# Linking two emulators Function constructs a linked GASP emulator of a composite computer model f2(f1). ```r link(f1_MLEs, f2_MLEs, test_input) ``` ## Arguments - `f1_MLEs`: Parameters of the emulator of a simulator f1. - `f2_MLEs`: Parameters of the emulator of a simulator f2. - `test_input`: Testing inputs. ## Details See examples which illustrate inputs specification to the function. ## Returns Four types of the linked GASP. - **em1**: Type 1 emulator, which uses MAP estimates of parameters. - **em2**: Type 2 emulator within partial objective Bayesian (POB) implementation. - **emT**: T-GASP emulator within objective Bayesian (OB) implementation. - **em3**: Approximated T-GASP emulator with the Gaussian distribution. ## References Ksenia N. Kyzyurova, James O. Berger, and Robert L. Wolpert. Coupling computer models through linking their statistical emulators. SIAM/ASA Journal on Uncertainty Quantification, 6(3): 1151-1171, 2018 ## Author(s) Ksenia N. Kyzyurova, kseniak.ucoz.net ## Examples ```r ## Function f1 is a simulator f1<-function(x){sin(pi*x)} ## Function f2 is a simulator f2<-function(x){cos(5*x)} ## Function f2(f1) is a simulator of a composite model f2f1 <- function(x){f2(f1(x))} ## One-dimensional inputs are x1 x1 <- seq(-1,1,.37) ## The following contains the list of data inputs (training) and outputs (fD) together with ## the assumed fixed smoothness of a computer model output. data.f1 <- list(training = x1,fD = f1(x1), smooth = 1.99) ## Evaluation of GASP parameters f1_MLEs = eval_GASP_RFP(data.f1,list(function(x){x^0},function(x){x^1}),1,FALSE) ## Evaluate the emulator xn = seq(-1,1,.01) GASP_type2_f1 <- eval_type2_GASP(as.matrix(xn),f1_MLEs) par(mfrow = c(1,1)) par(mar = c(6.1, 6.1, 5.1, 2.1)) ylim = c(-1.5,1.5) GASP_plot(GASP_type2_f1,f1,data.f1,"Type 2 GASP",ylab = " f",xlab = "x", ylim = ylim, plot_training = TRUE) s = GASP_type2_f1$mu s.var = diag(GASP_type2_f1$var) x2 = seq(-0.95,0.95,length = 6)#f1(x1) data.f2 <- list(training = x2,fD = f2(x2), smooth = 2) # linking requires this emulator # to have smoothness parameter equal to 2 f2_MLEs = eval_GASP_RFP(data.f2,list(function(x){x^0},function(x){x^1}),1,FALSE) GASP_type1_f2 <- eval_type1_GASP(as.matrix(seq(-3.5,3.5,.01)),f2_MLEs) GASP_type2_f2 <- eval_type2_GASP(as.matrix(seq(-1,1,.01)),f2_MLEs) TGASP_f2 <- eval_TGASP(as.matrix(seq(-1,1,.01)),f2_MLEs) ylim = c(-1.5,1.5) # labels = c(expression(phantom(x)*phantom(x)*phantom(x)*f(x[1])), # expression(f(x[2])*phantom(x)*phantom(x)*phantom(x)), # expression(f(x[3])),expression(f(x[4])), # expression(f(x[5])),expression(f(x[6]))) par(mar = c(6.1, 6.1, 5.1, 2.1)) GASP_plot(GASP_type2_f2,f2,data.f2, "Type 2 GASP",labels = x2,xlab= "z",ylab = " g", ylim = ylim,plot_training = TRUE) le <- link(f1_MLEs, f2_MLEs, as.matrix(xn)) ## Plot second type of the linked GASP data.f2f1 <- list(training = x1,fD = f2f1(x1), smooth = 2) par(mar = c(6.1, 6.1, 5.1, 2.1)) GASP_plot(le$em2,f2f1,data.f2f1,"Linked",labels = x1, ylab = expression("g" ~ scriptscriptstyle(O) ~ "f"),xlab = "x",ylim = ylim) ```

link function