gp.mcmc {GPBayes}

A wraper to fit a Gaussian stochastic process model with MCMC algorithms

Description

This function is a wraper to estimate parameters via MCMC algorithms in the GaSP model with different choices of priors.

Usage

gp.mcmc(
  obj,
  input.new = NULL,
  method = "Cauchy_prior",
  prior = list(),
  proposal = list(),
  nsample = 10000,
  verbose = TRUE
)

Arguments

obj

an S4 object gp

input.new

a matrix of prediction locations. Default value is NULL, indicating that prediction is not carried out along with parameter estimation in the MCMC algorithm.

method

a string indicating the Bayes estimation approaches with different choices of priors on correlation parameters:

Cauchy_prior

This indicates that a fully Bayesian approach with objective priors is used for parameter estimation, where location-scale parameters are assigned with constant priors and correlation parameters are assigned with half-Cauchy priors (default). If the smoothness parameter is estimated for isotropic covariance functions, the smoothness parameter is assigned with a uniform prior on (0, 4), indicating that the corresponding GP is at most four times mean-square differentiable. This is a reasonable prior belief for modeling spatial processes; If the smoothness parameter is estimated for tensor or ARD covariance functions, the smoothness parameter is assigned with a uniform prior on (0, 6).

Ref_prior

This indicates that a fully Bayesian approach with objective priors is used for parameter estimation, where location-scale parameters are assigned with constant priors and correlation parameters are assigned with reference priors. If the smoothness parameter is estimated for isotropic covariance functions, the smoothness parameter is assigned with a uniform prior on (0, 4), indicating that the corresponding GP is at most four times mean-square differentiable. This is a reasonable prior belief for modeling spatial processes; If the smoothness parameter is estimated for tensor or ARD covariance functions, the smoothness parameter is assigned with a uniform prior on (0, 6).

Beta_prior

This indicates that a fully Bayesian approach with subjective priors is used for parameter estimation, where location-scale parameters are assigned with constant priors and correlation parameters are assigned with beta priors parameterized as Beta(a, b, lb, ub). In the beta distribution, lb and ub are the support for correlation parameters, and they should be determined based on domain knowledge. a and b are two shape parameters with default values at 1, corresponding to the uniform prior over the support (lb, ub).

prior

a list containing tuning parameters in prior distributions. This is used only if a Bayes estimation method with subjective priors is used.

proposal

a list containing tuning parameters in proposal distributions. This is used only if a Bayes estimation method is used.

nsample

an integer indicating the number of MCMC samples.

verbose

a logical value. If it is TRUE, the MCMC progress bar is shown.

Value

a gp object with prior, proposal, MCMC samples included.

Author(s)

Pulong Ma [e-mail]

See Also

GPBayes-package, GaSP, gp, gp.optim

Examples

 
 
code = function(x){
y = (sin(pi*x/5) + 0.2*cos(4*pi*x/5))*(x<=9.6) + (x/10-1)*(x>9.6) 
return(y)
}
n=100
input = seq(0, 20, length=n)
XX = seq(0, 20, length=99)
Ztrue = code(input)
set.seed(1234)
output = Ztrue + rnorm(length(Ztrue), sd=0.1)
obj = gp(formula=~1, output, input, 
        param=list(range=4, nugget=0.1,nu=2.5),
        smooth.est=FALSE,
        cov.model=list(family="matern", form="isotropic"))
        
fit.mcmc = gp.mcmc(obj, method="Cauchy_prior",
                   proposal=list(range=0.3, nugget=0.8),
                   nsample=100, verbose=TRUE)