FunctionalChaosResult¶
- class FunctionalChaosResult(*args)¶
Functional chaos result.
Returned by functional chaos algorithms, see
FunctionalChaosAlgorithm
.- Parameters:
- sampleX2-d sequence of float
Input sample of .
- sampleY2-d sequence of float
Output sample of .
- distribution
Distribution
Distribution of the random vector
- transformation
Function
The function that maps the physical input to the standardized input .
- inverseTransformation
Function
The function that maps standardized input to the the physical input .
- orthogonalBasis
OrthogonalBasis
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,
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,
The relative error is the empirical error divided by the sample variance of the output.
Notes
Let be the sample size. Let be the dimension of the output of the physical model. For any and any , let be the output of the physical model and let be the output of the metamodel. For any , let be the sample output and let be the output predicted by the metamodel. The marginal residual is:
for , where is the marginal sum of squares:
The marginal relative error is:
for , where is the unbiased sample variance of the -th output.
This structure is created by the method run() of
FunctionalChaosAlgorithm
, and obtained thanks to the getResult() method.Methods
Draw the error history.
Draw the basis selection history.
Accessor to the object's name.
Get the coefficients.
The coefficients values selection history accessor.
Get the composed metamodel.
Get the input distribution.
The error history accessor.
Get the indices of the final basis.
The basis indices selection history accessor.
Accessor to the input sample.
Get the inverse isoprobabilistic transformation.
Accessor to the metamodel.
getName
()Accessor to the object's name.
Get the orthogonal basis.
Accessor to the output sample.
Get the reduced basis.
Accessor to the relative errors.
Accessor to the residuals.
Get residuals sample.
Get the isoprobabilistic transformation.
hasName
()Test if the object is named.
Get the model selection flag.
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:
- graph
Graph
The evolution of the error at each selection iteration
- graph
- drawSelectionHistory()¶
Draw the basis selection history.
This is only available with
LARS
, and when the output dimension is 1.- Returns:
- graph
Graph
The evolution of the basis coefficients at each selection iteration
- graph
- 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 .
- 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.
- getDistribution()¶
Get the input distribution.
- Returns:
- distribution
Distribution
Distribution of the input random vector .
- distribution
- 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:
- indices
Indices
Indices of the elements of the multivariate basis used in the decomposition.
- indices
- 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.
- getInverseTransformation()¶
Get the inverse isoprobabilistic transformation.
- Returns:
- invTransf
Function
such that .
- invTransf
- getName()¶
Accessor to the object’s name.
- Returns:
- namestr
The name of the object.
- getOrthogonalBasis()¶
Get the orthogonal basis.
- Returns:
- basis
OrthogonalBasis
Factory of the orthogonal basis.
- basis
- getReducedBasis()¶
Get the reduced basis.
- Returns:
- basislist of
Function
Collection of the K functions used in the decomposition.
- basislist of
- getRelativeErrors()¶
Accessor to the relative errors.
- Returns:
- relativeErrors
Point
The relative errors defined as follows for each output of the model: with the vector of the model’s values and the metamodel’s values.
- relativeErrors
- getResiduals()¶
Accessor to the residuals.
- Returns:
- residuals
Point
The residual values defined as follows for each output of the model: with the model’s values and the metamodel’s values.
- residuals
- getSampleResiduals()¶
Get residuals sample.
- Returns:
- residualsSample
Sample
The sample of residuals for and .
- residualsSample
- getTransformation()¶
Get the isoprobabilistic transformation.
- Returns:
- transformation
Function
Transformation such that .
- transformation
- 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:
- inputSample
Sample
The input sample.
- inputSample
- setIsLeastSquares(isLeastSquares)¶
Set the least squares flag.
- Parameters:
- isLeastSquaresbool
True if the coefficients were estimated from least squares.
- setName(name)¶
Accessor to the object’s name.
- Parameters:
- namestr
The name of the object.
- setOutputSample(sampleY)¶
Accessor to the output sample.
- Parameters:
- outputSample
Sample
The output sample.
- outputSample
- setRelativeErrors(relativeErrors)¶
Accessor to the relative errors.
- Parameters:
- relativeErrorssequence of float
The relative errors defined as follows for each output of the model: with the vector of the model’s values and 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: with the model’s values and 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.
Examples using the class¶
Compute grouped indices for the Ishigami function
Create a full or sparse polynomial chaos expansion
Create a polynomial chaos metamodel by integration on the cantilever beam
Create a polynomial chaos metamodel from a data set
Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos
Polynomial chaos expansion cross-validation
Polynomial chaos is sensitive to the degree
Create a sparse chaos by integration
Compute Sobol’ indices confidence intervals
Sobol’ sensitivity indices from chaos
Example of sensitivity analyses on the wing weight model
Compute leave-one-out error of a polynomial chaos expansion