dots: Kernel Functions
In kerndwd: Distance Weighted Discrimination (DWD) and Kernel Methods

Description Usage Arguments Details Value References Examples

Kernel functions provided in the R package kernlab. Details can be seen in the reference below.
The Gaussian RBF kernel k(x, x') = exp(-σ|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(-σ|x - x'|)
The Bessel kernel k(x, x') = (-Bessel{(ν+1)}^n σ|x - x'|^2)
The ANOVA RBF kernel k(x, x') = sum_{1 <= i_1, ..., < i_D <= 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) - (x_i + x_j)/2 min(x_i,x_j)^2 + (min(x_i,x_j)^3)/3. The parameter sigma used in rbfdot can be selected by sigest().

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)

`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()`.

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

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

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

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)