Given warpings for ordinal inputs.
warpNorm
warpUnorm
warpPower
warpSpline1
warpSpline2The format is a list of 6:
$ fun : the warping function. The second argument is the vector of
parameters. The function returns a numeric vector with an attribute
"gradient" giving the derivative w.r.t. the parameters.
$ parNames : names of warping parameters (character vector).
$ parDefault: default values of warping parameters (numeric vector).
$ parLower : lower bounds of warping parameters (numeric vector).
$ parUpper : upper bounds of warping parameters (numeric vector).
$ hasGrad : a boolean equal to TRUE if gradient is
supplied as an attribute of fun.
See covOrd for the definition of a warping in this
context. At this stage, two warpings corresponding to cumulative
density functions (cdf) are implemented:
Normal distribution, truncated to \([0,1]\): $$F(x) = [N(x) - N(0)] / [N(1) - N(0)]$$ where \(N(x) = \Phi([x - \mu] / \sigma)\) is the cdf of the normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
Power distribution on \([0, 1]\): \(F(x) = x^{pow}\).
Furthermore, a warping corresponding to unnormalized Normal cdf is implemented,
as well as spline warpings of degree 1 and 2.
Splines are defined by a sequence of k knots between 0 and 1. The first knot is 0, and the last is 1.
A spline warping of degree 1 is a continuous piecewise linear function.
It is parameterized by a positive vector of length k-1, representing the increments at knots.
A spline warping of degree 2 is a non-decreasing quadratic spline. It is obtained by integrating a spline of degree 1.
Its parameters form a positive vector of length k, representing the derivatives at knots. The implementation relies on the function scalingFun1d of DiceKriging package.