FunctionalChaosResult

class FunctionalChaosResult(*args)

Functional chaos result.

Returned by functional chaos algorithms, see FunctionalChaosAlgorithm.

Parameters
modelFunction

The physical model, that maps the physical input \vect{X} \in \mathbb{R}^{n_X} to the output \vect{Y} \in \mathbb{R}^{n_Y}.

distributionDistribution

Distribution of the random vector \vect{X}

transformationFunction

The function that maps the physical input \vect{X} to the standardized input \vect{\xi}.

inverseTransformationFunction

The function that maps standardized input \vect{\xi} to the the physical input \vect{X}.

composedModelFunction

The functional chaos expansion model, that maps the standardized input \vect{\xi} to the predicted output \vect{Y}. This is the composition of the model and the inverseTransformation.

orthogonalBasisOrthogonalBasis

The multivariate orthogonal basis.

indicessequence of int

The indices of the selected basis function within the orthogonal basis.

alpha_k2-d sequence of float

The coefficients of the functional chaos expansion.

Psi_ksequence of Function

The functions of the multivariate basis selected by the algorithm.

residualssequence of float, \hat{\vect{r}} \in \mathbb{R}^{n_Y}

For each output component, the residual is the square root of the sum of squared differences between the model and the meta model, divided by the sample size.

relativeErrorssequence of float, \widehat{\vect{re}} \in \mathbb{R}^{n_Y}

The relative error is the empirical error divided by the sample variance of the output.

Notes

Let n \in \mathbb{N} be the sample size. Let n_Y \in \mathbb{N} be the dimension of the output of the physical model. For any j = 1, ..., n and any i = 1, ..., n_Y, let y_{j, i} \in \mathbb{R} be the output of the physical model and let \hat{y}_{j, i} \in \mathbb{R} be the output of the metamodel. For any i = 1, ..., n_Y, let \vect{y}_i \in \mathbb{R}^n be the sample output and let \hat{\vect{y}}_i \in \mathbb{R}^n be the output predicted by the metamodel. The marginal residual is:

\hat{r}_i = \frac{\sqrt{SS_i}}{n}

for i = 1, ..., n_Y, where SS_i is the marginal sum of squares:

SS_i = \sum_{j = 1}^n (y_{j, i} - \hat{y}_{j, i})^2.

The marginal relative error is:

\widehat{re}_i = \frac{\hat{r}_i / n}{\hat{s}_{Y, i}^2}

for i = 1, ..., n_Y, where \hat{s}_{Y, i}^2 is the unbiased sample variance of the i-th output.

This structure is created by the method run() of FunctionalChaosAlgorithm, and obtained thanks to the getResult() method.

Methods

getClassName()

Accessor to the object's name.

getCoefficients()

Get the coefficients.

getComposedMetaModel()

Get the composed metamodel.

getComposedModel()

Get the composed model.

getDistribution()

Get the input distribution.

getId()

Accessor to the object's id.

getIndices()

Get the indices of the final basis.

getInverseTransformation()

Get the inverse isoprobabilistic transformation.

getMetaModel()

Accessor to the metamodel.

getModel()

Accessor to the model.

getName()

Accessor to the object's name.

getOrthogonalBasis()

Get the orthogonal basis.

getReducedBasis()

Get the reduced basis.

getRelativeErrors()

Accessor to the relative errors.

getResiduals()

Accessor to the residuals.

getShadowedId()

Accessor to the object's shadowed id.

getTransformation()

Get the isoprobabilistic transformation.

getVisibility()

Accessor to the object's visibility state.

hasName()

Test if the object is named.

hasVisibleName()

Test if the object has a distinguishable name.

setMetaModel(metaModel)

Accessor to the metamodel.

setModel(model)

Accessor to the model.

setName(name)

Accessor to the object's name.

setRelativeErrors(relativeErrors)

Accessor to the relative errors.

setResiduals(residuals)

Accessor to the residuals.

setShadowedId(id)

Accessor to the object's shadowed id.

setVisibility(visible)

Accessor to the object's visibility state.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns
class_namestr

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

getCoefficients()

Get the coefficients.

Returns
coefficients2-d sequence of float

Coefficients (\vect{\alpha_k})_{k \in K}.

getComposedMetaModel()

Get the composed metamodel.

Returns
composedMetamodelFunction

\tilde{f} = \sum_{k \in K} \vect{\alpha}_k \Psi_k

getComposedModel()

Get the composed model.

Returns
composedModelFunction

f = g\circ T^{-1}.

getDistribution()

Get the input distribution.

Returns
distributionDistribution

Distribution of the input random vector \vect{X}.

getId()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getIndices()

Get the indices of the final basis.

Returns
indicesIndices

Indices of the elements of the multivariate basis used in the decomposition.

getInverseTransformation()

Get the inverse isoprobabilistic transformation.

Returns
invTransfFunction

T^{-1} such that T(\vect{X}) = \vect{Z}.

getMetaModel()

Accessor to the metamodel.

Returns
metaModelFunction

Metamodel.

getModel()

Accessor to the model.

Returns
modelFunction

Physical model approximated by a metamodel.

getName()

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOrthogonalBasis()

Get the orthogonal basis.

Returns
basisOrthogonalBasis

Factory of the orthogonal basis.

getReducedBasis()

Get the reduced basis.

Returns
basislist of Function

Collection of the K functions (\Psi_k)_{k\in K} used in the decomposition.

getRelativeErrors()

Accessor to the relative errors.

Returns
relativeErrorsPoint

The relative errors defined as follows for each output of the model: \displaystyle \frac{\sum_{i=1}^N (y_i - \hat{y_i})^2}{N \Var{\vect{Y}}} with \vect{Y} the vector of the N model’s values y_i and \hat{y_i} the metamodel’s values.

getResiduals()

Accessor to the residuals.

Returns
residualsPoint

The residual values defined as follows for each output of the model: \displaystyle \frac{\sqrt{\sum_{i=1}^N (y_i - \hat{y_i})^2}}{N} with y_i the N model’s values and \hat{y_i} the metamodel’s values.

getShadowedId()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getTransformation()

Get the isoprobabilistic transformation.

Returns
transformationFunction

Transformation T such that T(\vect{X}) = \vect{Z}.

getVisibility()

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

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.

setMetaModel(metaModel)

Accessor to the metamodel.

Parameters
metaModelFunction

Metamodel.

setModel(model)

Accessor to the model.

Parameters
modelFunction

Physical model approximated by a metamodel.

setName(name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setRelativeErrors(relativeErrors)

Accessor to the relative errors.

Parameters
relativeErrorssequence of float

The relative errors defined as follows for each output of the model: \displaystyle \frac{\sum_{i=1}^N (y_i - \hat{y_i})^2}{N \Var{\vect{Y}}} with \vect{Y} the vector of the N model’s values y_i and \hat{y_i} the metamodel’s values.

setResiduals(residuals)

Accessor to the residuals.

Parameters
residualssequence of float

The residual values defined as follows for each output of the model: \displaystyle \frac{\sqrt{\sum_{i=1}^N (y_i - \hat{y_i})^2}}{N} with y_i the N model’s values and \hat{y_i} the metamodel’s values.

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.