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\SweaveOpts{engine=R,eps=FALSE}
%\VignetteIndexEntry{kernlab - An S4 Package for Kernel Methods in R}
%\VignetteDepends{kernlab}
%\VignetteKeywords{kernel methods, support vector machines, quadratic programming, ranking, clustering, S4, R}
%\VignettePackage{kernlab}

<<preliminaries,echo=FALSE,results=hide>>=
library(kernlab)
options(width = 70)
@

\title{\pkg{kernlab} -- An \proglang{S4} Package for Kernel Methods in \proglang{R}}
\Plaintitle{kernlab - An S4 Package for Kernel Methods in R}

\author{Alexandros Karatzoglou\\Technische Universit\"at Wien
   \And Alex Smola\\Australian National University, NICTA
   \And Kurt Hornik\\Wirtschaftsuniversit\"at Wien
   \And Achim Zeileis\\Wirtschaftsuniversit\"at Wien 
}
\Plainauthor{Alexandros Karatzoglou, Alex Smola, Kurt Hornik, Achim Zeileis}

\Abstract{
  A shorter version of this introduction to the \proglang{R} package \pkg{kernlab}
  is published as \cite{kernlab:Karatzoglou+Smola+Hornik:2004} in the
  \emph{Journal of Statistical Software}.

  \pkg{kernlab} is an extensible package for kernel-based machine
  learning methods in \proglang{R}.  It takes
  advantage of \proglang{R}'s new \proglang{S4} object model and provides a
  framework for creating and using kernel-based algorithms.  The package
  contains dot product primitives (kernels), implementations of support
  vector machines and the relevance vector machine, Gaussian processes,
  a ranking algorithm, kernel PCA, kernel CCA, kernel feature analysis,
  online kernel methods and a spectral clustering
  algorithm.  Moreover it provides a general purpose quadratic
  programming solver, and an incomplete Cholesky decomposition method.
}


\Keywords{kernel methods, support vector machines, quadratic
programming, ranking, clustering, \proglang{S4}, \proglang{R}}
\Plainkeywords{kernel methods, support vector machines, quadratic
programming, ranking, clustering, S4, R}

\begin{document}

\section{Introduction}

Machine learning is all about extracting structure from data, but it is
often difficult to solve problems like classification, regression and
clustering
in the space in which the underlying observations have been made.

Kernel-based learning methods use an implicit mapping of the input data
into a high dimensional feature space defined by a kernel function,
i.e., a function returning the inner product $ \langle \Phi(x),\Phi(y)
\rangle$ between the images of two data points $x, y$ in the feature
space.  The learning then takes place in the feature space, provided the
learning algorithm can be entirely rewritten so that the data points
only appear inside dot products with other points.  This is often
referred to as the ``kernel trick''
\citep{kernlab:Schoelkopf+Smola:2002}.  More precisely, if a projection
$\Phi: X \rightarrow H$ is used, the dot product
$\langle\Phi(x),\Phi(y)\rangle$ can be represented by a kernel
function~$k$
\begin{equation} \label{eq:kernel}
k(x,y)= \langle \Phi(x),\Phi(y) \rangle,
\end{equation}
which is computationally simpler than explicitly projecting $x$ and $y$
into the feature space~$H$.

One interesting property of kernel-based systems is that, once a valid
kernel function has been selected, one can practically work in spaces of
any dimension without paying any computational cost, since feature
mapping is never effectively performed. In fact, one does not even need
to know which features are being used.

Another advantage is the that one can design and use a kernel for a
particular problem that could be applied directly to the data without
the need for a feature extraction process.  This is particularly
important in problems where a lot of structure of the data is lost by
the feature extraction process (e.g., text processing).  The inherent
modularity of kernel-based learning methods allows one to use any valid
kernel on a kernel-based algorithm.

