LeastSquaresStrategy

class LeastSquaresStrategy(*args)

Least squares strategy for the approximation coefficients.

Available constructors:

LeastSquaresStrategy(weightedExp)

LeastSquaresStrategy(weightedExp, 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.

computeCoefficients
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

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.