# GP Prediction/Kriging Perform Gaussian processes prediction (under isotropic or separable formulation) at new `XX` locations using a GP object stored on the C-side ```r predGP(gpi, XX, lite = FALSE, nonug = FALSE) predGPsep(gpsepi, XX, lite = FALSE, nonug = FALSE) ``` ## Arguments - `gpi`: a C-side GP object identifier (positive integer); e.g., as returned by `newGP` - `gpsepi`: similar to `gpi` but indicating a separable GP object, as returned by `newGPsep` - `XX`: a `matrix` or `data.frame` containing a design of predictive locations - `lite`: a scalar logical indicating whether (`lite = FALSE`, default) or not (`lite = TRUE`) a full predictive covariance matrix should be returned, as would be required for plotting random sample paths, but substantially increasing computation time if only point-prediction is required - `nonug`: a scalar logical indicating if a (nonzero) nugget should be used in the predictive equations; this allows the user to toggle between visualizations of uncertainty due just to the mean, and a full quantification of predictive uncertainty. The latter (default `nonug = FALSE`) is the standard approach, but the former may work better in some sequential design contexts. See, e.g., `ieciGP` ## Details Returns the parameters of Student-t predictive equations. By default, these include a full predictive covariance matrix between all `XX` locations. However, this can be slow when `nrow(XX)` is large, so a `lite` options is provided, which only returns the diagonal of that matrix. GP prediction is sometimes called kriging , especially in the spatial statistics literature. So this function could also be described as returning evaluations of the kriging equations ## Returns The output is a `list` with the following components. - **mean**: a vector of predictive means of length `nrow(Xref)` - **Sigma**: covariance matrix for a multivariate Student-t distribution; alternately if `lite = TRUE`, then a field `s2` contains the diagonal of this matrix - **df**: a Student-t degrees of freedom scalar (applies to all `XX`) ## References For standard GP prediction, refer to any graduate text, e.g., Rasmussen & Williams **Gaussian Processes for Machine Learning**, or Gramacy, R. B. (2020) **Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences**. Boca Raton, Florida: Chapman Hall/CRC. (See Chapter 5.) [https://bobby.gramacy.com/surrogates/](https://bobby.gramacy.com/surrogates/) ## Author(s) Robert B. Gramacy rbg@vt.edu ## See Also `vignette("laGP")`, `newGP`, `mleGP`, `jmleGP`, ## Examples ```r ## a "computer experiment" -- a much smaller version than the one shown ## in the aGP docs ## Simple 2-d test function used in Gramacy & Apley (2015); ## thanks to Lee, Gramacy, Taddy, and others who have used it before f2d <- function(x, y=NULL) { if(is.null(y)) { if(!is.matrix(x) && !is.data.frame(x)) x <- matrix(x, ncol=2) y <- x[,2]; x <- x[,1] } g <- function(z) return(exp(-(z-1)^2) + exp(-0.8*(z+1)^2) - 0.05*sin(8*(z+0.1))) z <- -g(x)*g(y) } ## design with N=441 x <- seq(-2, 2, length=11) X <- expand.grid(x, x) Z <- f2d(X) ## fit a GP gpi <- newGP(X, Z, d=0.35, g=1/1000) ## predictive grid with NN=400 xx <- seq(-1.9, 1.9, length=20) XX <- expand.grid(xx, xx) ZZ <- f2d(XX) ## predict p <- predGP(gpi, XX, lite=TRUE) ## RMSE: compare to similar experiment in aGP docs sqrt(mean((p$mean - ZZ)^2)) ## visualize the result par(mfrow=c(1,2)) image(xx, xx, matrix(p$mean, nrow=length(xx)), col=heat.colors(128), xlab="x1", ylab="x2", main="predictive mean") image(xx, xx, matrix(p$mean-ZZ, nrow=length(xx)), col=heat.colors(128), xlab="x1", ylab="x2", main="bas") ## clean up deleteGP(gpi) ## see the newGP and mleGP docs for examples using lite = FALSE for ## sampling from the joint predictive distribution ```

predGP function