\subsection{Software review}

The most prominent kernel based learning algorithm is without doubt the
support vector machine (SVM), so the existence of many support vector
machine packages comes as little surprise.  Most of the existing SVM
software is written in \proglang{C} or \proglang{C++}, e.g.\ the award
winning
\pkg{libsvm}\footnote{\url{http://www.csie.ntu.edu.tw/~cjlin/libsvm/}}
\citep{kernlab:Chang+Lin:2001},
\pkg{SVMlight}\footnote{\url{http://svmlight.joachims.org}}
\citep{kernlab:joachim:1999},
\pkg{SVMTorch}\footnote{\url{http://www.torch.ch}}, Royal Holloway
Support Vector Machines\footnote{\url{http://svm.dcs.rhbnc.ac.uk}},
\pkg{mySVM}\footnote{\url{http://www-ai.cs.uni-dortmund.de/SOFTWARE/MYSVM/index.eng.html}},
and \pkg{M-SVM}\footnote{\url{http://www.loria.fr/~guermeur/}} with
many packages providing interfaces to \proglang{MATLAB} (such as
\pkg{libsvm}), and even some native \proglang{MATLAB}
toolboxes\footnote{
  \url{http://www.isis.ecs.soton.ac.uk/resources/svminfo/}}\,\footnote{
  \url{http://asi.insa-rouen.fr/~arakotom/toolbox/index}}\,\footnote{
  \url{http://www.cis.tugraz.at/igi/aschwaig/software.html}}.

Putting SVM specific software aside and considering the abundance of
other kernel-based algorithms published nowadays, there is little
software available implementing a wider range of kernel methods with
some exceptions like the
\pkg{Spider}\footnote{\url{http://www.kyb.tuebingen.mpg.de/bs/people/spider/}}
software which provides a \proglang{MATLAB} interface to various
\proglang{C}/\proglang{C++} SVM libraries and \proglang{MATLAB}
implementations of various kernel-based algorithms,
\pkg{Torch} \footnote{\url{http://www.torch.ch}} which also includes more
traditional machine learning algorithms, and the occasional
\proglang{MATLAB} or \proglang{C} program found on a personal web page where
an author includes code from a published paper.

\subsection[R software]{\proglang{R} software}

The \proglang{R} package \pkg{e1071} offers an interface to the award
winning \pkg{libsvm} \citep{kernlab:Chang+Lin:2001}, a very efficient
SVM implementation.  \pkg{libsvm} provides a robust and fast SVM
implementation and produces state of the art results on most
classification and regression problems
\citep{kernlab:Meyer+Leisch+Hornik:2003}.  The \proglang{R} interface
provided in \pkg{e1071} adds all standard \proglang{R} functionality like
object orientation and formula interfaces to \pkg{libsvm}.  Another
SVM related \proglang{R} package which was made recently available is
\pkg{klaR} \citep{kernlab:Roever:2004} which includes an interface to
\pkg{SVMlight}, a popular SVM implementation along with other
classification tools like Regularized Discriminant Analysis.

However, most of the \pkg{libsvm} and \pkg{klaR} SVM code is in
\proglang{C++}.  Therefore, if one would like to extend or enhance the
code with e.g.\ new kernels or different optimizers, one would have to
modify the core \proglang{C++} code.

\section[kernlab]{\pkg{kernlab}}

\pkg{kernlab} aims to provide the \proglang{R} user with basic kernel
functionality (e.g., like computing a kernel matrix using a particular
kernel), along with some utility functions commonly used in kernel-based
methods like a quadratic programming solver, and modern kernel-based
algorithms based on the functionality that the package provides. Taking
advantage of the inherent modularity of kernel-based methods,
\pkg{kernlab} aims to allow the user to switch between kernels on an
existing algorithm and even create and use own kernel functions for the
kernel methods provided in the package.


\subsection[S4 objects]{\proglang{S4} objects}

\pkg{kernlab} uses \proglang{R}'s new object model described in
``Programming with Data'' \citep{kernlab:Chambers:1998} which is known
as the \proglang{S4} class system and is implemented in the
\pkg{methods} package.

