Note

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Use the ANCOVA indices

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

ANCOVA allows one to estimate the Sobol’ indices, and thanks to a functional decomposition of the model it allows one 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.

import openturns as ot
import openturns.viewer as otv

Create the model (x1,x2) –> (y) = (4.*x1+5.*x2)

model = ot.SymbolicFunction(["x1", "x2"], ["4.*x1+5.*x2"])

Create the input independent joint distribution

distribution = ot.Normal(2)
distribution.setDescription(["X1", "X2"])

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.JointDistribution([ot.Normal()] * 2, copula)

ANCOVA needs a functional decomposition of the model

enumerateFunction = ot.LinearEnumerateFunction(2)
productBasis = ot.OrthogonalProductPolynomialFactory(
    [ot.HermiteFactory()] * 2, enumerateFunction
)
adaptiveStrategy = ot.FixedStrategy(
    productBasis, enumerateFunction.getStrataCumulatedCardinal(4)
)
samplingSize = 250
experiment = ot.MonteCarloExperiment(distribution, samplingSize)
X = experiment.generate()
Y = model(X)
algo = ot.FunctionalChaosAlgorithm(X, Y, distribution, adaptiveStrategy)
algo.run()
result = algo.getResult()

Create the input sample taking account the correlation

size = 2000
sample = distribution_corr.getSample(size)

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

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  [0.407531,0.592469]
ANCOVA uncorrelated indices  [0.297877,0.482816]
ANCOVA correlated indices  [0.109654,0.109654]

graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
    indices, distribution.getDescription(), "ANCOVA indices (Sobol')"
)
view = otv.View(graph)
ANCOVA indices (Sobol')
graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
    uncorrelatedIndices,
    distribution.getDescription(),
    "ANCOVA uncorrelated indices\n(part of physical variance in the model)",
)
view = otv.View(graph)
ANCOVA uncorrelated indices (part of physical variance in the model)
graph = ot.SobolIndicesAlgorithm.DrawImportanceFactors(
    correlatedIndices,
    distribution.getDescription(),
    "ANCOVA correlated indices\n(part of variance due to the correlation)",
)
view = otv.View(graph)
ANCOVA correlated indices (part of variance due to the correlation)

Display all figures

otv.View.ShowAll()