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.