancova_11_0.svg
[57]:
ot.SobolIndicesAlgorithm.DrawImportanceFactors(correlatedIndices, distribution.getDescription(), 'ANCOVA correlated indices\n(part of variance due to the correlation)')
[57]:
../../_images/examples_reliability_sensitivity_sensitivity_ancova_12_0.svg