GPfit (version 1.0-8)

# predict: Model Predictions from GPfit

## Description

Computes the regularized predicted response $$\hat{y}_{\delta_{lb},M}(x)$$ and the mean squared error $$s^2_{\delta_{lb},M}(x)$$ for a new set of inputs using the fitted GP model.

The value of M determines the number of iterations (or terms) in approximating $$R^{-1} \approx R^{-1}_{\delta_{lb},M}$$. The iterative use of the nugget $$\delta_{lb}$$, as outlined in Ranjan et al. (2011), is used in calculating $$\hat{y}_{\delta_{lb},M}(x)$$ and $$s^2_{\delta_{lb},M}(x)$$, where $$R_{\delta,M}^{-1} = \sum_{t = 1}^{M} \delta^{t - 1}(R+\delta I)^{-t}$$.

## Usage

# S3 method for GP
predict(object, xnew = object\$X, M = 1, ...)

# S3 method for GP fitted(object, ...)

## Arguments

object

a class GP object estimated by GP_fit

xnew

the (n_new x d) design matrix of test points where model predictions and MSEs are desired

M

the number of iterations. See 'Details'

for compatibility with generic method predict

## Value

Returns a list containing the predicted values (Y_hat), the mean squared errors of the predictions (MSE), and a matrix (complete_data) containing xnew, Y_hat, and MSE

## Methods (by class)

• GP: The predict method returns a list of elements Y_hat (fitted values), Y (dependent variable), MSE (residuals), and completed_data (the matrix of independent variables, Y_hat, and MSE).

• GP: The fitted method extracts the complete data.

## 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.

GP_fit for estimating the parameters of the GP model; plot for plotting the predicted and error surfaces.

## Examples

# NOT RUN {
## 1D Example
n <- 5
d <- 1
computer_simulator <- function(x){
x <- 2*x+0.5
sin(10*pi*x)/(2*x) + (x-1)^4
}
set.seed(3)
library(lhs)
x <- maximinLHS(n,d)
y <- computer_simulator(x)
xvec <- seq(from = 0, to = 1, length.out = 10)
GPmodel <- GP_fit(x, y)

## 1D Example 2
n <- 7
d <- 1
computer_simulator <- function(x) {
log(x+0.1)+sin(5*pi*x)
}
set.seed(1)
library(lhs)
x <- maximinLHS(n,d)
y <- computer_simulator(x)
xvec <- seq(from = 0,to = 1, length.out = 10)
GPmodel <- GP_fit(x, y)
predict(GPmodel, xvec)

## 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)
GPmodel <- GP_fit(x,y)
# fitted values