# Minimization of the Kriging quantile. Minimization, based on the package rgenoud of the kriging quantile. ```r min_quantile( model, beta = 0.1, type = "UK", lower, upper, parinit = NULL, control = NULL ) ``` ## Arguments - `model`: a Kriging model of "km" class - `beta`: Quantile level (default value is 0.1) - `type`: Kriging type: "SK" or "UK" - `lower`: vector containing the lower bounds of the variables to be optimized over - `upper`: vector containing the upper bounds of the variables to be optimized over - `parinit`: optional vector containing the initial values for the variables to be optimized over - `control`: optional list of control parameters for optimization. One can control `"pop.size"` (default : [N=3*2^dim for dim<6 and N=32*dim otherwise]), `"max.generations"` (12), `"wait.generations"` (2) and `"BFGSburnin"` (2) of function `"genoud"` (see `genoud`). Numbers into brackets are the default values ## Returns A list with components: - **par**: the best set of parameters found. - **value**: the value of the krigign quantile at par. ## Examples ```r ########################################################################## ### KRIGING QUANTILE SURFACE AND OPTIMIZATION PERFORMED BY GENOUD #### ### FOR AN ORDINARY KRIGING MODEL #### ### OF THE BRANIN FUNCTION KNOWN AT A 12-POINT LATIN HYPERCUBE DESIGN #### ########################################################################## set.seed(10) # Set test problem parameters doe.size <- 10 dim <- 2 test.function <- get("branin2") lower <- rep(0,1,dim) upper <- rep(1,1,dim) noise.var <- 0.2 # Generate DOE and response doe <- as.data.frame(matrix(runif(doe.size*dim),doe.size)) y.tilde <- rep(0, 1, doe.size) for (i in 1:doe.size) {y.tilde[i] <- test.function(doe[i,]) + sqrt(noise.var)*rnorm(n=1)} y.tilde <- as.numeric(y.tilde) # Create kriging model model <- km(y~1, design=doe, response=data.frame(y=y.tilde), covtype="gauss", noise.var=rep(noise.var,1,doe.size), lower=rep(.1,dim), upper=rep(1,dim), control=list(trace=FALSE)) # Optimisation using max_kriging.quantile res <- min_quantile(model, beta=0.1, type = "UK", lower=c(0,0), upper=c(1,1)) X.genoud <- res$par # Compute actual function and criterion on a grid n.grid <- 12 # Change to 21 for a nicer picture x.grid <- y.grid <- seq(0,1,length=n.grid) design.grid <- expand.grid(x.grid, y.grid) names(design.grid) <- c("V1","V2") nt <- nrow(design.grid) crit.grid <- apply(design.grid, 1, kriging.quantile, model=model, beta=.1) # # 2D plots z.grid <- matrix(crit.grid, n.grid, n.grid) tit <- "Green: best point found by optimizer" filled.contour(x.grid,y.grid, z.grid, nlevels=50, color = topo.colors, plot.axes = {title(tit);points(model@X[,1],model@X[,2],pch=17,col="blue"); points(X.genoud[1],X.genoud[2],pch=17,col="green"); axis(1); axis(2)}) ``` ## Author(s) Victor Picheny David Ginsbourger

min_quantile function