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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 \mathbb{R}^4 \rightarrow \mathbb{R}:

g(\mathbf{x}) = 1+x_1 x_2 + 2 x_3^2+x_4^4

for any x_1,x_2,x_3,x_4\in\mathbb{R}.

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

X_1 \sim \mathcal{N}(0,1), \qquad X_2 \sim \mathcal{U}(-1,1), \qquad X_3 \sim \mathcal{G}(2.75,1), \qquad X_4 \sim \mathcal{B}(2.5,1,-1,2),

and X_1, X_2, X_3 and X_4 are independent.

Define the model and the input distribution

import openturns as ot

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

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)
multivariateBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
multivariateBasis
  • dimension: 4
  • enumerate function: class=LinearEnumerateFunction dimension=4
Index Name Distribution Univariate polynomial
0 X0 Normal HermiteFactory
1 X1 Uniform LegendreFactory
2 X2 Gamma LaguerreFactory
3 X3 Beta JacobiFactory


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

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

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

multivariateBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
multivariateBasis
  • dimension: 4
  • enumerate function: class=HyperbolicAnisotropicEnumerateFunction derived from class=HyperbolicAnisotropicEnumerateFunction dimension=4 q=0.4 weights=class=Point name=Unnamed dimension=4 values=[1,1,1,1]
Index Name Distribution Univariate polynomial
0 X0 Normal HermiteFactory
1 X1 Uniform LegendreFactory
2 X2 Gamma LaguerreFactory
3 X3 Beta JacobiFactory


Ask how many basis terms there are in the 6-th strata. In the special case of the Linear enumerate function this is also the strata with all the multi-indices of total degree 5.

k = 5
enumerateFunction.getStrataCardinal(k)

4

Ask how many basis multi-indices have total degrees lower or equal to k=5.

enumerateFunction.getBasisSizeFromTotalDegree(k)

27

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()
distributionStandard

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)

CleaningStrategy : among the maximumConsideredTerms = 500 first polynomials, those which have the mostSignificant = 50 most significant contributions with significance criterion significanceFactor equal to 10^{-4} 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()
experiment = ot.MonteCarloExperiment(distribution, 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(approximationAlgorithm)
experiment_2 = experiment

Try integration.

marginalSizes = [2] * inputDimension
evaluationCoeffStrategy_3 = ot.IntegrationStrategy()
experiment_3 = ot.GaussProductExperiment(distribution, marginalSizes)

Evaluate design of experiments. For the Gauss product we need to specify the non-uniform weights.

X, W = experiment.generateWithWeights()
Y = model(X)

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(
    X, W, Y, distribution, truncatureBasisStrategy, evaluationCoeffStrategy
)