`kumaraswamy` — Kumaraswamy warping

Description

The Kumaraswamy CDF maps \([0,1] \to [0,1]\):

\[w(x; a, b) = 1 - (1 - x^a)^b, \quad a,b > 0.\]

It generalises the Beta CDF with a closed-form inverse, making it ideal for inputs bounded in \([0,1]\). Shape is S-curve, left-skewed or right-skewed depending on \((a,b)\).

Specification

warp_kumaraswamy()   # returns "kumaraswamy"

Parameters

Symbol	Role	Range
\(a\)	first shape	\(>0\)
\(b\)	second shape	\(>0\)

Warping shape

kuma <- function(x, a, b) 1 - (1 - x^a)^b
x <- seq(0, 1, length.out = 300)
params <- list(c(0.5, 0.5), c(2, 5), c(5, 2), c(2, 2), c(1, 1))
cols   <- c("steelblue","darkorange","red","darkgreen","grey50")
labels <- sapply(params, function(p) sprintf("a=%.1f, b=%.1f", p[1], p[2]))

plot(0:1, 0:1, type="n", xlab="x", ylab="w(x)",
     main="Kumaraswamy warping shapes")
for (i in seq_along(params))
  lines(x, kuma(x, params[[i]][1], params[[i]][2]), col=cols[i], lwd=2)
legend("topleft", labels, col=cols, lwd=2, cex=0.8)
abline(0, 1, lty=3, col="grey70")

Kumaraswamy warping shapes

Regression example

library(rlibkriging)
# Nonlinear function on [0,1]
f <- function(x) sin(pi * x^2) + 0.3 * cos(8 * pi * x)
set.seed(42)
X <- as.matrix(runif(15))
y <- f(X)

wk <- WarpKriging(y, X,
                  warping = warp_kumaraswamy(),
                  kernel  = "matern5_2",
                  optim   = "BFGS+Adam")

x <- as.matrix(seq(0, 1, length.out = 200))
p <- wk$predict(x, return_stdev = TRUE)

plot(f, xlim = c(0,1), col = "grey", lty = 2, ylab = "y",
     main = "kumaraswamy warping")
points(X, y, pch = 19)
lines(x, p$mean, col = "steelblue", lwd = 2)
polygon(c(x, rev(x)),
        c(p$mean - 2*p$stdev, rev(p$mean + 2*p$stdev)),
        border = NA, col = rgb(0.27, 0.51, 0.71, 0.2))

Kumaraswamy warping regression

References

Snoek, J., Swersky, K., Zemel, R. S., & Adams, R. P. (2014). Input Warping for Bayesian Optimization of Non-Stationary Functions. Proceedings of ICML 2014, PMLR 32(2):1674–1682. arXiv: 1402.0929

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1–2), 79–88. DOI: 10.1016/0022-1694(80)90008-X

kumaraswamy — Kumaraswamy warping