In contrast with the older \proglang{S3} model for objects in \proglang{R},
classes, slots, and methods relationships must be declared explicitly
when using the \proglang{S4} system.  The number and types of slots in an
instance of a class have to be established at the time the class is
defined.  The objects from the class are validated against this
definition and have to comply to it at any time.  \proglang{S4} also
requires formal declarations of methods, unlike the informal system of
using function names to identify a certain method in \proglang{S3}.

An \proglang{S4} method is declared by a call to \code{setMethod} along
with the name and a ``signature'' of the arguments.  The signature is
used to identify the classes of one or more arguments of the method.
Generic functions can be declared using the \code{setGeneric}
function.  Although such formal declarations require package authors to
be more disciplined than when using the informal \proglang{S3} classes,
they provide assurance that each object in a class has the required
slots and that the names and classes of data in the slots are
consistent.

An example of a class used in \pkg{kernlab} is shown below.
Typically, in a return object we want to include information on the
result of the method along with additional information and parameters.
Usually \pkg{kernlab}'s classes include slots for the kernel function
used and the results and additional useful information.
\begin{smallexample}
setClass("specc",
          representation("vector",  # the vector containing the cluster
                          centers="matrix",     # the cluster centers
                          size="vector",        # size of each cluster
                          kernelf="function",   # kernel function used
                          withinss = "vector"), # within cluster sum of squares
          prototype = structure(.Data = vector(), 
                                centers = matrix(),
                                size = matrix(),
                                kernelf = ls,
                                withinss = vector()))
\end{smallexample}

Accessor and assignment function are defined and used to access the
content of each slot which can be also accessed with the \verb|@|
operator.

\subsection{Namespace}

Namespaces were introduced in \proglang{R} 1.7.0 and provide a means for
packages to control the way global variables and methods are being made
available.  Due to the number of assignment and accessor function
involved, a namespace is used to control the methods which are being
made visible outside the package.  Since \proglang{S4} methods are being
used, the \pkg{kernlab} namespace also imports methods and variables
from the \pkg{methods} package.

\subsection{Data}

The \pkg{kernlab} package also includes data set which will be used
to illustrate the methods included in the package.  The \code{spam}
data set \citep{kernlab:Hastie:2001} set collected at Hewlett-Packard
Labs contains data on 2788 and 1813 e-mails classified as non-spam and
spam, respectively. The 57 variables of
each data vector indicate the frequency of certain words and characters
in the e-mail.

Another data set included in \pkg{kernlab}, the \code{income} data
set \citep{kernlab:Hastie:2001}, is taken by a marketing survey in the
San Francisco Bay concerning the income of shopping mall customers.  It
consists of 14 demographic attributes (nominal and ordinal variables)
including the income and 8993 observations.

The \code{ticdata} data set \citep{kernlab:Putten:2000} was used in
the 2000 Coil Challenge and contains information on customers of an
insurance company.  The data consists of 86 variables and includes
product usage data and socio-demographic data derived from zip area
codes.  The data was collected to answer the following question: Can you
predict who would be interested in buying a caravan insurance policy and
give an explanation why?

The \code{promotergene} is a data set of
E. Coli promoter gene sequences (DNA) with 106 observations and 58
variables available at the UCI Machine Learning repository.
Promoters have a region where a protein (RNA polymerase) must make
contact and the helical DNA sequence must have a valid conformation so that
the two pieces of the contact region spatially align. The data contains
DNA sequences of promoters and non-promoters.

The \code{spirals} data set was created by the
\code{mlbench.spirals} function in the \pkg{mlbench} package
\citep{kernlab:Leisch+Dimitriadou}.  This two-dimensional data set with
300 data points consists of two spirals where Gaussian noise is added to
each data point.

\subsection{Kernels}

A kernel function~$k$ calculates the inner product of two vectors $x$,
$x'$ in a given feature mapping $\Phi: X \rightarrow H$.  The notion of
a kernel is obviously central in the making of any kernel-based
algorithm and consequently also in any software package containing
kernel-based methods.

