Polynomial chaos over database

In this example we are going to create a global approximation of a model response using functional chaos over a design of experiment.

You will need to specify the distribution of the input parameters. If not known, statistical inference can be used to select a possible candidate, and fitting tests can validate such an hypothesis.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)

Create a function R^n –> R^p For example R^4 –> R

myModel = ot.SymbolicFunction(['x1', 'x2', 'x3', 'x4'], [
                              '1+x1*x2 + 2*x3^2+x4^4'])

# Create a distribution of dimension n
# for example n=3 with indpendent components
distribution = ot.ComposedDistribution(
    [ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)])

Prepare the input/output samples

sampleSize = 250
X = distribution.getSample(sampleSize)
Y = myModel(X)
dimension = X.getDimension()

build the orthogonal basis

coll = [ot.StandardDistributionPolynomialFactory(
    distribution.getMarginal(i)) for i in range(dimension)]
enumerateFunction = ot.LinearEnumerateFunction(dimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)

create the algorithm

degree = 6
adaptiveStrategy = ot.FixedStrategy(
    productBasis, enumerateFunction.getStrataCumulatedCardinal(degree))
projectionStrategy = ot.LeastSquaresStrategy()
algo = ot.FunctionalChaosAlgorithm(
    X, Y, distribution, adaptiveStrategy, projectionStrategy)
algo.run()

get the metamodel function

result = algo.getResult()
metamodel = result.getMetaModel()

Print residuals

result.getResiduals()

[2.64115e-15]