LeastSquaresStrategy¶

class LeastSquaresStrategy(*args)¶

Least squares strategy for the approximation coefficients.

Available constructors:

LeastSquaresStrategy(weightedExp)

LeastSquaresStrategy(weightedExp, approxAlgoImpFact)

LeastSquaresStrategy(approxAlgoImpFact)

LeastSquaresStrategy(measure, approxAlgoImpFact)

LeastSquaresStrategy(measure, weightedExp, approxAlgoImpFact)

LeastSquaresStrategy(inputSample, outputSample, approxAlgoImpFact)

LeastSquaresStrategy(inputSample, weights, outputSample, approxAlgoImpFact)

Parameters:

weightedExpWeightedExperiment: Experimental design used for the transformed input data. By default the class MonteCarloExperiment is used.
approxAlgoImpFactApproximationAlgorithmImplementationFactory: The factory that builds the desired ApproximationAlgorithm. By default the class PenalizedLeastSquaresAlgorithmFactory is used.
measureDistribution: Distribution $\mu$ with respect to which the basis is orthonormal. By default, the limit measure defined within the class WeightedExperiment is used.
inputSample, outputSample2-d sequence of float: The input random variables $\vect{X}=(X_1, \dots, X_{n_X})^T$ and the output samples $\vect{Y}$ that describe the model.
weightssequence of float: Numerical point that are the weights associated to the input sample points such that the corresponding weighted experiment is a good approximation of $\mu$ . If not precised, all weights are equals to $\omega_i = \frac{1}{size}$ , where $size$ is the size of the sample.

See also

FunctionalChaosAlgorithm, ProjectionStrategy, IntegrationStrategy

Notes

This class is not usable because it has sense only within the FunctionalChaosAlgorithm : the least squares strategy evaluates the coefficients $(a_k)_{k \in K}$ of the polynomials decomposition as follows:

$\vect{a} = \argmin_{\vect{b} \in \Rset^P} E_{\mu} \left[ \left( g \circ T^{-1} (\vect{U}) - \vect{b}^{\intercal} \vect{\Psi}(\vect{U}) \right)^2 \right]$

where $\vect{U} = T(\vect{X})$ .

The mean expectation $E_{\mu}$ is approximated by a relation of type:

$E_{\mu} \left[ f(\vect{U}) \right] \approx \sum_{i \in I} \omega_i f(\Xi_i)$

where is a function $L_1(\mu)$ defined as:

$f(\vect{U} = \left( g \circ T^{-1} (\vect{U}) - \vect{b}^{\intercal} \vect{\Psi}(\vect{U}) \right)^2$

In the approximation of the mean expectation, the set I, the points $(\Xi_i)_{i \in I}$ and the weights $(\omega_i)_{i \in I}$ are evaluated from methods implemented in the WeightedExperiment.

Methods

`getClassName`()	Accessor to the object's name.
`getCoefficients`()	Accessor to the coefficients.
`getExperiment`()	Accessor to the experiments.
`getId`()	Accessor to the object's id.
`getInputSample`()	Accessor to the input sample.
`getMeasure`()	Accessor to the measure.
`getName`()	Accessor to the object's name.
`getOutputSample`()	Accessor to the output sample.
`getRelativeError`()	Accessor to the relative error.
`getResidual`()	Accessor to the residual.
`getShadowedId`()	Accessor to the object's shadowed id.
`getVisibility`()	Accessor to the object's visibility state.
`getWeights`()	Accessor to the weights.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`setExperiment`(weightedExperiment)	Accessor to the design of experiment.
`setInputSample`(inputSample)	Accessor to the input sample.
`setMeasure`(measure)	Accessor to the measure.
`setName`(name)	Accessor to the object's name.
`setOutputSample`(outputSample)	Accessor to the output sample.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.
`setWeights`(weights)	Accessor to the weights.

__init__(*args)¶

getClassName()¶

Accessor to the object’s name.

Returns:

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

getCoefficients()¶

Accessor to the coefficients.

Returns:

coefPoint: Coefficients $(\alpha_k)_{k \in K}$ .

getExperiment()¶

Accessor to the experiments.

Returns:

expWeightedExperiment: Weighted experiment used to evaluate the coefficients.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getInputSample()¶

Accessor to the input sample.

Returns:

XSample: Input Sample.

getMeasure()¶

Accessor to the measure.

Returns:

muDistribution: Measure $\mu$ defining the scalar product.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOutputSample()¶

Accessor to the output sample.

Returns:

YSample: Output Sample.

getRelativeError()¶

Accessor to the relative error.

Returns:

efloat: Relative error.

getResidual()¶

Accessor to the residual.

Returns:

erfloat: Residual error.

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()¶

Accessor to the weights.

Returns:

wPoint: Weights of the design of experiments.

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.

setExperiment(weightedExperiment)¶

Accessor to the design of experiment.

Parameters:

expWeightedExperiment: Weighted design of experiment.

setInputSample(inputSample)¶

Accessor to the input sample.

Parameters:

XSample: Input Sample.

setMeasure(measure)¶

Accessor to the measure.

Parameters:

mDistribution: Measure $\mu$ defining the scalar product.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOutputSample(outputSample)¶

Accessor to the output sample.

Parameters:

YSample: Output Sample.

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.

setWeights(weights)¶

Accessor to the weights.

Parameters:

wPoint: Weights of the design of experiments.

Examples using the class¶

Polynomial chaos over database

Compute grouped indices for the Ishigami function

Validate a polynomial chaos

Advanced polynomial chaos construction

Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos

Chaos cross-validation

Polynomial chaos is sensitive to the degree

Compute Sobol’ indices confidence intervals

Compute Sobol' indices confidence intervals

Metamodel of a field function

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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LeastSquaresStrategy¶

Examples using the class¶