From the celebrated Mercer's Theorem, we know that for a mapping $$\phi$$, there exists a kernel function - or, symmetric bilinear form, $$K$$ such that $$K(x,y) = <\phi(x),\phi(y)>$$ 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 $$\phi(x_i),i=1,2,\dots,n$$ are not centered after transformation. Since some kernels require parameters - up to 2, its usage will be listed in arguments section.

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)

c("rq",c)

c("mq",c)

c("iq",c)

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)

## Value

a named list containing

K

a $$(p\times p)$$ kernelizd gram matrix.

Kcenter

a $$(p\times p)$$ centered version of K.

## 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

$$=\sum_k exp(-c(x_k-y_k)^2)^d$$, $$c>0,d\ge 1$$

sigmoid

$$=tanh(a<x,y>+b)$$

$$=1-(\|x-y\|^2)/(\|x-y\|^2+c)$$

$$=\sqrt{\|x-y\|^2 + c^2}$$

$$=1/(\|x-y\|^2+c^2)$$

$$=1/\sqrt{\|x-y\|^2+c^2}$$

circular

$$= \frac{2}{\pi} arccos(-\frac{\|x-y\|}{c}) - \frac{2}{\pi} \frac{\|x-y\|}{c}\sqrt{1-(\|x-y\|/c)^2}$$, $$c>0$$

spherical

$$= 1-1.5\frac{\|x-y\|}{c}+0.5(\|x-y\|/c)^3$$, $$c>0$$

power/triangular

$$=-\|x-y\|^d$$, $$d\ge 1$$

log

$$=-\log (\|x-y\|^d+1)$$

spline

$$= \prod_i ( 1+x_i y_i(1+min(x_i,y_i)) - \frac{x_i + y_i}{2} min(x_i,y_i)^2 + \frac{min(x_i,y_i)^3}{3} )$$

Cauchy

$$=\frac{c^2}{c^2+\|x-y\|^2}$$

Chi-squared

$$=\sum_i \frac{2x_i y_i}{x_i+y_i}$$

histogram intersection

$$=\sum_i min(x_i,y_i)$$

generalized histogram intersection

$$=sum_i min( |x_i|^c,|y_i|^d )$$

generalized Student-t

$$=1/(1+\|x-y\|^d)$$, $$d\ge 1$$

## References

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

Kisung You

## Examples

# \donttest{
## 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
image(Klin$K, main="kernel=linear") image(Kgau$K, main="kernel=gaussian")