GPCE.lar: Adaptive sparse generalized polynomial chaos expansion based...
In GPC: Generalized Polynomial Chaos

Description Usage Arguments Value Author(s) References See Also Examples

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.

  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))

`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

`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

Munoz Zuniga Miguel

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

tell.GPCE.lar

### 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.