FunctionalChaosAlgorithm¶

class FunctionalChaosAlgorithm(*args)¶

Functional chaos algorithm.

Refer to Functional Chaos Expansion, Least squares polynomial response surface.

Available constructors:

FunctionalChaosAlgorithm(inputSample, outputSample)

FunctionalChaosAlgorithm(inputSample, outputSample, distribution)

FunctionalChaosAlgorithm(inputSample, outputSample, distribution, adaptiveStrategy)

FunctionalChaosAlgorithm(inputSample, outputSample, distribution, adaptiveStrategy, projectionStrategy)

FunctionalChaosAlgorithm(inputSample, weights, outputSample, distribution, adaptiveStrategy)

FunctionalChaosAlgorithm(inputSample, weights, outputSample, distribution, adaptiveStrategy, projectionStrategy)

Parameters:

inputSample: 2-d sequence of float: Sample of the input random vectors with size $\sampleSize$ and dimension $\inputDim$ .
outputSample: 2-d sequence of float: Sample of the output random vectors with size $\sampleSize$ and dimension $\outputDim$ .
distributionDistribution: Distribution of the random vector $\inputRV$ of dimension $\inputDim$ . When the distribution is unspecified, the BuildDistribution() static method is evaluated on the inputSample.
adaptiveStrategyAdaptiveStrategy: Strategy of selection of the different terms of the multivariate basis.
projectionStrategyProjectionStrategy: Strategy of evaluation of the coefficients $a_k$
weightssequence of float: Weights $(w_i)_{i = 1, ..., \sampleSize}$ associated to the output sample, where $\sampleSize$ is the sample size. Default values are $w_i = \frac{1}{\sampleSize}$ for $i = 1, ..., \sampleSize$ .

Methods

`BuildDistribution`(inputSample)	Recover the distribution, with metamodel performance in mind.
`getAdaptiveStrategy`()	Get the adaptive strategy.
`getClassName`()	Accessor to the object's name.
`getDistribution`()	Accessor to the joint probability density function of the physical input vector.
`getInputSample`()	Accessor to the input sample.
`getMaximumResidual`()	Get the maximum residual.
`getName`()	Accessor to the object's name.
`getOutputSample`()	Accessor to the output sample.
`getProjectionStrategy`()	Get the projection strategy.
`getResult`()	Get the results of the metamodel computation.
`getWeights`()	Return the weights of the input sample.
`hasName`()	Test if the object is named.
`run`()	Compute the metamodel.
`setDistribution`(distribution)	Accessor to the joint probability density function of the physical input vector.
`setMaximumResidual`(residual)	Set the maximum residual.
`setName`(name)	Accessor to the object's name.
`setProjectionStrategy`(projectionStrategy)	Set the projection strategy.

See also

FunctionalChaosResult

Notes

This class creates a functional chaos expansion or polynomial chaos expansion (PCE) based on an input and output sample of the physical model. More details on this type of surrogate models are presented in Functional Chaos Expansion. Once the expansion is computed, the FunctionalChaosRandomVector class provides methods to get the mean and variance of the PCE. Moreover, the FunctionalChaosSobolIndices provides Sobol’ indices of the PCE.

Default settings for the adaptive strategy

When the adaptiveStrategy is unspecified, the FunctionalChaosAlgorithm-QNorm parameter of the ResourceMap is used. If this parameter is equal to 1, then the LinearEnumerateFunction class is used. Otherwise, the HyperbolicAnisotropicEnumerateFunction class is used. If the FunctionalChaosAlgorithm-BasisSize key of the ResourceMap is nonzero, then this parameter sets the basis size. Otherwise, the FunctionalChaosAlgorithm-MaximumTotalDegree key of the ResourceMap is used to compute the basis size using the getBasisSizeFromTotalDegree method of the orthogonal basis (with a maximum of $n$ terms due to the sample size). Finally, the FixedStrategy class is used.

Default settings for the projection strategy

When the projectionStrategy is unspecified, the FunctionalChaosAlgorithm-Sparse key of the ResourceMap is used. If it is false, then the LeastSquaresStrategy class is used, which produces a full PCE, without model selection. Otherwise, a LARS PCE is created, i.e. a sparse PCE is computed using model selection. In this case, the FunctionalChaosAlgorithm-FittingAlgorithm key of the ResourceMap is used.

If this key is equal to ‘CorrectedLeaveOneOut’, then the CorrectedLeaveOneOut criteria is used.
If this key is equal to ‘KFold’, then the KFold criteria is used.
Otherwise, an exception is produced.

