rchemo (version 0.1-3)

krbf: Kernel functions

Description

Building Gram matrices for different kernels (e.g. Scholkopf & Smola 2002).

- radial basis: exp(-gamma * |x - y|^2)

- polynomial: (gamma * x' * y + coef0)^degree

- sigmoid: tanh(gamma * x' * y + coef0)

Usage

krbf(X, Y = NULL, gamma = 1)
kpol(X, Y = NULL, degree = 1, gamma = 1, coef0 = 0)
ktanh(X, Y = NULL, gamma = 1, coef0 = 0)

Value

Gram matrix

Arguments

X

Dataset (\(n, p\)).

Y

Dataset (\(m, p\)). The resulting Gram matrix \(K(X, Y)\) has dimensionnality (\(n, m\)). If NULL (default), Y is set equal to X.

gamma

value of the gamma parameter in the kernel calculation.

degree

For kpol: value of the degree parameter in the polynomial kernel calculation.

coef0

For kpol and ktanh: value of the coef0 parameter in the polynomial or sigmoid kernel calculation.

References

Scholkopf, B., Smola, A.J., 2002. Learning with kernels: support vector machines, regularization, optimization, and beyond, Adaptive computation and machine learning. MIT Press, Cambridge, Mass.

Examples

n <- 5 ; p <- 3
Xtrain <- matrix(rnorm(n * p), ncol = p)
Xtest <- Xtrain[1:2, , drop = FALSE] 

gamma <- .8
krbf(Xtrain, gamma = gamma)

krbf(Xtest, Xtrain, gamma = gamma)
exp(-.5 * euclsq(Xtest, Xtrain) / gamma^2)

kpol(Xtrain, degree = 2, gamma = .5, coef0 = 1)