Kernels in \pkg{kernlab} are \proglang{S4} objects of class
\code{kernel} extending the \code{function} class with one
additional slot containing a list with the kernel hyper-parameters.
Package \pkg{kernlab} includes 7 different kernel classes which all
contain the class \code{kernel} and are used to implement the existing
kernels.  These classes are used in the function dispatch mechanism of
the kernel utility functions described below.  Existing kernel functions
are initialized by ``creator'' functions.  All kernel functions take two
feature vectors as parameters and return the scalar dot product of the
vectors.  An example of the functionality of a kernel in
\pkg{kernlab}:

<<rbf1,echo=TRUE>>=
## create a RBF kernel function with sigma hyper-parameter 0.05 
rbf <- rbfdot(sigma = 0.05)
rbf
## create two random feature vectors
x <- rnorm(10)
y <- rnorm(10)
## compute dot product between x,y
rbf(x, y)
@
The package includes implementations of the following kernels:

\begin{itemize}
 \item the linear \code{vanilladot} kernel implements the simplest of all
  kernel functions
  \begin{equation}
    k(x,x') = \langle x, x' \rangle
  \end{equation}
  which is useful specially when dealing with large sparse data
  vectors~$x$ as is usually the case in text categorization.
  
 \item the Gaussian radial basis function \code{rbfdot}
  \begin{equation}
    k(x,x') = \exp(-\sigma \|x - x'\|^2)
  \end{equation} 
  which is a general purpose kernel and is typically used when no
  further prior knowledge is available about the data.

 \item the polynomial kernel \code{polydot}
  \begin{equation}
    k(x, x') = 
    \left(
      \mathrm{scale} \cdot \langle x, x' \rangle + 
      \mathrm{offset}
    \right)^\mathrm{degree}.
  \end{equation}
  which is used in classification of images.

 \item the hyperbolic tangent kernel \code{tanhdot}
  \begin{equation}
    k(x, x') = 
    \tanh 
    \left(
      \mathrm{scale} \cdot \langle x, x' \rangle + \mathrm{offset}
    \right)
  \end{equation}
  which is mainly used as a proxy for neural networks.
  
 \item the Bessel function of the first kind kernel \code{besseldot}
  \begin{equation}
    k(x, x') = 
    \frac{\mathrm{Bessel}_{(\nu+1)}^n(\sigma \|x - x'\|)}
    {(\|x-x'\|)^{-n(\nu+1)}}.
  \end{equation}
  is a general purpose kernel and is typically used when no further
  prior knowledge is available and mainly popular in the Gaussian
  process community.

 \item the Laplace radial basis kernel \code{laplacedot}
  \begin{equation}
    k(x, x') = \exp(-\sigma \|x - x'\|)
  \end{equation}
  which is a general purpose kernel and is typically used when no
  further prior knowledge is available.

 \item the ANOVA radial basis kernel \code{anovadot} performs well in multidimensional regression problems
  \begin{equation}
    k(x, x') = \left(\sum_{k=1}^{n}\exp(-\sigma(x^k-{x'}^k)^2)\right)^{d}
  \end{equation}
  where $x^k$ is the $k$th component of $x$.
\end{itemize}

\subsection{Kernel utility methods}

The package also includes methods for computing commonly used kernel
expressions (e.g., the Gram matrix).  These methods are written in such
a way that they take functions (i.e., kernels) and matrices (i.e.,
vectors of patterns) as arguments.  These can be either the kernel
functions already included in \pkg{kernlab} or any other function
implementing a valid dot product (taking two vector arguments and
returning a scalar).  In case one of the already implemented kernels is
used, the function calls a vectorized implementation of the
corresponding function.  Moreover, in the case of symmetric matrices
(e.g., the dot product matrix of a Support Vector Machine) they only
require one argument rather than having to pass the same matrix twice
(for rows and columns).

