GPC (version 0.1)

GPCE.lar: Adaptive sparse generalized polynomial chaos expansion based on least angle regression

Description

The GPCE.lar function implements a polynomial chaos expansion of a given model or an external model. The strategy for the expansion of the model into a polynomial chaos basis is the adaptive sparse method based on the least angle regression. A statistical and a global sensitivity analysis of the model is then carried out.

Usage

GPCE.lar(Model = NULL, PCSpace = "Uniform",  InputDim = 3, InputDistrib = c(), ParamDistrib = NULL, Q2tgt = 1 - 10^(-6),  Eps = 10^(-2) * (1 - Q2tgt), EpsForw = Eps,  EpsBack = Eps, EnrichStep = 50,  jmax= InputDim, pmaxi = 6,  DesignLength = 8, SeedSob =sample(1:1000, 1))

Arguments

Model

a function defining the model to analyze or NULL if the model is external

PCSpace

The space where the expansion is achieved. Options are Gaussian, Uniform and Physic. Physic use the same distributions as the input ones for the expansion

InputDim

Dimension of the input

InputDistrib

Distribution of the input. Options are Gaussian, Uniform, Beta, Gamma

ParamDistrib

Parameters of the input distributions

Q2tgt

Fix the accuracy of the expansion fitting. By default 1-10(^-6)

Eps

Common epsilon for the selection of the basis. By default set to 10^(-2)*(1-Q2tgt)

EpsForw

Epsilon used for the forward selection of the basis. By default set to Eps

EpsBack

Epsilon used for the forward selection of the basis. By default set to Eps

EnrichStep

Number of samples to add to the experimental design. By default set to 50

jmax

The maximum interaction order between the input variables

pmaxi

The maximum degree of the polynomial basis

DesignLength

The length of the input design. By default set to 8

SeedSob

Seed for the Sobol design generation

Value

Designs

A list containing the Sobol design, the input distributions design, the polynomial chaos design and the design length

Output

Vector of the model output

TruncSet

Matrix of the kept sparse polynomial basis. TruncSet_ij is the jth polynomial degree associated to the ith variable

CoeffPCE

Vector of the expansion coefficients associated to the TruncSet. CoeffPCE_j is the jth coefficient associaed to the jth polynomial basis.

R2

The R2 PCE approximation oerror

Q2

The Q2 PCE approximation error

Moments

A list containing the fourth first moments of the output: mean, variance, standard deviation, skweness and kurtosis

Sensitivity

A list containing the sobol sensitivity indices and the sobol total sensitivity indices

OutputDistrib

A list containing a kernel estimation of the output distribution and the associated bandwidth

References

G. Blatman and B. Sudret, 2011, Adapive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, 230, 2345--2367.

See Also

tell.GPCE.lar

Examples

### CASE 1: model is a R function. 
### Model definition: y= 1 + Phi_1(x1)*Phi_1(x2)
Model <- function(x){
  PHerm = hermite.he.polynomials(5, normalized=FALSE)
  y=1+unlist(polynomial.values(PHerm[2],x[,1]))*unlist(polynomial.values(PHerm[2],x[,2]))
  return(y)
}

### Run the algorithm with the lar regression method
ResultObject=GPCE.lar(Model=Model, PCSpace="Gaussian", InputDim=3, InputDistrib=rep("Gaussian",3))
names(ResultObject)
###

### CASE 2: external model (for the example the function Model will be used externaly).

### initialized Output
Output=c()

### Get a first design
ResultObject=GPCE.lar(PCSpace="Gaussian",InputDim=3,InputDistrib=rep("Gaussian",3))
names(ResultObject)

### Calculate the model output for the given design and concatenate the model output results
### into the output vector 
Output=c(Output,Model(ResultObject$Design2Eval))

### Give the design and the calculated ouput to the tell function
ResultObject=tell(ResultObject,Output)
names(ResultObject)

### If the expansion has been calculated the function tell return the full expansion 
### paramaters, the moments analysis, the sensitivity analysis and the output distribution
### If not the function tell() return an enriched design. 
### In the later case the user calculate the output externally and give them
### to the tell function with the previous ResultObject for further calculation. 
### See GPCE.sparse documentation for an example.