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.
scalingFun(X, knots, eta, plot=FALSE)
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. |
The image of X by a scaling transformation of parameters knots and eta
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.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.