Validate a polynomial chaos

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


Model description

We load the Ishigami test function from the usecases module :

im = ishigami_function.IshigamiModel()

The IshigamiModel data class contains the input distribution X=(X_1, X_2, X_3) 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()
N = 100
inputTrain = im.distributionX.getSample(N)
outputTrain = im.model(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.

chaosalgo = ot.FunctionalChaosAlgorithm(inputTrain, outputTrain)

Since the input distribution is known in our particular case, we instead create the multivariate basis from the distribution, that is three independent variables X1, X2 and X3.

multivariateBasis = ot.OrthogonalProductPolynomialFactory([im.X1, im.X2, im.X3])
totalDegree = 8
enumfunc = multivariateBasis.getEnumerateFunction()
P = enumfunc.getStrataCumulatedCardinal(totalDegree)
adaptiveStrategy = ot.FixedStrategy(multivariateBasis, P)
selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
projectionStrategy = ot.LeastSquaresStrategy(
    inputTrain, outputTrain, selectionAlgorithm
chaosalgo = ot.FunctionalChaosAlgorithm(
    inputTrain, outputTrain, im.distributionX, adaptiveStrategy, projectionStrategy
result = chaosalgo.getResult()
metamodel = result.getMetaModel()

Validation of the metamodel

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

n_valid = 1000
inputTest = im.distributionX.getSample(n_valid)
outputTest = im.model(inputTest)
val = ot.MetaModelValidation(inputTest, outputTest, metamodel)
Q2 = val.computePredictivityFactor()[0]

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

graph = val.drawValidation()
graph.setTitle("Q2=%.2f%%" % (Q2 * 100))
view = viewer.View(graph)

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