The computations for the kernels already available in the package are
vectorized whenever possible which guarantees good performance and
acceptable memory requirements.  Users can define their own kernel by
creating a function which takes two vectors as arguments (the data
points) and returns a scalar (the dot product).  This function can then
be based as an argument to the kernel utility methods.  For a user
defined kernel the dispatch mechanism calls a generic method
implementation which calculates the expression by passing the kernel
function through a pair of \code{for} loops.  The kernel methods
included are:

\begin{description}
  
 \item[\code{kernelMatrix}] This is the most commonly used function.
  It computes $k(x, x')$, i.e., it computes the matrix $K$ where $K_{ij}
  = k(x_i, x_j)$ and $x$ is a \emph{row} vector.  In particular,
\begin{verbatim}
K <- kernelMatrix(kernel, x)
\end{verbatim}
  computes the matrix $K_{ij} = k(x_i, x_j)$ where the $x_i$ are the
  columns of $X$ and
\begin{verbatim}
K <- kernelMatrix(kernel, x1, x2)
\end{verbatim}
  computes the matrix $K_{ij} = k(x1_i, x2_j)$.
  
  \item[\code{kernelFast}]
  This method is different to \code{kernelMatrix} for \code{rbfdot}, \code{besseldot}, 
  and the \code{laplacedot} kernel, which are all RBF kernels. 
  It is identical to \code{kernelMatrix}, 
  except that it also requires the squared norm of the 
  first argument as additional input.
  It is mainly used in kernel algorithms, where columns
  of the kernel matrix are computed per invocation. In these cases,
  evaluating the norm of each column-entry as it is done on a \code{kernelMatrix} 
  invocation on an RBF kernel, over and over again would cause
  significant computational overhead. Its invocation is via
\begin{verbatim}
K = kernelFast(kernel, x1, x2, a)
\end{verbatim}
  Here $a$ is a vector containing the squared norms of $x1$.

 \item[\code{kernelMult}] is a convenient way of computing kernel
  expansions.  It returns the vector $f = (f(x_1), \dots, f(x_m))$ where
  \begin{equation}
    f(x_i) = \sum_{j=1}^{m} k(x_i, x_j) \alpha_j, 
    \mbox{~hence~} f = K \alpha.
  \end{equation}
  The need for such a function arises from the fact that $K$ may
  sometimes be larger than the memory available.  Therefore, it is
  convenient to compute $K$ only in stripes and discard the latter after
  the corresponding part of $K \alpha$ has been computed.  The parameter
  \code{blocksize} determines the number of rows in the stripes.  In
  particular,
\begin{verbatim}
f <- kernelMult(kernel, x, alpha)
\end{verbatim}
  computes $f_i = \sum_{j=1}^m k(x_i, x_j) \alpha_j$ and
\begin{verbatim}
f <- kernelMult(kernel, x1, x2, alpha)
\end{verbatim}
  computes $f_i = \sum_{j=1}^m k(x1_i, x2_j) \alpha_j$. 

 \item[\code{kernelPol}]
  is a method very similar to \code{kernelMatrix} with the only
  difference that rather than computing $K_{ij} = k(x_i, x_j)$ it
  computes $K_{ij} = y_i y_j k(x_i, x_j)$.  This means that 
\begin{verbatim}
K <- kernelPol(kernel, x, y)
\end{verbatim}
  computes the matrix $K_{ij} = y_i y_j k(x_i, x_j)$ where the $x_i$ are
  the columns of $x$ and $y_i$ are elements of the vector~$y$.  Moreover,
\begin{verbatim}
K <- kernelPol(kernel, x1, x2, y1, y2)
\end{verbatim}
  computes the matrix $K_{ij} = y1_i y2_j k(x1_i, x2_j)$.  Both
  \code{x1} and \code{x2} may be matrices and \code{y1} and
  \code{y2} vectors.
\end{description}

An example using these functions :
<<kernelMatrix,echo=TRUE>>=
## create a RBF kernel function with sigma hyper-parameter 0.05 
poly <- polydot(degree=2)
## create artificial data set
x <- matrix(rnorm(60), 6, 10)
y <- matrix(rnorm(40), 4, 10)
## compute kernel matrix
kx <- kernelMatrix(poly, x)
kxy <- kernelMatrix(poly, x, y)
@

\section{Kernel methods}

Providing a solid base for creating kernel-based methods is part of what
we are trying to achieve with this package, the other being to provide a
wider range of kernel-based methods in \proglang{R}.  In the rest of the
paper we present the kernel-based methods available in \pkg{kernlab}.
All the methods in \pkg{kernlab} can be used with any of the kernels
included in the package as well as with any valid user-defined kernel.
User defined kernel functions can be passed to existing kernel-methods
in the \code{kernel} argument.

