OpenTURNS
Note
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Create a full or sparse polynomial chaos expansion¶
In this example we create a global approximation of a model using polynomial chaos expansion based on a design of experiment. The goal of this example is to show how we can create a full or sparse polynomial chaos expansion depending on our needs and depending on the number of observations we have. In general, we should have more observations than parameters to estimate. This is why a sparse polynomial chaos may be interesting: by carefully selecting the coefficients we estimate, we may reduce overfitting and increase the predictions of the metamodel.
import openturns as ot
ot.Log.Show(ot.Log.NONE)
Define the model¶
Create the function.
myModel = ot.SymbolicFunction(
["x1", "x2", "x3", "x4"], ["1 + x1 * x2 + 2 * x3^2 + x4^4"]
)
Create a multivariate distribution.
distribution = ot.JointDistribution(
[ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)]
)
In order to create the PCE, we can 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. Please read Fit a distribution from an input sample for an example of this method.
Create a training sample¶
Create a pair of input and output samples.
sampleSize = 250
inputSample = distribution.getSample(sampleSize)
outputSample = myModel(inputSample)
Build the orthogonal basis¶
In the next cell, we create the univariate orthogonal polynomial basis for each marginal.
inputDimension = inputSample.getDimension()
coll = [
ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
for i in range(inputDimension)
]
enumerateFunction = ot.LinearEnumerateFunction(inputDimension)
productBasis = ot.OrthogonalProductPolynomialFactory(coll, enumerateFunction)
We can achieve the same result using OrthogonalProductPolynomialFactory.
marginalDistributionCollection = [
distribution.getMarginal(i) for i in range(inputDimension)
]
multivariateBasis = ot.OrthogonalProductPolynomialFactory(
marginalDistributionCollection
)
multivariateBasis
Create a full PCE¶
Create the algorithm.
We compute the basis size from the total degree.
The next lines use the LeastSquaresStrategy class
with default parameters (the default is the
PenalizedLeastSquaresAlgorithmFactory class).
This creates a full polynomial chaos expansion, i.e.
we keep all the candidate coefficients produced by the enumeration
rule.
In order to create a sparse polynomial chaos expansion, we
must use the LeastSquaresMetaModelSelectionFactory
class instead.
totalDegree = 3
candidateBasisSize = enumerateFunction.getBasisSizeFromTotalDegree(totalDegree)
print("Candidate basis size = ", candidateBasisSize)
adaptiveStrategy = ot.FixedStrategy(productBasis, candidateBasisSize)
projectionStrategy = ot.LeastSquaresStrategy()
algo = ot.FunctionalChaosAlgorithm(
inputSample, outputSample, distribution, adaptiveStrategy, projectionStrategy
)
algo.run()
result = algo.getResult()
result
Candidate basis size = 35
Get the number of coefficients in the PCE.
selectedBasisSizeFull = result.getIndices().getSize()
print("Selected basis size = ", selectedBasisSizeFull)
Selected basis size = 35
We see that the number of coefficients in the selected basis is equal to the number of coefficients in the candidate basis. This is, indeed, a full PCE.
Use the PCE¶
Get the metamodel function.
metamodel = result.getMetaModel()
In order to evaluate the metamodel on a single point, we just
use it as any other openturns.Function.
xPoint = distribution.getMean()
yPoint = metamodel(xPoint)
print("Value at ", xPoint, " is ", yPoint)
Value at [0,0,2.75,1.14286] is [17.7455]
Print residuals.
result.getResiduals()
Based on these results, we may want to validate our metamodel. More details on this topic are presented in Validate a polynomial chaos.
Create a sparse PCE¶
In order to create a sparse polynomial chaos expansion, we
use the LeastSquaresMetaModelSelectionFactory
class instead.
totalDegree = 6
candidateBasisSize = enumerateFunction.getBasisSizeFromTotalDegree(totalDegree)
print("Candidate basis size = ", candidateBasisSize)
adaptiveStrategy = ot.FixedStrategy(productBasis, candidateBasisSize)
selectionAlgorithm = ot.LeastSquaresMetaModelSelectionFactory()
projectionStrategy = ot.LeastSquaresStrategy(selectionAlgorithm)
algo = ot.FunctionalChaosAlgorithm(
inputSample, outputSample, distribution, adaptiveStrategy, projectionStrategy
)
algo.run()
result = algo.getResult()
result
Candidate basis size = 210
Get the number of coefficients in the PCE.
selectedBasisSizeSparse = result.getIndices().getSize()
print("Selected basis size = ", selectedBasisSizeSparse)
Selected basis size = 24
We see that the number of selected coefficients is lower than the number of candidate coefficients. This may reduce overfitting and can produce a PCE with more accurate predictions.