GP_deviance function

Computes the Deviance of a GP model

Evaluates the deviance (negative 2*log-likelihood), as defined in Ranjan et al. (2011), however the correlation is reparametrized and can be either power exponential or Matern as discussed in corr_matrix.

GP_deviance(
  beta,
  X,
  Y,
  nug_thres = 20,
  corr = list(type = "exponential", power = 1.95)
)

Arguments

• beta: a (d x 1) vector of correlation hyper-parameters, as described in corr_matrix
• X: the (n x d) design matrix
• Y: the (n x 1) vector of simulator outputs
• nug_thres: a parameter used in computing the nugget. See GP_fit.
• corr: a list of parameters for the specifing the correlation to be used. See corr_matrix.

Returns

the deviance (negative 2 * log-likelihood)

Examples

## 1D Example 1
n = 5
d = 1
computer_simulator <- function(x) {
    x = 2 * x + 0.5
    y = sin(10 * pi * x)/(2 * x) + (x - 1)^4
    return(y)
}
set.seed(3)
library(lhs)
x = maximinLHS(n,d)
y = computer_simulator(x)
beta =  rnorm(1)
GP_deviance(beta,x,y)

## 1D Example 2
n = 7
d = 1
computer_simulator <- function(x) {
    y <- log(x + 0.1) + sin(5 * pi * x)
    return(y)
}
set.seed(1)
library(lhs)
x = maximinLHS(n, d)
y = computer_simulator(x)
beta = rnorm(1)
GP_deviance(beta, x, y,
    corr = list(type = "matern", nu = 5/2))

## 2D Example: GoldPrice Function
computer_simulator <- function(x) {
    x1 = 4 * x[, 1] - 2
    x2 = 4 * x[, 2] - 2
    t1 = 1 + (x1 + x2 + 1)^2 * 
        (19 - 14 * x1 + 3 * x1^2 - 
        14 * x2 + 6 * x1 * x2 + 3 * x2^2)
    t2 = 30 + (2 * x1 - 3 * x2)^2 * 
        (18 - 32 * x1 + 12 * x1^2 + 
        48 * x2 - 36 * x1 * x2 + 27 * x2^2)
    y = t1 * t2
    return(y)
}
n = 10
d = 2
set.seed(1)
library(lhs)
x = maximinLHS(n, d) 
y = computer_simulator(x)
beta = rnorm(2)
GP_deviance(beta, x, y)

References

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.

See Also

corr_matrix for computing the correlation matrix;

GP_fit for estimating the parameters of the GP model.

Author(s)

Blake MacDonald, Hugh Chipman, Pritam Ranjan