\subsection{Support vector machine}

Support vector machines \citep{kernlab:Vapnik:1998} have gained
prominence in the field of machine learning and pattern classification
and regression.  The solutions to classification and regression problems
sought by kernel-based algorithms such as the SVM are linear functions
in the feature space:
\begin{equation}
f(x) = w^\top \Phi(x)
\end{equation} 
for some weight vector $w \in F$.  The kernel trick can be exploited in
this whenever the weight vector~$w$ can be expressed as a linear
combination of the training points, $w = \sum_{i=1}^{n} \alpha_i
\Phi(x_i)$, implying that $f$ can be written as
\begin{equation}
f(x) = \sum_{i=1}^{n}\alpha_i k(x_i, x)
\end{equation}

A very important issue that arises is that of choosing a kernel~$k$ for
a given learning task.  Intuitively, we wish to choose a kernel that
induces the ``right'' metric in the space.  Support Vector Machines
choose a function $f$ that is linear in the feature space by optimizing
some criterion over the sample.  In the case of the 2-norm Soft Margin
classification the optimization problem takes the form:
 \begin{eqnarray} \nonumber
    \mathrm{minimize} 
    && t(w,\xi) = \frac{1}{2}{\|w\|}^2+\frac{C}{m}\sum_{i=1}^{m}\xi_i \\
    \mbox{subject to~}
    && y_i ( \langle x_i , w \rangle +b ) \geq 1- \xi_i \qquad (i=1,\dots,m)\\
    \nonumber && \xi_i \ge 0 \qquad (i=1,\dots, m)
\end{eqnarray}
Based on similar methodology, SVMs deal with the problem of novelty
detection (or one class classification) and regression.

\pkg{kernlab}'s implementation of support vector machines,
\code{ksvm}, is based on the optimizers found in
\pkg{bsvm}\footnote{\url{http://www.csie.ntu.edu.tw/~cjlin/bsvm}}
\citep{kernlab:Hsu:2002} and \pkg{libsvm}
\citep{kernlab:Chang+Lin:2001} which includes a very efficient version
of the Sequential Minimization Optimization (SMO).  SMO decomposes the
SVM Quadratic Problem (QP) without using any numerical QP optimization
steps.  Instead, it chooses to solve the smallest possible optimization
problem involving two elements of $\alpha_i$ because they must obey one
linear equality constraint.  At every step, SMO chooses two $\alpha_i$
to jointly optimize and finds the optimal values for these $\alpha_i$
analytically, thus avoiding numerical QP optimization, and updates the
SVM to reflect the new optimal values.

The SVM implementations available in \code{ksvm} include the C-SVM
classification algorithm along with the $\nu$-SVM classification
formulation which is equivalent to the former but has a more natural
($\nu$) model parameter taking values in $[0,1]$ and is proportional to
the fraction of support vectors found in the data set and the training
error.

For classification problems which include more than two classes
(multi-class) a one-against-one or pairwise classification method
\citep{kernlab:Knerr:1990, kernlab:Kressel:1999} is used.  This method
constructs ${k \choose 2}$ classifiers where each one is trained on data
from two classes.  Prediction is done by voting where each classifier
gives a prediction and the class which is predicted more often wins
(``Max Wins'').  This method has been shown to produce robust results
when used with SVMs \citep{kernlab:Hsu2:2002}.  Furthermore the
\code{ksvm} implementation provides the ability to produce class
probabilities as output instead of class labels.  This is done by an
improved implementation \citep{kernlab:Lin:2001} of Platt's posteriori
probabilities \citep{kernlab:Platt:2000} where a sigmoid function
\begin{equation}
  P(y=1\mid f) = \frac{1}{1+ e^{Af+B}}
\end{equation}
is fitted on the decision values~$f$ of the binary SVM classifiers, $A$
and $B$ are estimated by minimizing the negative log-likelihood
function.  To extend the class probabilities to the multi-class case,
each binary classifiers class probability output is combined by the
\code{couple} method which implements methods for combing class
probabilities proposed in \citep{kernlab:Wu:2003}.

