# Gaussian Process Laboratory Laboratory Package for Gaussian Process interpolation, regression and simulation, with an emphasis on user-defined covariance kernels. UTF8 package ## Note This package was developed within the frame of the ReDice Consortium, gathering industrial partners (CEA, EDF, IFPEN, IRSN, Renault) and academic partners (Mines Saint-Étienne, INRIA, and the University of Bern) around advanced methods for Computer Experiments. ## Details tools:::Rd_package_DESCRIPTION("kergp") ## Warning As a lab, `kergp` may strongly evolve in its future life. Users interested in stable software for the Analysis of Computer Experiments are encouraged to use other packages such as `DiceKriging` instead. ## Author(s) Yves Deville (Alpestat), David Ginsbourger (University of Bern), Olivier Roustant (INSA Toulouse), with contributions from Nicolas Durrande (Mines Saint-Étienne). Maintainer: Olivier Roustant, ## References Nicolas Durrande, David Ginsbourger, Olivier Roustant (2012). "Additive covariance kernels for high-dimensional gaussian process modeling". **Annales de la Faculté des Sciences de Toulouse**, 21 (3): 481-499, tools:::Rd_expr_doi("10.5802/afst.1342") . Nicolas Durrande, David Ginsbourger, Olivier Roustant, Laurent Carraro (2013). "ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis". **Journal of Multivariate Analysis**, 115, 57-67, tools:::Rd_expr_doi("10.1016/j.jmva.2012.08.016") . David Ginsbourger, Xavier Bay, Olivier Roustant, Laurent Carraro (2012). "Argumentwise invariant kernels for the approximation of invariant functions". **Annales de la Faculté des Sciences de Toulouse**, 21 (3): 501-527, tools:::Rd_expr_doi("10.5802/afst.1343") . David Ginsbourger, Nicolas Durrande, Olivier Roustant (2013). "Kernels and designs for modelling invariant functions: From group invariance to additivity" **mODa 10 - Advances in Model-Oriented Design and Analysis. Contributions to Statistics**, 107-115, tools:::Rd_expr_doi("10.1007/978-3-319-00218-7_13") . Olivier Roustant, David Ginsbourger, Yves Deville (2012). "DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization". **Journal of Statistical Software**, 51(1), 1-55, tools:::Rd_expr_doi("10.18637/jss.v051.i01") . ## Examples ```r ## ------------------------------------------------------------------ ## Gaussian process modelling of function with invariance properties, ## by using an argumentwise invariant kernel ## ------------------------------------------------------------------ ## -- define manually an argumentwise invariant kernel -- kernFun <- function(x1, x2, par) { h <- (abs(x1) - abs(x2)) / par[1] S <- sum(h^2) d2 <- exp(-S) K <- par[2] * d2 d1 <- 2 * K * S / par[1] attr(K, "gradient") <- c(theta = d1, sigma2 = d2) return(K) } ## --------------------------------------------------------------- ## quicker: with Rcpp; see also an example with package inline ## in "gp" doc. file. Note that the Rcpp "sugar" fucntions are ## vectorized, so no for loops is required. ## --------------------------------------------------------------- ## Not run: cppFunction(' NumericVector cppKernFun(NumericVector x1, NumericVector x2, NumericVector par){ int n1 = x1.size(); double S, d1, d2; NumericVector K(1), h(n1); h = (abs(x1) - abs(x2)) / par[0]; // sugar function "abs" S = sum(h * h); // sugar "*" and "sum" d2 = exp(-S); K[0] = par[1] * d2; d1 = 2 * K[0] * S / par[0]; K.attr("gradient") = NumericVector::create(Named("theta", d1), Named("sigma2", d2)); return K; }') ## End(Not run) ## --------------------------------------------------------------- ## Below: with the R-based code for the kernel namely 'kernFun'. ## You can also replace 'kernFun' by 'cppKernFun' for speed. ## --------------------------------------------------------------- covSymGauss <- covMan(kernel = kernFun, hasGrad = TRUE, label = "argumentwise invariant", d = 2, parLower = c(theta = 0.0, sigma2 = 0.0), parUpper = c(theta = Inf, sigma2 = Inf), parNames = c("theta", "sigma2"), par = c(theta = 0.5, sigma2 = 2)) covSymGauss ## -- simulate a path from the corresponding GP -- nGrid <- 24; n <- nGrid^2; d <- 2 xGrid <- seq(from = -1, to = 1, length.out = nGrid) Xgrid <- expand.grid(x1 = xGrid, x2 = xGrid) Kmat <- covMat(object = covSymGauss, X = Xgrid, compGrad = FALSE, index = 1L) library(MASS) set.seed(1) ygrid <- mvrnorm(mu = rep(0, n), Sigma = Kmat) ## -- extract a design and the corr. response from the grid -- nDesign <- 25 tab <- subset(cbind(Xgrid, ygrid), x1 > 0 & x2 > 0) rowIndex <- seq(1, nrow(tab), length = nDesign) X <- tab[rowIndex, 1:2] y <- tab[rowIndex, 3] opar <- par(mfrow = c(1, 3)) contour(x = xGrid, y = xGrid, z = matrix(ygrid, nrow = nGrid, ncol = nGrid), nlevels = 15) abline(h = 0, v = 0, col = "SpringGreen3") points(x2 ~ x1, data = X, type = "p", pch = 21, col = "orangered", bg = "yellow", cex = 0.8) title("GRF Simulation") ## -- Fit the Gaussian process model (trend + covariance parameters) -- covSymGauss symgp <- gp(formula = y ~ 1, data = data.frame(y, X), inputs = names(X), cov = covSymGauss, parCovIni = c(0.1, 2), varNoiseIni = 1.0e-8, varNoiseLower = 0.9e-8, varNoiseUpper = 1.1e-8) # mind that the noise is not a symmetric kernel # so varNoiseUpper should be chosen as small as possible. summary(symgp) ## -- predict and compare -- predSymgp <- predict(object = symgp, newdata = Xgrid, type = "UK") contour(x = xGrid, y = xGrid, z = matrix(predSymgp$mean, nrow = nGrid, ncol = nGrid), nlevels = 15) abline(h = 0, v = 0, col = "SpringGreen3") points(x2 ~ x1, data = X, type = "p", pch = 21, col = "orangered", bg = "yellow", cex = 0.8) title("Kriging mean") contour(x = xGrid, y = xGrid, z = matrix(predSymgp$sd, nrow = nGrid, ncol = nGrid), nlevels = 15) abline(h = 0, v = 0, col = "SpringGreen3") points(x2 ~ x1, data = X, type = "p", pch = 21, col = "orangered", bg = "yellow", cex = 0.8) title("Kriging s.d.") par(opar) ```

kergp-package