# IntegrationExpansion¶

class IntegrationExpansion(*args)

L2 approximation on an orthonormal basis using least-squares and a fixed basis.

Available constructors:

IntegrationExpansion(inputSample, outputSample, distribution)

IntegrationExpansion(inputSample, weights, outputSample, distribution)

IntegrationExpansion(inputSample, outputSample, distribution, basis, basisSize)

IntegrationExpansion(inputSample, weights, outputSample, distribution, basis, basisSize)

Parameters:
inputSample2-d sequence of float of size

Input sample.

weightsequence of float of size , optional

Input weights. By default, a uniform weight equals to is used.

outputSample2-d sequence of floats of size

Output sample.

distributionDistribution

The distribution associated to the weighted sample (inputSample, weights)

basisOrthogonalBasis, optional

The orthonormal basis of the vector space on which the approximation is built. By default, a basis is built using an hyperbolic enumerate function HyperbolicAnisotropicEnumerateFunction associated to the norm specified by the FunctionalChaosAlgorithm-QNorm key in ResourceMap. The basis is obtained through a tensorization of the univariate orthonormal polynomial bases associated to each marginal distributions of distribution.

basisSizeint, optional

The size of the basis on which the approximation is built. If not given, it is given by the FunctionalChaosAlgorithm-BasisSize key in ResourceMap if this value is positive, or as the cardinal of the set of basis functions up to a degree given by the FunctionalChaosAlgorithm-MaximumTotalDegree key in ResourceMap.

LeastSquaresExpansion, FunctionalChaosAlgorithm, IntegrationMethod

Examples

Solves a functional least-squares approximation using a projection on an orthonormal basis:

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> g = ot.SymbolicFunction('x', 'sin(x)')
>>> distribution = ot.Uniform(-2.0, 2.0)
>>> inSample = distribution.getSample(10)
>>> outSample = g(inSample)
>>> basis = ot.OrthogonalProductPolynomialFactory([ot.LegendreFactory()])
>>> basisSize = 5
>>> algo = otexp.IntegrationExpansion(inSample, outSample, distribution, basis, basisSize)
>>> algo.run()


Methods

 BuildDistribution(inputSample) Recover the distribution, with metamodel performance in mind. Accessor to the active functions in the basis. Get the adaptive strategy. Accessor to the object's name. Accessor to the joint probability density function of the physical input vector. Accessor to the object's id. Accessor to the input sample. Get the maximum residual. Accessor to the object's name. Accessor to the output sample. Get the projection strategy. Get the results of the metamodel computation. Accessor to the object's shadowed id. Accessor to the object's visibility state. Return the weights of the input sample. Test if the object is named. Test if the object has a distinguishable name. Compute the metamodel. setActiveFunctions(activeFunctions) Accessor to the active functions in the basis. 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. Accessor to the object's shadowed id. setVisibility(visible) Accessor to the object's visibility state.
__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:
sampleSample

Input sample.

Returns:
distributionDistribution

Input distribution.

getActiveFunctions()

Accessor to the active functions in the basis.

Returns:
activeFunctionssequence of int

The indices of the functions used for the approximation in the basis.

Returns:
adaptiveStrategyAdaptiveStrategy

Strategy of selection of the different terms of the multivariate basis.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getDistribution()

Accessor to the joint probability density function of the physical input vector.

Returns:
distributionDistribution

Joint probability density function of the physical input vector.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getInputSample()

Accessor to the input sample.

Returns:
inputSampleSample

Input sample of a model evaluated apart.

getMaximumResidual()

Get the maximum residual.

Returns:
residualfloat

Residual value needed in the projection strategy.

Default value is .

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getOutputSample()

Accessor to the output sample.

Returns:
outputSampleSample

Output sample of a model evaluated apart.

getProjectionStrategy()

Get the projection strategy.

Returns:
strategyProjectionStrategy

Projection strategy.

Notes

The projection strategy selects the different terms of the multivariate basis to define the subset K.

getResult()

Get the results of the metamodel computation.

Returns:
resultFunctionalChaosResult

Result structure, created by the method run().

Accessor to the object’s shadowed id.

Returns:
idint

Internal unique identifier.

getVisibility()

Accessor to the object’s visibility state.

Returns:
visiblebool

Visibility flag.

getWeights()

Return the weights of the input sample.

Returns:
weightssequence of float

The weights of the points in the input sample.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:
hasVisibleNamebool

True if the name is not empty and not the default one.

run()

Compute the metamodel.

Notes

Evaluates the metamodel and stores all the results in a result structure.

setActiveFunctions(activeFunctions)

Accessor to the active functions in the basis.

Parameters:
activeFunctionssequence of int

The indices of the functions used for the approximation in the basis.

setDistribution(distribution)

Accessor to the joint probability density function of the physical input vector.

Parameters:
distributionDistribution

Joint probability density function of the physical input vector.

setMaximumResidual(residual)

Set the maximum residual.

Parameters:
residualfloat

Residual value needed in the projection strategy.

Default value is .

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setProjectionStrategy(projectionStrategy)

Set the projection strategy.

Parameters:
strategyProjectionStrategy

Strategy to estimate the coefficients .

Accessor to the object’s shadowed id.

Parameters:
idint

Internal unique identifier.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:
visiblebool

Visibility flag.