# Scaling function Parametric transformation of the input space variables. The transformation is obtained coordinatewise by integrating piecewise affine marginal "densities" parametrized by a vector of knots and a matrix of density values at the knots. See references for more detail. ```r scalingFun(X, knots, eta, plot=FALSE) ``` ## Arguments - `X`: an n*d matrix standing for a design of n experiments in d-dimensional space - `knots`: a list of knots parametrizing the transformation. - `eta`: a list of coefficients parametrizing the d marginal transformations. Each element stands for a set of marginal density values at the knots defined above. - `plot`: if TRUE plots the image of the columns of X according to the corresponding marginal transformations. ## Returns The image of X by a scaling transformation of parameters knots and eta ## References Y. Xiong, W. Chen, D. Apley, and X. Ding (2007), **Int. J. Numer. Meth. Engng**, A non-stationary covariance-based Kriging method for metamodelling in engineering design. ## See Also `scalingGrad` ## Examples ```r ## 1D Transform of Xiong et al. knots <- c(0, 0.3, 0.8, 1); eta <- c(2, 0.4, 1.4, 1.1) nk <- length(knots) t <- seq(from = 0, to = 1, length = 200) f <- scalingFun(X = matrix(t), knots = list(knots), eta = list(eta)) ## for text positions only itext <- round(length(t) * 0.7) xtext <- t[itext]; ftext <- f[itext] / 2; etamax <- max(eta) ## plot the transform function opar <- par(mfrow = c(2, 1)) par(mar = c(0, 4, 5, 4)) plot(x = t, y = f, type = "l", lwd = 2, col = "orangered", main = "scaling transform f(x) and density g(x)", xlab = "", ylab = "", xaxt = "n", yaxt = "n") axis(side = 4) abline(v = knots, lty = "dotted"); abline(h = 0) text(x = xtext, y = ftext, cex = 1.4, labels = expression(f(x) == integral(g(t)*dt, 0, x))) ## plot the density function, which is piecewise linear scalingDens1d <- approxfun(x = knots, y = eta) g <- scalingDens1d(t) gtext <- 0.5 * g[itext] + 0.6 * etamax par(mar = c(5, 4, 0, 4)) plot(t, g, type = "l", lwd = 2, ylim = c(0, etamax * 1.2), col = "SpringGreen4", xlab = expression(x), ylab ="") abline(v = knots, lty = "dotted") lines(x = knots, y = eta, lty = 1, lwd = 2, type = "h", col = "SpringGreen4") abline(h = 0) text(x = 0.7, y = gtext, cex = 1.4, labels = expression(g(x))) ## show knots with math symbols eta, zeta for (i in 1:nk) { text(x = knots[i], y = eta[i] + 0.12 * etamax, cex = 1.4, labels = substitute(eta[i], list(i = i))) mtext(side = 1, cex = 1.4, at = knots[i], line = 2.4, text = substitute(zeta[i], list(i = i))) } polygon(x = c(knots, knots[nk], knots[1]), y = c(eta, 0, 0), density = 15, angle = 45, col = "SpringGreen", border = NA) par(opar) ```

scalingFun function