FunctionalChaosResult¶

class FunctionalChaosResult(*args)¶

Functional chaos result.

Returned by functional chaos algorithms, see FunctionalChaosAlgorithm.

Parameters:

sampleX2-d sequence of float: Input sample of $\vect{X} \in \mathbb{R}^{n_X}$ .
sampleY2-d sequence of float: Output sample of $\vect{Y} \in \mathbb{R}^{n_Y}$ .
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 $\vect{X}$ .
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

`drawErrorHistory`()	Draw the error history.
`drawSelectionHistory`()	Draw the basis selection history.
`getClassName`()	Accessor to the object's name.
`getCoefficients`()	Get the coefficients.
`getCoefficientsHistory`()	The coefficients values selection history accessor.
`getComposedMetaModel`()	Get the composed metamodel.
`getDistribution`()	Get the input distribution.
`getErrorHistory`()	The error history accessor.
`getIndices`()	Get the indices of the final basis.
`getIndicesHistory`()	The basis indices selection history accessor.
`getInputSample`()	Accessor to the input sample.
`getInverseTransformation`()	Get the inverse isoprobabilistic transformation.
`getMetaModel`()	Accessor to the metamodel.
`getName`()	Accessor to the object's name.
`getOrthogonalBasis`()	Get the orthogonal basis.
`getOutputSample`()	Accessor to the output sample.
`getReducedBasis`()	Get the reduced basis.
`getRelativeErrors`()	Accessor to the relative errors.
`getResiduals`()	Accessor to the residuals.
`getSampleResiduals`()	Get residuals sample.
`getTransformation`()	Get the isoprobabilistic transformation.
`hasName`()	Test if the object is named.
`involvesModelSelection`()	Get the model selection flag.
`isLeastSquares`()	Get the least squares flag.
`setErrorHistory`(errorHistory)	The error history accessor.
`setInputSample`(sampleX)	Accessor to the input sample.
`setIsLeastSquares`(isLeastSquares)	Set the least squares flag.
`setMetaModel`(metaModel)	Accessor to the metamodel.
`setName`(name)	Accessor to the object's name.
`setOutputSample`(sampleY)	Accessor to the output sample.
`setRelativeErrors`(relativeErrors)	Accessor to the relative errors.
`setResiduals`(residuals)	Accessor to the residuals.
`setSelectionHistory`(indicesHistory, ...)	The basis coefficients and indices accessor.

setInvolvesModelSelection

__init__(*args)¶

drawErrorHistory()¶

Draw the error history.

This is only available with LARS, and when the output dimension is 1.

Returns:

graphGraph: The evolution of the error at each selection iteration

drawSelectionHistory()¶

Draw the basis selection history.

This is only available with LARS, and when the output dimension is 1.

Returns:

graphGraph: The evolution of the basis coefficients at each selection iteration

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

getCoefficientsHistory()¶

The coefficients values selection history accessor.

This is only available with LARS, and when the output dimension is 1.

Returns:

coefficientsHistory2-d sequence of float: The coefficients values selection history, for each iteration. Each inner list gives the coefficients values of the basis terms at i-th iteration.

getComposedMetaModel()¶

Get the composed metamodel.

Returns:

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

getDistribution()¶

Get the input distribution.

Returns:

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

getErrorHistory()¶

The error history accessor.

This is only available with LARS, and when the output dimension is 1.

Returns:

errorHistorysequence of float: The error history

getIndices()¶

Get the indices of the final basis.

Returns:

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

getIndicesHistory()¶

The basis indices selection history accessor.

This is only available with LARS, and when the output dimension is 1.

Returns:

indicesHistory2-d sequence of int: The basis indices selection history, for each iteration. Each inner list gives the indices of the basis terms at i-th iteration.

getInputSample()¶

Accessor to the input sample.

Returns:

inputSampleSample: The input sample.

getInverseTransformation()¶

Get the inverse isoprobabilistic transformation.

Returns:

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

getMetaModel()¶

Accessor to the metamodel.

Returns:

metaModelFunction: 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.

getOutputSample()¶

Accessor to the output sample.

Returns:

outputSampleSample: The output sample.

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.

getSampleResiduals()¶

Get residuals sample.

Returns:

residualsSampleSample: The sample of residuals $r_{ji} = y_{ji} - \metamodel_i(\vect{x^{(j)}})$ for $i = 1, ..., n_Y$ and $j = 1, ..., n$ .

getTransformation()¶

Get the isoprobabilistic transformation.

Returns:

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

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.

Returns:

isLeastSquaresbool: True if the coefficients were estimated from least squares.

setErrorHistory(errorHistory)¶

The error history accessor.

Parameters:

errorHistorysequence of float: The error history

setInputSample(sampleX)¶

Accessor to the input sample.

Parameters:

inputSampleSample: The input sample.

setIsLeastSquares(isLeastSquares)¶

Set the least squares flag.

Parameters:

isLeastSquaresbool: True if the coefficients were estimated from least squares.

setMetaModel(metaModel)¶

Accessor to the metamodel.

Parameters:

metaModelFunction: Metamodel.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOutputSample(sampleY)¶

Accessor to the output sample.

Parameters:

outputSampleSample: The output sample.

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.

setSelectionHistory(indicesHistory, coefficientsHistory)¶

The basis coefficients and indices accessor.

Parameters:

indicesHistory2-d sequence of int: The basis indices selection history
coefficientsHistory2-d sequence of float: The coefficients values selection history Must be of same size as indicesHistory.