GPCE.sparse: Adaptive sparse generalized polynomial chaos expansion

In GPC: Generalized Polynomial Chaos

Description

The GPCE.sparse 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 where the sampled design is iteratively enriched until the expansion coefficients estimation, by regression, is accurate. Meanwile a relevant selection of the polynomial basis to keep in the expansion is carried out. A statistical and a global sensitivity analysis of the model is then carried out.

Usage

GPCE.sparse(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 = 12, 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
DesignLength	The length of the input design. By default set to 8
jmax	The maximum interaction order between the input variables
pmaxi	The maximum degree of the polynomial basis
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 error
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

Author(s)

Munoz Zuniga Miguel

References

G. Blatman and B. Sudret, 2010, An adapive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, 25, 183–197.

See Also

tell.GPCE.sparse

Examples

### CASE 1: model is a R function. 
### Model definition: y= 1 + Phi_1(x1)*Phi_1(x2) + Phi_3(x2)
Model <- function(x){
  PHerm = hermite.he.polynomials(7, normalized=FALSE)
  y=1+unlist(polynomial.values(PHerm[2],x[,1]))*unlist(polynomial.values(PHerm[2],x[,2]))+
  unlist(polynomial.values(PHerm[4],x[,2]))/sqrt(factorial(3))
  return(y)
}

### Run the algorithm with the sparse regression method
ResultObject=GPCE.sparse(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.sparse(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. 
Output=c(Output,Model(ResultObject$Design2Eval))

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

Example output

Loading required package: randtoolbox
Loading required package: rngWELL
This is randtoolbox. For an overview, type 'help("randtoolbox")'.
Loading required package: orthopolynom
Loading required package: polynom
Loading required package: ks
Loading required package: lars
Loaded lars 1.2

[1] "h problem"
[1] "Entering PostProcessReg"
[1] "Entering EvalPCE"
 [1] "TruncSet"     "R2"           "Q2"           "Designs"      "CoeffPCE"    
 [6] "Moments"      "Sensitivity"  "OutputDistib" "Args"         "Output"      
[1] "Designs"     "Design2Eval" "Args"       
[1] "Designs"     "Design2Eval" "Output"      "Args"       
[1] "Entering PostProcessReg"
[1] "Entering EvalPCE"
 [1] "TruncSet"     "R2"           "Q2"           "Designs"      "CoeffPCE"    
 [6] "Moments"      "Sensitivity"  "OutputDistib" "Args"         "Output"      

GPC documentation built on May 30, 2017, 12:50 a.m.