predict function

Model Predictions from GPfit

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 c("$R_{\\delta,M}^{-1} = \\sum_{t = 1}^{M}\n$", "$\\delta^{t - 1}(R+\\delta I)^{-t}$").

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

## S3 method for class '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

Returns

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)

• predict(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).

Functions

• fitted(GP): The fitted method extracts the complete data.

Examples

## 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)
head(fitted(GPmodel))
lapply(predict(GPmodel, xvec), head)

## 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)
head(fitted(GPmodel))
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
head(fitted(GPmodel))
# new data
xvector <- seq(from = 0,to = 1, length.out = 10)
xdf <- expand.grid(x = xvector, y = xvector)
predict(GPmodel, xdf)

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

GP_fit for estimating the parameters of the GP model;

plot for plotting the predicted and error surfaces.

Author(s)

Blake MacDonald, Hugh Chipman, Pritam Ranjan