GaSP: Building, fitting, predicting for a GaSP model

In GPBayes: Tools for Gaussian Process Modeling in Uncertainty Quantification

Building, fitting, predicting for a GaSP model

Description

This function serves as a wrapper to build, fit, and make prediction for a Gaussian process model. It calls on functions gp, gp.mcmc, gp.optim, gp.predict.

Usage

GaSP(
  formula = ~1,
  output,
  input,
  param,
  smooth.est = FALSE,
  input.new = NULL,
  cov.model = list(family = "CH", form = "isotropic"),
  model.fit = "Cauchy_prior",
  prior = list(),
  proposal = list(range = 0.35, tail = 2, nugget = 0.8, nu = 0.8),
  nsample = 5000,
  burnin = 1000,
  opt = NULL,
  bound = NULL,
  dtype = "Euclidean",
  verbose = TRUE
)

Arguments

formula	an object of formula class that specifies regressors; see formula for details.
output	a numerical vector including observations or outputs in a GaSP
input	a matrix including inputs in a GaSP
param	a list including values for regression parameters, covariance parameters, and nugget variance parameter. The specification of param should depend on the covariance model. The regression parameters are denoted by coeff. Default value is \mathbf{0}. The marginal variance or partial sill is denoted by sig2. Default value is 1. The nugget variance parameter is denoted by nugget for all covariance models. Default value is 0. For the Confluent Hypergeometric class, range is used to denote the range parameter \beta. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu. For the generalized Cauchy class, range is used to denote the range parameter \phi. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu. For the Matérn class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter \nu. When \nu=0.5, the Matérn class corresponds to the exponential covariance. For the powered-exponential class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter. When \nu=2, the powered-exponential class corresponds to the Gaussian covariance.
smooth.est	a logical value indicating whether smoothness parameter will be estimated.
input.new	a matrix of new input locations
cov.model	a list of two strings: family, form, where family indicates the family of covariance functions including the Confluent Hypergeometric class, the Matérn class, the Cauchy class, the powered-exponential class. form indicates the specific form of covariance structures including the isotropic form, tensor form, automatic relevance determination form. family CH The Confluent Hypergeometric correlation function is given by C(h) = \frac{\Gamma(\nu+\alpha)}{\Gamma(\nu)} \mathcal{U}\left(\alpha, 1-\nu, \left(\frac{h}{\beta}\right)^2\right), where \alpha is the tail decay parameter. \beta is the range parameter. \nu is the smoothness parameter. \mathcal{U}(\cdot) is the confluent hypergeometric function of the second kind. For details about this covariance, see Ma and Bhadra (2023; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2022.2027775")}). cauchy The generalized Cauchy covariance is given by C(h) = \left\{ 1 + \left( \frac{h}{\phi} \right)^{\nu} \right\}^{-\alpha/\nu}, where \phi is the range parameter. \alpha is the tail decay parameter. \nu is the smoothness parameter with default value at 2. matern The Matérn correlation function is given by C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{h}{\phi} \right)^{\nu} \mathcal{K}_{\nu}\left( \frac{h}{\phi} \right), where \phi is the range parameter. \nu is the smoothness parameter. \mathcal{K}_{\nu}(\cdot) is the modified Bessel function of the second kind of order \nu. exp The exponential correlation function is given by C(h)=\exp(-h/\phi), where \phi is the range parameter. This is the Matérn correlation with \nu=0.5. matern_3_2 The Matérn correlation with \nu=1.5. matern_5_2 The Matérn correlation with \nu=2.5. powexp The powered-exponential correlation function is given by C(h)=\exp\left\{-\left(\frac{h}{\phi}\right)^{\nu}\right\}, where \phi is the range parameter. \nu is the smoothness parameter. gauss The Gaussian correlation function is given by C(h)=\exp\left(-\frac{h^2}{\phi^2}\right), where \phi is the range parameter. form isotropic This indicates the isotropic form of covariance functions. That is, C(\mathbf{h}) = C^0(\|\mathbf{h}\|; \boldsymbol \theta), where \| \mathbf{h}\| denotes the Euclidean distance or the great circle distance for data on sphere. C^0(\cdot) denotes any isotropic covariance family specified in family. tensor This indicates the tensor product of correlation functions. That is, C(\mathbf{h}) = \prod_{i=1}^d C^0(|h_i|; \boldsymbol \theta_i), where d is the dimension of input space. h_i is the distance along the ith input dimension. This type of covariance structure has been often used in Gaussian process emulation for computer experiments. ARD This indicates the automatic relevance determination form. That is, C(\mathbf{h}) = C^0\left(\sqrt{\sum_{i=1}^d\frac{h_i^2}{\phi^2_i}}; \boldsymbol \theta \right), where \phi_i denotes the range parameter along the ith input dimension.
model.fit	a string indicating the choice 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). 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. This is only supported for isotropic covariance functions. For details, see gp.mcmc. 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). MPLE This indicates that the maximum profile likelihood estimation (MPLE) is used. MMLE This indicates that the maximum marginal likelihood estimation (MMLE) is used. MAP This indicates that the marginal/integrated posterior is maximized.
prior	a list containing tuning parameters in prior distribution. This is used only if a subjective Bayes estimation method with informative priors is used.
proposal	a list containing tuning parameters in proposal distribution. This is used only if a Bayes estimation method is used.
nsample	an integer indicating the number of MCMC samples.
burnin	an integer indicating the burn-in period.
opt	a list of arguments to setup the optim routine. Current implementation uses three arguments: method The optimization method: Nelder-Mead or L-BFGS-B. lower The lower bound for parameters. upper The upper bound for parameters.
bound	Default value is NULL. Otherwise, it should be a list containing the following elements depending on the covariance class: nugget a list of bounds for the nugget parameter. It is a list containing lower bound lb and upper bound ub with default value list(lb=0, ub=Inf). range a list of bounds for the range parameter. It has default value range=list(lb=0, ub=Inf) for the Confluent Hypergeometric covariance, the Matérn covariance, exponential covariance, Gaussian covariance, powered-exponential covariance, and Cauchy covariance. The log of range parameterization is used: \log(\phi). tail a list of bounds for the tail decay parameter. It has default value list(lb=0, ub=Inf) for the Confluent Hypergeometric covariance and the Cauchy covariance. nu a list of bounds for the smoothness parameter. It has default value list(lb=0, ub=Inf) for the Confluent Hypergeometric covariance and the Matérn covariance. when the powered-exponential or Cauchy class is used, it has default value nu=list(lb=0, ub=2). This can be achieved by specifying the lower bound in opt.
dtype	a string indicating the type of distance: Euclidean Euclidean distance is used. This is the default choice. GCD Great circle distance is used for data on sphere.
verbose	a logical value. If it is TRUE, the MCMC progress bar is shown.

Value

a list containing the S4 object gp and prediction results

Author(s)

Pulong Ma [e-mail]

See Also

GPBayes-package, gp, gp.mcmc, gp.optim, gp.predict

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)

# fitting a GaSP model with the objective Bayes approach
fit = GaSP(formula=~1, output, input,  
          param=list(range=3, nugget=0.1, nu=2.5), 
          smooth.est=FALSE, input.new=XX,
          cov.model=list(family="matern", form="isotropic"),
          proposal=list(range=.35, nugget=.8, nu=0.8),
          dtype="Euclidean", model.fit="Cauchy_prior", nsample=50, 
          burnin=10, verbose=TRUE)

GPBayes documentation built on May 29, 2024, 7:30 a.m.