aux_kernelcov: Build a centered kernel matrix K
In Rdimtools: Dimension Reduction and Estimation Methods

aux.kernelcov

R Documentation

Build a centered kernel matrix K

Description

From the celebrated Mercer's Theorem, we know that for a mapping φ, there exists a kernel function - or, symmetric bilinear form, K such that

K(x,y) = <φ(x),φ(y)>

where <,> is standard inner product. aux.kernelcov is a collection of 20 such positive definite kernel functions, as well as centering of such kernel since covariance requires a mean to be subtracted and a set of transformed values φ(x_i),i=1,2,…,n are not centered after transformation. Since some kernels require parameters - up to 2, its usage will be listed in arguments section.

Usage

aux.kernelcov(X, ktype)

Arguments

X

an (n\times p) data matrix

ktype

a vector containing the type of kernel and parameters involved. Below the usage is consistent with description

linear: c("linear",c)
polynomial: c("polynomial",c,d)
gaussian: c("gaussian",c)
laplacian: c("laplacian",c)
anova: c("anova",c,d)
sigmoid: c("sigmoid",a,b)
rational quadratic: c("rq",c)
multiquadric: c("mq",c)
inverse quadric: c("iq",c)
inverse multiquadric: c("imq",c)
circular: c("circular",c)
spherical: c("spherical",c)
power/triangular: c("power",d)
log: c("log",d)
spline: c("spline")
Cauchy: c("cauchy",c)
Chi-squared: c("chisq")
histogram intersection: c("histintx")
generalized histogram intersection: c("ghistintx",c,d)
generalized Student-t: c("t",d)

Details

There are 20 kernels supported. Belows are the kernels when given two vectors x,y, K(x,y)

linear: =<x,y>+c
polynomial: =(<x,y>+c)^d
gaussian: =exp(-c\|x-y\|^2), c>0
laplacian: =exp(-c\|x-y\|), c>0
anova: =∑_k exp(-c(x_k-y_k)^2)^d, c>0,d≥ 1
sigmoid: =tanh(a<x,y>+b)
rational quadratic: =1-(\|x-y\|^2)/(\|x-y\|^2+c)
multiquadric: =√{\|x-y\|^2 + c^2}
inverse quadric: =1/(\|x-y\|^2+c^2)
inverse multiquadric: =1/√{\|x-y\|^2+c^2}
circular: = \frac{2}{π} arccos(-\frac{\|x-y\|}{c}) - \frac{2}{π} \frac{\|x-y\|}{c}√{1-(\|x-y\|/c)^2} , c>0
spherical: = 1-1.5\frac{\|x-y\|}{c}+0.5(\|x-y\|/c)^3 , c>0
power/triangular: =-\|x-y\|^d, d≥ 1
log: =-\log (\|x-y\|^d+1)
spline: = ∏_i ( 1+x_i y_i(1+min(x_i,y_i)) - \frac{x_i + y_i}{2} min(x_i,y_i)^2 + \frac{min(x_i,y_i)^3}{3} )
Cauchy: =\frac{c^2}{c^2+\|x-y\|^2}
Chi-squared: =∑_i \frac{2x_i y_i}{x_i+y_i}
histogram intersection: =∑_i min(x_i,y_i)
generalized histogram intersection: =sum_i min( |x_i|^c,|y_i|^d )
generalized Student-t: =1/(1+\|x-y\|^d), d≥ 1

Value

a named list containing

K: a (p\times p) kernelizd gram matrix.
Kcenter: a (p\times p) centered version of K.

Author(s)

Kisung You

References

Hofmann, T., Scholkopf, B., and Smola, A.J. (2008) Kernel methods in machine learning. arXiv:math/0701907.

Examples


## generate a toy data
set.seed(100)
X = aux.gensamples(n=100)

## compute a few kernels
Klin = aux.kernelcov(X, ktype=c("linear",0))
Kgau = aux.kernelcov(X, ktype=c("gaussian",1))
Klap = aux.kernelcov(X, ktype=c("laplacian",1))

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(Klin$K, main="kernel=linear")
image(Kgau$K, main="kernel=gaussian")
image(Klap$K, main="kernel=laplacian")
par(opar)

Rdimtools documentation built on Dec. 28, 2022, 1:44 a.m.