kernel functions {kerndwd}

Kernel Functions

Description

Kernel functions provided in the R package kernlab. Details can be seen in the reference below.
The Gaussian RBF kernel k(x,x') = \exp(-\sigma \|x - x'\|^2)
The Polynomial kernel k(x,x') = (scale <x, x'> + offset)^{degree}
The Linear kernel k(x,x') = <x, x'>
The Laplacian kernel k(x,x') = \exp(-\sigma \|x - x'\|)
The Bessel kernel k(x,x') = (- \mathrm{Bessel}_{(\nu+1)}^n \sigma \|x - x'\|^2)
The ANOVA RBF kernel k(x,x') = \sum_{1\leq i_1 \ldots < i_D \leq N} \prod_{d=1}^D k(x_{id}, {x'}_{id}) where k(x, x) is a Gaussian RBF kernel.
The Spline kernel \prod_{d=1}^D 1 + x_i x_j + x_i x_j \min(x_i, x_j) - \frac{x_i + x_j}{2} \min(x_i,x_j)^2 + \frac{\min(x_i,x_j)^3}{3}. The parameter sigma used in rbfdot can be selected by sigest().

Usage

rbfdot(sigma = 1)
polydot(degree = 1, scale = 1, offset = 1)
vanilladot()
laplacedot(sigma = 1)
besseldot(sigma = 1, order = 1, degree = 1)
anovadot(sigma = 1, degree = 1)
splinedot()
sigest(x)

Arguments

sigma

The inverse kernel width used by the Gaussian, the Laplacian, the Bessel, and the ANOVA kernel.

degree

The degree of the polynomial, bessel or ANOVA kernel function. This has to be an positive integer.

scale

The scaling parameter of the polynomial kernel function.

offset

The offset used in a polynomial kernel.

order

The order of the Bessel function to be used as a kernel.

x

The design matrix used in kerndwd when sigest is called to estimate sigma in rbfdot().

Details

These R functions and descriptions are directly duplicated and/or adapted from the R package kernlab.

Value

Return an S4 object of class kernel which can be used as the argument of kern when fitting a kerndwd model.

References

Wang, B. and Zou, H. (2018) “Another Look at Distance Weighted Discrimination," Journal of Royal Statistical Society, Series B, 80(1), 177–198.
https://rss.onlinelibrary.wiley.com/doi/10.1111/rssb.12244

Karatzoglou, A., Smola, A., Hornik, K., and Zeileis, A. (2004) “kernlab – An S4 Package for Kernel Methods in R", Journal of Statistical Software, 11(9), 1–20.
https://www.jstatsoft.org/v11/i09/paper

Examples

data(BUPA)
# generate a linear kernel
kfun = vanilladot()

# generate a Laplacian kernel function with sigma = 1
kfun = laplacedot(sigma=1)

# generate a Gaussian kernel function with sigma estimated by sigest()
kfun = rbfdot(sigma=sigest(BUPA$X))

# set kern=kfun when fitting a kerndwd object
data(BUPA)
BUPA$X = scale(BUPA$X, center=TRUE, scale=TRUE)
lambda = 10^(seq(-3, 3, length.out=10))
m1 = kerndwd(BUPA$X, BUPA$y, kern=kfun,
  qval=1, lambda=lambda, eps=1e-5, maxit=1e5)