GPfit: AnRPackage for Fitting a Gaussian Process Model to Deterministic Simulator Outputs

Abstract

Gaussian process (GP) models are commonly used statistical metamodels for emulating expensive computer simulators. Fitting a GP model can be numerically unstable if any pair of design points in the input space are close together. Ranjan, Haynes, and Karsten (2011) proposed a computationally stable approach for fitting GP models to deterministic computer simulators. They used a genetic algorithm based approach that is robust but computationally intensive for maximizing the likelihood. This paper implements a slightly modified version of the model proposed by Ranjan et al. (2011) in the R package GPfit. A novel parameterization of the spatial correlation function and a clustering based multistart gradient based optimization algorithm yield robust optimization that is typically faster than the genetic algorithm based approach. We present two examples with R codes to illustrate the usage of the main functions in GPfit. Several test functions are used for performance comparison with the popular R package mlegp. We also use GPfit for a real application, i.e., for emulating the tidal kinetic energy model for the Bay

Adaptive design of an X-ray magnetic circular dichroism spectroscopy experiment with Gaussian process modelling

“…Our implementation of the proposed method is based on the R 33 package GPfit. 34 Let the i-th energy and corresponding output spectral be denoted by x i and y i = y(x i ), respectively. The observed data points are denoted by ; x n Þ T 2 R n , and the corresponding responses are denoted by Y = y(X) = (y 1 ,…,y n ) Τ .…”

Efficient optimization of the likelihood function in Gaussian process modelling

“…Finding the global maximum of the likelihood function for a GP model is typically very challenging as the likelihood surface often has multiple local optima, and an explicit expression for the gradient of the likelihood function is typically unavailable. Previous methods for optimizing the likelihood function (e.g., MacDonald et al (2013)) have proven to be robust and accurate, though relatively inefficient. We propose several likelihood optimization techniques, including two modified multi-start local search techniques, based on the method implemented by MacDonald et al (2013), that are equally as reliable, and significantly more efficient.…”

“…Previous methods for optimizing the likelihood function (e.g., MacDonald et al (2013)) have proven to be robust and accurate, though relatively inefficient. We propose several likelihood optimization techniques, including two modified multi-start local search techniques, based on the method implemented by MacDonald et al (2013), that are equally as reliable, and significantly more efficient. A hybridization of the global search algorithm Dividing Rectangles (DIRECT) with the local optimization algorithm BFGS provides a comparable GP model quality for a fraction of the computational cost, and is the preferred optimization technique when computational resources are limited.…”

“…A hybridization of the global search algorithm Dividing Rectangles (DIRECT) with the local optimization algorithm BFGS provides a comparable GP model quality for a fraction of the computational cost, and is the preferred optimization technique when computational resources are limited. We use several test functions and a real application from an oil reservoir simulation to test and compare the performance of the proposed methods with the one implemented by MacDonald et al (2013) in the R library GPfit. The proposed method is implemented in a Matlab package, GPMfit.…”

“…Rasmussen and Williams (2006) proposed the use of either a randomized multi-start conjugate gradient method or Newton's method for this problem. Explicit information about the gradient of deviance, however, cannot easily be obtained, and the deviance surface often has multiple local optima, making the optimization problem challenging (MacDonald et al, 2013). Derivative-free optimization techniques, such as the genetic algorithm used by Ranjan et al (2011), or the differential evolution algorithm used by Petelin et al (2011), are robust, but can be computationally inefficient.…”