Another approach for multIn order to create a similar probability output for regression, following
\cite{kernlab:Weng:2004}, we suppose that the SVM is trained on data from the model 
\begin{equation}
y_i = f(x_i) + \delta_i
\end{equation}
where $f(x_i)$ is the underlying function and $\delta_i$ is independent and identical distributed
random noise. Given a test data $x$ the distribution of $y$ given $x$ and allows 
one to draw probabilistic inferences about $y$ e.g. one can construct 
a predictive interval $\Phi = \Phi(x)$ such that $y \in \Phi$ with a certain probability. 
If $\hat{f}$ is the estimated (predicted) function of the SVM on new data
then $\eta = \eta(x) = y - \hat{f}(x)$ is the prediction error and $y \in \Phi$ is equivalent to 
$\eta \in \Phi $. Empirical observation shows that the distribution of the residuals $\eta$ can be 
modeled both by a Gaussian and a Laplacian distribution with zero mean. In this implementation the 
Laplacian with zero mean is used :
\begin{equation}
p(z)  = \frac{1}{2\sigma}e^{-\frac{|z|}{\sigma}}
\end{equation} 

Assuming that $\eta$ are independent the scale parameter $\sigma$ is estimated by maximizing the
likelihood. The data for the estimation is produced by a three-fold cross-validation.
For the Laplace distribution the maximum likelihood estimate is :
\begin{equation}
\sigma = \frac{\sum_{i=1}^m|\eta_i|}{m}
\end{equation}

i-class classification supported by the
\code{ksvm} function is the one proposed in
\cite{kernlab:Crammer:2000}.  This algorithm works by solving a single
optimization problem including the data from all classes:

\begin{eqnarray} \nonumber
  \mathrm{minimize}
  && t(w_n,\xi) =
  \frac{1}{2}\sum_{n=1}^k{\|w_n\|}^2+\frac{C}{m}\sum_{i=1}^{m}\xi_i \\
  \mbox{subject to~}
  && \langle x_i , w_{y_i} \rangle - \langle x_i , w_{n} \rangle  \geq
  b_i^n - \xi_i \qquad (i=1,\dots,m) \\ 
  \mbox{where} && b_i^n = 1 - \delta_{y_i,n} 
\end{eqnarray}
where the decision function is 
\begin{equation}
  \mathrm{argmax}_{m=1,\dots,k} \langle x_i , w_{n} \rangle
\end{equation}

This optimization problem is solved by a decomposition method proposed
in \cite{kernlab:Hsu:2002} where optimal working sets are found (that
is, sets of $\alpha_i$ values which have a high probability of being
non-zero).  The QP sub-problems are then solved by a modified version of
the
\pkg{TRON}\footnote{\url{http://www-unix.mcs.anl.gov/~more/tron/}}
\citep{kernlab:more:1999} optimization software.

One-class classification or novelty detection
\citep{kernlab:Williamson:1999, kernlab:Tax:1999}, where essentially an
SVM detects outliers in a data set, is another algorithm supported by
\code{ksvm}.  SVM novelty detection works by creating a spherical
decision boundary around a set of data points by a set of support
vectors describing the spheres boundary. The $\nu$ parameter is used to
control the volume of the sphere and consequently the number of outliers
found.  Again, the value of $\nu$ represents the fraction of outliers
found.  Furthermore, $\epsilon$-SVM \citep{kernlab:Vapnik2:1995} and
$\nu$-SVM \citep{kernlab:Smola1:2000} regression are also available.

