Advanced polynomial chaos construction

In this example we are going to expose advanced elements in the construction of a polynomial chaos algorithm:

• construction of the multivariate orthonormal basis,

• truncature strategy of the multivariate orthonormal basis,

• evaluation strategy of the approximation coefficients.

In this example, we consider the following function :

for any .

We assume that the inputs have normal, uniform, gamma and beta distributions :

and , , and are independent.

Define the model and the input distribution

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

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

Create a distribution of dimension 4.

distribution = ot.ComposedDistribution(
    [ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)])

inputDimension = distribution.getDimension()
inputDimension

Out:

4

STEP 1: Construction of the multivariate orthonormal basis

Create the univariate polynomial family collection which regroups the polynomial families for each direction.

polyColl = ot.PolynomialFamilyCollection(inputDimension)

We could use the Krawtchouk and Charlier families (for discrete distributions).

polyColl[0] = ot.KrawtchoukFactory()
polyColl[1] = ot.CharlierFactory()

We could also use the automatic selection of the polynomial which corresponds to the distribution: this is done with the StandardDistributionPolynomialFactory class.

for i in range(inputDimension):
    marginal = distribution.getMarginal(i)
    polyColl[i] = ot.StandardDistributionPolynomialFactory(marginal)

In our specific case, we use specific polynomial factories.

polyColl[0] = ot.HermiteFactory()
polyColl[1] = ot.LegendreFactory()
polyColl[2] = ot.LaguerreFactory(2.75)
# Parameter for the Jacobi factory : 'Probabilty' encoded with 1
polyColl[3] = ot.JacobiFactory(2.5, 3.5, 1)

Create the enumeration function.

The first possibility is to use the LinearEnumerateFunction.

enumerateFunction = ot.LinearEnumerateFunction(inputDimension)

Another possibility is to use the HyperbolicAnisotropicEnumerateFunction, which gives less weight to interactions.

q = 0.4
enumerateFunction_1 = ot.HyperbolicAnisotropicEnumerateFunction(
    inputDimension, q)

Create the multivariate orthonormal basis which is the the cartesian product of the univariate basis.

multivariateBasis = ot.OrthogonalProductPolynomialFactory(
    polyColl, enumerateFunction)

Ask how many polynomials have total degrees equal to k=5.

k = 5
enumerateFunction.getStrataCardinal(k)

Out:

56

Ask how many polynomials have degrees lower or equal to k=5.

enumerateFunction.getStrataCumulatedCardinal(k)

Out:

126

Give the k-th term of the multivariate basis. To calculate its degree, add the integers.

k = 5
enumerateFunction(k)

[2,0,0,0]

Build a term of the basis as a Function. Generally, we do not need to construct manually any term, all terms are constructed automatically by a strategy of construction of the basis.

i = 5
Psi_i = multivariateBasis.build(i)
Psi_i

-0.707107 + 0.707107 * x0^2

Get the measure mu associated to the multivariate basis.

distributionStandard = multivariateBasis.getMeasure()
print(distributionStandard)

Out:

ComposedDistribution(Normal(mu = 0, sigma = 1), Uniform(a = -1, b = 1), Gamma(k = 3.75, lambda = 1, gamma = 0), Beta(alpha = 2.5, beta = 1, a = -1, b = 1), IndependentCopula(dimension = 4))

STEP 2: Truncature strategy of the multivariate orthonormal basis

FixedStrategy : all the polynomials af degree lower or equal to 2 which corresponds to the 15 first ones.

p = 15
truncatureBasisStrategy = ot.FixedStrategy(multivariateBasis, p)

SequentialStrategy : among the maximumCardinalBasis = 100 first polynomials of the multivariate basis those verfying the convergence criterion.

maximumCardinalBasis = 100
truncatureBasisStrategy_1 = ot.SequentialStrategy(
    multivariateBasis, maximumCardinalBasis)

CleaningStrategy : among the maximumConsideredTerms = 500 first polynomials, those which have the mostSignificant = 50 most significant contributions with significance criterion significanceFactor equal to The True boolean indicates if we are interested in the online monitoring of the current basis update (removed or added coefficients).

maximumConsideredTerms = 500
mostSignificant = 50
significanceFactor = 1.0e-4
truncatureBasisStrategy_2 = ot.CleaningStrategy(
    multivariateBasis, maximumConsideredTerms, mostSignificant, significanceFactor, True)

STEP 3: Evaluation strategy of the approximation coefficients

The technique illustrated is the Least Squares technique where the points come from an design of experiments. Here : the Monte Carlo sampling technique.

sampleSize = 100
evaluationCoeffStrategy = ot.LeastSquaresStrategy(
    ot.MonteCarloExperiment(sampleSize))

You can specify the approximation algorithm. This is the algorithm that generates a sequence of basis using Least Angle Regression.

basisSequenceFactory = ot.LARS()

This algorithm estimates the empirical error on each sub-basis using Leave One Out strategy.

fittingAlgorithm = ot.CorrectedLeaveOneOut()
# Finally the metamodel selection algorithm embbeded in LeastSquaresStrategy
approximationAlgorithm = ot.LeastSquaresMetaModelSelectionFactory(
    basisSequenceFactory, fittingAlgorithm)
evaluationCoeffStrategy_2 = ot.LeastSquaresStrategy(
    ot.MonteCarloExperiment(sampleSize), approximationAlgorithm)

Try integration.

marginalSizes = [2] * inputDimension
evaluationCoeffStrategy_3 = ot.IntegrationStrategy(
    ot.GaussProductExperiment(distributionStandard, marginalSizes))

STEP 4: Creation of the Functional Chaos Algorithm

The FunctionalChaosAlgorithm class combines

• the model : model

• the distribution of the input random vector : distribution

• the truncature strategy of the multivariate basis

• and the evaluation strategy of the coefficients

polynomialChaosAlgorithm = ot.FunctionalChaosAlgorithm(
    model, distribution, truncatureBasisStrategy, evaluationCoeffStrategy)