corr_matrix {GPfit}

Power Exponential or Matern Correlation Matrix

Description

Computes the power exponential or Matern correlation matrix for a set of n design points in d-dimensional input region and a vector of d correlation hyper-parameters (beta).

Usage

corr_matrix(X, beta, corr = list(type = "exponential", power = 1.95))

Arguments

X

the (n x d) design matrix

beta

a (d x 1) vector of correlation hyper-parameters in (-\infty, \infty)

corr

a list that specifies the type of correlation function along with the smoothness parameter. The default corresponds to power exponential correlation with smoothness parameter "power=1.95". One can specify a different power (between 1.0 and 2.0) for the power exponential, or use the Matern correlation function, specified as corr=list(type = "matern", nu=(2*k+1)/2), where k \in \{0,1,2,...\}

Details

The power exponential correlation function is given by

R_{ij} = \prod_{k=1}^{d} \exp({-10^{\beta_k}|x_{ik}-x_{jk}|^{power}}).

The Matern correlation function given by Santner, Williams, and Notz (2003) is

R_{ij} = \prod_{k=1}^{d} \frac{1}{\Gamma(\nu)2^{\nu-1}}(2\sqrt{\nu}|x_{ik} - x_{jk}|10^{\beta_k})^\nu \kappa_{\nu}(2\sqrt{\nu}|x_{ik} - x_{jk}|10^{\beta_k}),

where \kappa_{\nu} is the modified Bessel function of order \nu.

Value

The (n x n) correlation matrix, R, for the design matrix (X) and the hyper-parameters (beta).

Note

Both Matern and power exponential correlation functions use the new \beta parametrization of hyper-parameters given by \theta_k = 10^{\beta_k} for easier likelihood optimization. That is, beta is a log scale parameter (see MacDonald et al. (2015)).

Author(s)

Blake MacDonald, Hugh Chipman, Pritam Ranjan

References

MacDonald, K.B., Ranjan, P. and Chipman, H. (2015). GPfit: An R Package for Fitting a Gaussian Process Model to Deterministic Simulator Outputs. Journal of Statistical Software, 64(12), 1-23. http://www.jstatsoft.org/v64/i12/

Ranjan, P., Haynes, R., and Karsten, R. (2011). A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data, Technometrics, 53(4), 366 - 378.

Santner, T.J., Williams, B., and Notz, W. (2003), The design and analysis of computer experiments, Springer Verlag, New York.

Examples

## 1D Example - 1
n = 5
d = 1
set.seed(3)
library(lhs)
x = maximinLHS(n,d)
beta =  rnorm(1)
corr_matrix(x,beta)

## 1D Example - 2
beta = rnorm(1)
corr_matrix(x,beta,corr = list(type = "matern"))

## 2D example - 1
n = 10
d = 2
set.seed(2)
library(lhs)
x = maximinLHS(n,d) 
beta = rnorm(2)
corr_matrix(x, beta,
    corr = list(type = "exponential", power = 2))

## 2D example - 2
beta = rnorm(2)
R = corr_matrix(x,beta,corr = list(type = "matern", nu = 5/2))
print(R)