1\name{dots} 2\alias{dots} 3\alias{kernels} 4\alias{rbfdot} 5\alias{polydot} 6\alias{tanhdot} 7\alias{vanilladot} 8\alias{laplacedot} 9\alias{besseldot} 10\alias{anovadot} 11\alias{fourierdot} 12\alias{splinedot} 13\alias{kpar} 14\alias{kfunction} 15\alias{show,kernel-method} 16\title{Kernel Functions} 17\description{ 18 The kernel generating functions provided in kernlab. \cr 19 The Gaussian RBF kernel \eqn{k(x,x') = \exp(-\sigma \|x - x'\|^2)} \cr 20 The Polynomial kernel \eqn{k(x,x') = (scale <x, x'> + offset)^{degree}}\cr 21 The Linear kernel \eqn{k(x,x') = <x, x'>}\cr 22 The Hyperbolic tangent kernel \eqn{k(x, x') = \tanh(scale <x, x'> + offset)}\cr 23 The Laplacian kernel \eqn{k(x,x') = \exp(-\sigma \|x - x'\|)} \cr 24 The Bessel kernel \eqn{k(x,x') = (- Bessel_{(\nu+1)}^n \sigma \|x - x'\|^2)} \cr 25 The ANOVA RBF kernel \eqn{k(x,x') = \sum_{1\leq i_1 \ldots < i_D \leq 26 N} \prod_{d=1}^D k(x_{id}, {x'}_{id})} where k(x,x) is a Gaussian 27 RBF kernel. \cr 28 The Spline kernel \eqn{ \prod_{d=1}^D 1 + x_i x_j + x_i x_j min(x_i, 29 x_j) - \frac{x_i + x_j}{2} min(x_i,x_j)^2 + 30 \frac{min(x_i,x_j)^3}{3}} \\ 31 The String kernels (see \code{stringdot}. 32} 33\usage{ 34rbfdot(sigma = 1) 35 36polydot(degree = 1, scale = 1, offset = 1) 37 38tanhdot(scale = 1, offset = 1) 39 40vanilladot() 41 42laplacedot(sigma = 1) 43 44besseldot(sigma = 1, order = 1, degree = 1) 45 46anovadot(sigma = 1, degree = 1) 47 48splinedot() 49} 50 51\arguments{ 52 \item{sigma}{The inverse kernel width used by the Gaussian the 53 Laplacian, the Bessel and the ANOVA kernel } 54 \item{degree}{The degree of the polynomial, bessel or ANOVA 55 kernel function. This has to be an positive integer.} 56 \item{scale}{The scaling parameter of the polynomial and tangent 57 kernel is a convenient way of normalizing 58 patterns without the need to modify the data itself} 59 \item{offset}{The offset used in a polynomial or hyperbolic tangent 60 kernel} 61 \item{order}{The order of the Bessel function to be used as a kernel} 62} 63\details{ 64 The kernel generating functions are used to initialize a kernel 65 function 66 which calculates the dot (inner) product between two feature vectors in a 67 Hilbert Space. These functions can be passed as a \code{kernel} argument on almost all 68 functions in \pkg{kernlab}(e.g., \code{ksvm}, \code{kpca} etc). 69 70 Although using one of the existing kernel functions as a 71 \code{kernel} argument in various functions in \pkg{kernlab} has the 72 advantage that optimized code is used to calculate various kernel expressions, 73 any other function implementing a dot product of class \code{kernel} can also be used as a kernel 74 argument. This allows the user to use, test and develop special kernels 75 for a given data set or algorithm. 76 For details on the string kernels see \code{stringdot}. 77 } 78\value{ 79 Return an S4 object of class \code{kernel} which extents the 80 \code{function} class. The resulting function implements the given 81 kernel calculating the inner (dot) product between two vectors. 82 \item{kpar}{a list containing the kernel parameters (hyperparameters) 83 used.} 84 The kernel parameters can be accessed by the \code{kpar} function. 85 } 86 87\author{Alexandros Karatzoglou\cr 88 \email{alexandros.karatzoglou@ci.tuwien.ac.at}} 89 90\note{If the offset in the Polynomial kernel is set to $0$, we obtain homogeneous polynomial 91 kernels, for positive values, we have inhomogeneous 92 kernels. Note that for negative values the kernel does not satisfy Mercer's 93 condition and thus the optimizers may fail. \cr 94 95 In the Hyperbolic tangent kernel if the offset is negative the likelihood of obtaining a kernel 96 matrix that is not positive definite is much higher (since then even some 97 diagonal elements may be negative), hence if this kernel has to be used, the 98 offset should always be positive. Note, however, that this is no guarantee 99 that the kernel will be positive. 100} 101 102 103 104 105\seealso{\code{stringdot}, \code{\link{kernelMatrix} }, \code{\link{kernelMult}}, \code{\link{kernelPol}}} 106\examples{ 107rbfkernel <- rbfdot(sigma = 0.1) 108rbfkernel 109 110kpar(rbfkernel) 111 112## create two vectors 113x <- rnorm(10) 114y <- rnorm(10) 115 116## calculate dot product 117rbfkernel(x,y) 118 119} 120\keyword{symbolmath} 121 122