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.
inputSample2-d sequence of float: The input random observations $\left\{\vect{X}^{(1)}, ..., \vect{X}^{(n)}\right\}$ where $\vect{X}=(X_1, \dots, X_{n_X})^T$ is the input of the physical model, $n_X$ is the input dimension and $n$ is the sample size.
outputSample2-d sequence of float: The output random observations $\left\{\vect{Y}^{(1)}, ..., \vect{Y}^{(n)}\right\}$ where $\vect{Y}=(Y_1, \dots, Y_{n_Y})^T$ is the output of the physical model, $n_Y$ is the output dimension and $n$ is the sample size.
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 $\omega_i = \frac{1}{n}$ , where $n$ 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.
`getDesignProxy`()	Accessor to the design proxy.
`getExperiment`()	Accessor to the experiments.
`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.
`getWeights`()	Accessor to the weights.
`hasName`()	Test if the object is named.
`involvesModelSelection`()	Get the model selection flag.
`isLeastSquares`()	Get the least squares flag.
`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.
`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 $(\alpha_k)_{k \in K}$ .

getDesignProxy()¶

Accessor to the design proxy.

Parameters:

designProxyDesignProxy: The design matrix.

getExperiment()¶

Accessor to the experiments.

Returns:

expWeightedExperiment: Weighted experiment used to evaluate the coefficients.

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.

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.

involvesModelSelection()¶

Get the model selection flag.

A model selection method can be used to select the coefficients of the decomposition which enable to best predict the output. Model selection can lead to a sparse functional chaos expansion.

Returns: