# EQI's Gradient Analytical gradient of the Expected Quantile Improvement (EQI) criterion. ```r EQI.grad( x, model, new.noise.var = 0, beta = 0.9, q.min = NULL, type = "UK", envir = NULL ) ``` ## Arguments - `x`: the input vector at which one wants to evaluate the criterion - `model`: a Kriging model of "km" class - `new.noise.var`: (scalar) noise variance of the future observation. Default value is 0 (noise-free observation). - `beta`: Quantile level (default value is 0.9) - `q.min`: Best kriging quantile. If not provided, this quantity is evaluated. - `type`: Kriging type: "SK" or "UK" - `envir`: environment for inheriting intermediate calculations from EQI ## Returns Gradient of the Expected Quantile Improvement ## Examples ```r set.seed(421) # Set test problem parameters doe.size <- 12 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)) # Compute actual function and criterion on a grid n.grid <- 9 # change to 21 for nicer visuals x.grid <- y.grid <- seq(0,1,length=n.grid) design.grid <- expand.grid(x.grid, y.grid) nt <- nrow(design.grid) crit.grid <- apply(design.grid, 1, EQI, model=model, new.noise.var=noise.var, beta=.9) crit.grad <- t(apply(design.grid, 1, EQI.grad, model=model, new.noise.var=noise.var, beta=.9)) z.grid <- matrix(crit.grid, n.grid, n.grid) contour(x.grid,y.grid, z.grid, 30) title("EQI and its gradient") points(model@X[,1],model@X[,2],pch=17,col="blue") for (i in 1:nt) { x <- design.grid[i,] suppressWarnings(arrows(x$Var1,x$Var2, x$Var1+crit.grad[i,1]*.2,x$Var2+crit.grad[i,2]*.2, length=0.04,code=2,col="orange",lwd=2)) } ``` ## Author(s) Victor Picheny David Ginsbourger

EQI.grad function