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kernelMatrix

Kernel Matrix functions

Description

kernelMatrix calculates the kernel matrix K_{ij} = k(x_i,x_j) or K_{ij} = k(x_i,y_j).
kernelPol computes the quadratic kernel expression H = z_i z_j k(x_i,x_j), H = z_i k_j k(x_i,y_j).
kernelMult calculates the kernel expansion f(x_i) = ∑_{i=1}^m z_i k(x_i,x_j)
kernelFast computes the kernel matrix, identical to kernelMatrix, except that it also requires the squared norm of the first argument as additional input, useful in iterative kernel matrix calculations.

Usage

## S4 method for signature 'kernel'
kernelMatrix(kernel, x, y = NULL)

## S4 method for signature 'kernel'
kernelPol(kernel, x, y = NULL, z, k = NULL)

## S4 method for signature 'kernel'
kernelMult(kernel, x, y = NULL, z, blocksize = 256)

## S4 method for signature 'kernel'
kernelFast(kernel, x, y, a)

Arguments

`kernel`	the kernel function to be used to calculate the kernel matrix. This has to be a function of class `kernel`, i.e. which can be generated either one of the build in kernel generating functions (e.g., `rbfdot` etc.) or a user defined function of class `kernel` taking two vector arguments and returning a scalar.
`x`	a data matrix to be used to calculate the kernel matrix, or a list of vector when a `stringkernel` is used
`y`	second data matrix to calculate the kernel matrix, or a list of vector when a `stringkernel` is used
`z`	a suitable vector or matrix
`k`	a suitable vector or matrix
`a`	the squared norm of `x`, e.g., `rowSums(x^2)`
`blocksize`	the kernel expansion computations are done block wise to avoid storing the kernel matrix into memory. `blocksize` defines the size of the computational blocks.

Details

Common functions used during kernel based computations.
The kernel parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments. kernlab provides the most popular kernel functions which can be initialized by using the following functions:

rbfdot Radial Basis kernel function
polydot Polynomial kernel function
vanilladot Linear kernel function
tanhdot Hyperbolic tangent kernel function
laplacedot Laplacian kernel function
besseldot Bessel kernel function
anovadot ANOVA RBF kernel function
splinedot the Spline kernel

(see example.)

kernelFast is mainly used in situations where columns of the kernel matrix are computed per invocation. In these cases, evaluating the norm of each row-entry over and over again would cause significant computational overhead.

Value

kernelMatrix returns a symmetric diagonal semi-definite matrix.
kernelPol returns a matrix.
kernelMult usually returns a one-column matrix.

Author(s)

Alexandros Karatzoglou
alexandros.karatzoglou@ci.tuwien.ac.at

Examples

## use the spam data
data(spam)
dt <- as.matrix(spam[c(10:20,3000:3010),-58])

## initialize kernel function 
rbf <- rbfdot(sigma = 0.05)
rbf

## calculate kernel matrix
kernelMatrix(rbf, dt)

yt <- as.matrix(as.integer(spam[c(10:20,3000:3010),58]))
yt[yt==2] <- -1

## calculate the quadratic kernel expression
kernelPol(rbf, dt, ,yt)

## calculate the kernel expansion
kernelMult(rbf, dt, ,yt)

kernlab

Kernel-Based Machine Learning Lab

v0.9-29

GPL-2

Authors

Alexandros Karatzoglou [aut, cre], Alex Smola [aut], Kurt Hornik [aut], National ICT Australia (NICTA) [cph], Michael A. Maniscalco [ctb, cph], Choon Hui Teo [ctb]

Initial release