Examples

Create the model:

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> inputDimension = 1
>>> model = ot.SymbolicFunction(['x'], ['x * sin(x)'])
>>> distribution = ot.JointDistribution([ot.Uniform()] * inputDimension)

Build the multivariate orthonormal basis:

>>> polyColl = [0.0] * inputDimension
>>> for i in range(distribution.getDimension()):
...     polyColl[i] = ot.StandardDistributionPolynomialFactory(distribution.getMarginal(i))
>>> enumerateFunction = ot.LinearEnumerateFunction(inputDimension)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)

Define the strategy to truncate the multivariate orthonormal basis: We choose all the polynomials of degree lower or equal to 4.

>>> degree = 4
>>> indexMax = enumerateFunction.getBasisSizeFromTotalDegree(degree)
>>> print(indexMax)
5

We keep all the polynomials of degree lower or equal to 4 (which corresponds to the 5 first ones):

>>> adaptiveStrategy = ot.FixedStrategy(productBasis, indexMax)

Define the evaluation strategy of the coefficients:

>>> samplingSize = 50
>>> experiment = ot.MonteCarloExperiment(distribution, samplingSize)
>>> inputSample = experiment.generate()
>>> outputSample = model(inputSample)
>>> projectionStrategy = ot.LeastSquaresStrategy()

Create the chaos algorithm:

>>> algo = ot.FunctionalChaosAlgorithm(
...     inputSample, outputSample, distribution, adaptiveStrategy, projectionStrategy
... )
>>> algo.run()

Get the result:

>>> functionalChaosResult = algo.getResult()
>>> # print(functionalChaosResult)  # Pretty-print
>>> metamodel = functionalChaosResult.getMetaModel()

Test it:

>>> X = [0.5]
>>> print(model(X))
[0.239713]
>>> print(metamodel(X))
[0.239514]

There are several methods to define the algorithm: default settings are used when the information is not provided by the user. The simplest is to set only the input and output samples. In this case, the distribution and its parameters are estimated from the inputSample using the BuildDistribution class. See the Fit a distribution from an input sample example for more details on this topic.

>>> algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample)

In many cases, the distribution is known and it is best to use this information when we have it.

>>> algo = ot.FunctionalChaosAlgorithm(inputSample, outputSample, distribution)

A more involved method is to define the method to set the orthogonal basis of functions or polynomials. We use the OrthogonalProductPolynomialFactory class to define the orthogonal basis of polynomials. Then we use the FixedStrategy to define the maximum number of candidate polynomials to consider in the expansion, up to the total degree equal to 10.

>>> enumerateFunction = ot.LinearEnumerateFunction(inputDimension)
>>> maximumTotalDegree = 10
>>> totalSize = enumerateFunction.getBasisSizeFromTotalDegree(maximumTotalDegree)
>>> polynomialsList = []
>>> for i in range(inputDimension):
...     marginalDistribution = distribution.getMarginal(i)
...     marginalPolynomial = ot.StandardDistributionPolynomialFactory(marginalDistribution)
...     polynomialsList.append(marginalPolynomial)
>>> basis = ot.OrthogonalProductPolynomialFactory(polynomialsList, enumerateFunction)
>>> adaptiveStrategy = ot.FixedStrategy(basis, totalSize)
>>> algo = ot.FunctionalChaosAlgorithm(
...     inputSample, outputSample, distribution, adaptiveStrategy
... )

The most involved method is to define the way to compute the coefficients, thanks to the ProjectionStrategy. In the next example, we use LARS to create a sparse PCE.

>>> selection = ot.LeastSquaresMetaModelSelectionFactory(
...     ot.LARS(),
...     ot.CorrectedLeaveOneOut()
... )
>>> projectionStrategy = ot.LeastSquaresStrategy(inputSample, outputSample, selection)
>>> algo = ot.FunctionalChaosAlgorithm( 
...     inputSample, outputSample, distribution, adaptiveStrategy, projectionStrategy
... )

__init__(*args)¶

static BuildDistribution(inputSample)¶

Recover the distribution, with metamodel performance in mind.

For each marginal, find the best 1-d continuous parametric model else fallback to the use of a nonparametric one.

The selection is done as follow:

We start with a list of all parametric models (all factories)

For each model, we estimate its parameters if feasible.

We check then if model is valid, ie if its Kolmogorov score exceeds a threshold fixed in the MetaModelAlgorithm-PValueThreshold ResourceMap key. Default value is 5%

We sort all valid models and return the one with the optimal criterion.

For the last step, the criterion might be BIC, AIC or AICC. The specification of the criterion is done through the MetaModelAlgorithm-ModelSelectionCriterion ResourceMap key. Default value is fixed to BIC. Note that if there is no valid candidate, we estimate a non-parametric model (KernelSmoothing or Histogram). The MetaModelAlgorithm-NonParametricModel ResourceMap key allows selecting the preferred one. Default value is Histogram

One each marginal is estimated, we use the Spearman independence test on each component pair to decide whether an independent copula. In case of non independence, we rely on a NormalCopula.

Parameters: