LeastSquaresExpansion¶
- class LeastSquaresExpansion(*args)¶
L2 approximation on an orthonormal basis using least-squares and a fixed basis.
- Available constructors:
LeastSquaresExpansion(inputSample, outputSample, distribution)
LeastSquaresExpansion(inputSample, weights, outputSample, distribution)
LeastSquaresExpansion(inputSample, outputSample, distribution, basis, basisSize, methodName)
LeastSquaresExpansion(inputSample, weights, outputSample, distribution, basis, basisSize, methodName)
- 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.
- distribution
Distribution
The distribution associated to the weighted sample (inputSample, weights)
- basis
OrthogonalBasis
, 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 inResourceMap
. 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 inResourceMap
.- methodNamestr, optional
The decomposition method used to solve the least-squares problem, of type
LeastSquaresMethod
. If not given, it is given by the LeastSquaresExpansion-DecompositionMethod key inResourceMap
.
See also
IntegrationExpansion
,FunctionalChaosAlgorithm
,LeastSquaresMethod
Examples
Solves a functional linear least squares approximation using residual minimization with QR method:
>>> 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.LeastSquaresExpansion(inSample, outSample, distribution, basis, basisSize, 'QR') >>> 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.
getId
()Accessor to the object's id.
Accessor to the input sample.
Get the maximum residual.
getName
()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.
hasName
()Test if the object is named.
Test if the object has a distinguishable name.
run
()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.
setShadowedId
(id)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
orHistogram
). The MetaModelAlgorithm-NonParametricModel ResourceMap key allows selecting the preferred one. Default value is HistogramOne 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:
- sample
Sample
Input sample.
- sample
- Returns:
- distribution
Distribution
Input distribution.
- 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.
- getAdaptiveStrategy()¶
Get the adaptive strategy.
- Returns:
- adaptiveStrategy
AdaptiveStrategy
Strategy of selection of the different terms of the multivariate basis.
- adaptiveStrategy
- 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:
- distribution
Distribution
Joint probability density function of the physical input vector.
- distribution
- getId()¶
Accessor to the object’s id.
- Returns:
- idint
Internal unique identifier.
- getInputSample()¶
Accessor to the input sample.
- Returns:
- inputSample
Sample
Input sample of a model evaluated apart.
- inputSample
- 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:
- outputSample
Sample
Output sample of a model evaluated apart.
- outputSample
- getProjectionStrategy()¶
Get the projection strategy.
- Returns:
- strategy
ProjectionStrategy
Projection strategy.
- 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:
- result
FunctionalChaosResult
Result structure, created by the method
run()
.
- result
- getShadowedId()¶
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:
- distribution
Distribution
Joint probability density function of the physical input vector.
- distribution
- 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:
- strategy
ProjectionStrategy
Strategy to estimate the coefficients .
- strategy
- setShadowedId(id)¶
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.