# Validate a polynomial chaos¶

In this example, we show how to perform the draw validation of a polynomial chaos for the Ishigami function.

[1]:

import openturns as ot
from math import pi


Create the Ishigami test function.

[2]:

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

[3]:

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


Create a training sample

[4]:

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


Create the chaos.

We could use only the input and output training samples: in this case, the distribution of the input sample is computed by selecting the best distribution that fits the data.

[5]:

chaosalgo = ot.FunctionalChaosAlgorithm(inputTrain, outputTrain)


Since the input distribution is known in our particular case, we instead create the multivariate basis from the distribution.

[6]:

multivariateBasis = ot.OrthogonalProductPolynomialFactory(distributionList)
totalDegree = 8
enumfunc = multivariateBasis.getEnumerateFunction()
P = enumfunc.getStrataCumulatedCardinal(totalDegree)
adaptiveStrategy = ot.FixedStrategy(multivariateBasis, P)

[7]:

selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
projectionStrategy = ot.LeastSquaresStrategy(inputTrain, outputTrain, selectionAlgorithm)

[8]:

chaosalgo = ot.FunctionalChaosAlgorithm(inputTrain, outputTrain, distribution, adaptiveStrategy, projectionStrategy)

[9]:

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


In order to validate the metamodel, we generate a test sample.

[10]:

n_valid = 1000
inputTest = distribution.getSample(n_valid)
outputTest = g(inputTest)
val = ot.MetaModelValidation(inputTest, outputTest, metamodel)
Q2 = val.computePredictivityFactor()
Q2

[10]:

0.9992361845215688


The Q2 is very close to 1: the metamodel is excellent.

[11]:

graph = val.drawValidation()
graph.setLegends([""])
graph.setTitle("Q2=%.2f%%" % (Q2*100))
graph

[11]:


The metamodel has a good predictivity, since the points are almost on the first diagonal.