# Analytical expression of the Expected Improvement criterion Computes the Expected Improvement at current location. The current minimum of the observations can be replaced by an arbitrary value (plugin), which is usefull in particular in noisy frameworks. ```r EI( x, model, plugin = NULL, type = "UK", minimization = TRUE, envir = NULL, proxy = FALSE ) ``` ## Arguments - `x`: a vector representing the input for which one wishes to calculate EI, - `model`: an object of class `km`, - `plugin`: optional scalar: if provided, it replaces the minimum of the current observations, - `type`: "SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account, - `minimization`: logical specifying if EI is used in minimiziation or in maximization, - `envir`: an optional environment specifying where to assign intermediate values for future gradient calculations. Default is NULL. - `proxy`: an optional Boolean, if TRUE EI is replaced by the kriging mean (to minimize) ## Returns The expected improvement, defined as $$ EI(x) := E[( min(Y(X)) -Y(x))^{+} | Y(X)=y(X)], $$ where X is the current design of experiments and Y is the random process assumed to have generated the objective function y. If a plugin is specified, it replaces $$ min(Y(X)) $$ in the previous formula. ## Examples ```r set.seed(123) ########################################################################## ### EI SURFACE ASSOCIATED WITH AN ORDINARY KRIGING MODEL #### ### OF THE BRANIN FUNCTION KNOWN AT A 9-POINTS FACTORIAL DESIGN #### ########################################################################## # a 9-points factorial design, and the corresponding response d <- 2; n <- 9 design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3)) names(design.fact)<-c("x1", "x2") design.fact <- data.frame(design.fact) names(design.fact)<-c("x1", "x2") response.branin <- apply(design.fact, 1, branin) response.branin <- data.frame(response.branin) names(response.branin) <- "y" # model identification fitted.model1 <- km(~1, design=design.fact, response=response.branin, covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5)) # graphics n.grid <- 12 x.grid <- y.grid <- seq(0,1,length=n.grid) design.grid <- expand.grid(x.grid, y.grid) #response.grid <- apply(design.grid, 1, branin) EI.grid <- apply(design.grid, 1, EI,fitted.model1) z.grid <- matrix(EI.grid, n.grid, n.grid) contour(x.grid,y.grid,z.grid,25) title("Expected Improvement for the Branin function known at 9 points") points(design.fact[,1], design.fact[,2], pch=17, col="blue") ``` ## References D.R. Jones, M. Schonlau, and W.J. Welch (1998), Efficient global optimization of expensive black-box functions, **Journal of Global Optimization**, 13, 455-492. J. Mockus (1988), **Bayesian Approach to Global Optimization**. Kluwer academic publishers. T.J. Santner, B.J. Williams, and W.J. Notz (2003), **The design and analysis of computer experiments**, Springer. M. Schonlau (1997), **Computer experiments and global optimization**, Ph.D. thesis, University of Waterloo. ## See Also `max_EI`, `EGO.nsteps`, `qEI` ## Author(s) David Ginsbourger Olivier Roustant Victor Picheny

EI function