IntegrationStrategy

class IntegrationStrategy(*args)

Integration strategy for the approximation coefficients.

Available constructors:

IntegrationStrategy(measure)

IntegrationStrategy(weightedExp)

IntegrationStrategy(measure, weightedExp)

IntegrationStrategy(inputSample, outputSample)

IntegrationStrategy(inputSample, weights, outputSample)

Parameters:
weightedExpWeightedExperiment

Experimental design used for the transformed input data. When not precised, OpenTURNS uses a MonteCarloExperiment.

measureDistribution

Distribution \mu with respect to which the basis is orthonormal. When not precised, OpenTURNS uses the limit measure defined within the WeightedExperiment.

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 provided, all weights are equal to \omega_i = \frac{1}{n}, where n is the size of the sample.

See also

FunctionalChaosAlgorithm, ProjectionStrategy, LeastSquaresStrategy

Notes

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

\vect{a} = E_{\mu} \left[ g \circ T^{-1} (\vect{U}) \vect{\Psi}(\vect{U}) \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} = g \circ T^{-1} (\vect{U}) \vect{\Psi}(\vect{U})

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:
involvesModelSelection: bool

True if the method involves a model selection method.

isLeastSquares()

Get the least squares flag.

There are two methods to compute the coefficients: integration or least squares.

Returns:
isLeastSquares: bool

True if the coefficients are estimated from least squares.

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.

setWeights(weights)

Accessor to the weights.

Parameters:
wPoint

Weights of the design of experiments.

Examples using the class

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Advanced polynomial chaos construction

Advanced polynomial chaos construction

Create a sparse chaos by integration

Create a sparse chaos by integration