The problem of model selection is partially addressed by an empirical
observation for the popular Gaussian RBF kernel
\citep{kernlab:Caputo:2002}, where the optimal values of the
hyper-parameter of sigma are shown to lie in between the 0.1 and 0.9
quantile of the $\|x- x'\| $ statistics.  The \code{sigest} function
uses a sample of the training set to estimate the quantiles and returns
a vector containing the values of the quantiles.  Pretty much any value
within this interval leads to good performance.

An example for the \code{ksvm} function is shown below. 

<<ksvm, echo = TRUE>>=
## simple example using the promotergene data set
data(promotergene)
## create test and training set
tindex <- sample(1:dim(promotergene)[1],5)
genetrain <- promotergene[-tindex, ]
genetest <- promotergene[tindex,]
## train a support vector machine
gene <- ksvm(Class~.,data=genetrain,kernel="rbfdot",kpar="automatic",C=60,cross=3,prob.model=TRUE)
gene
predict(gene, genetest)
predict(gene, genetest, type="probabilities")
@

\begin{figure}
\centering
<<fig=TRUE,echo=TRUE,width=10,height=10,results=hide>>=
set.seed(123)
x <- rbind(matrix(rnorm(120),,2),matrix(rnorm(120,mean=3),,2))
y <- matrix(c(rep(1,60),rep(-1,60)))

svp <- ksvm(x,y,type="C-svc")
plot(svp)
@  
\caption{A contour plot of the SVM decision values for a toy binary classification problem using the
  \code{plot} function}
\label{fig:ksvm Plot}
\end{figure}

\subsection{Relevance vector machine}

The relevance vector machine \citep{kernlab:Tipping:2001} is a
probabilistic sparse kernel model identical in functional form to the
SVM making predictions based on a function of the form
\begin{equation}
  y(x) = \sum_{n=1}^{N} \alpha_n K(\mathbf{x},\mathbf{x}_n) + a_0
\end{equation}
where $\alpha_n$ are the model ``weights'' and $K(\cdotp,\cdotp)$ is a
kernel function.  It adopts a Bayesian approach to learning, by
introducing a prior over the weights $\alpha$
\begin{equation}
  p(\alpha, \beta) = 
  \prod_{i=1}^m N(\beta_i \mid 0 , a_i^{-1})
  \mathrm{Gamma}(\beta_i\mid \beta_\beta , \alpha_\beta) 
\end{equation}
governed by a set of hyper-parameters $\beta$, one associated with each
weight, whose most probable values are iteratively estimated for the
data.  Sparsity is achieved because in practice the posterior
distribution in many of the weights is sharply peaked around zero.
Furthermore, unlike the SVM classifier, the non-zero weights in the RVM
are not associated with examples close to the decision boundary, but
rather appear to represent ``prototypical'' examples.  These examples
are termed \emph{relevance vectors}.

\pkg{kernlab} currently has an implementation of the RVM based on a
type~II maximum likelihood method which can be used for regression.
The functions returns an \proglang{S4} object containing the model
parameters along with indexes for the relevance vectors and the kernel
function and hyper-parameters used.

<<rvm,echo=FALSE>>=
x <- seq(-20, 20, 0.5)
y <- sin(x)/x + rnorm(81, sd = 0.03)
y[41] <- 1
@
<<rvm2,echo=TRUE>>=
rvmm <- rvm(x, y)
rvmm
ytest <- predict(rvmm, x)
@

\begin{figure}
\centering
<<fig=TRUE,echo=FALSE,width=12,height=7>>=
plot(x, y, cex=0.5)
lines(x, ytest, col = "red")
points(x[RVindex(rvmm)],y[RVindex(rvmm)],pch=21)
@  
\caption{Relevance vector regression on data points created by the
  $sinc(x)$ function, relevance vectors are shown circled.}
\label{fig:RVM sigmoid}
\end{figure}


\subsection{Gaussian processes}

Gaussian processes \citep{kernlab:Williams:1995} are based on the
``prior'' assumption that adjacent observations should convey
information about each other. In particular, it is assumed that the
observed variables are normal, and that the coupling between them takes
place by means of the covariance matrix of a normal distribution. Using
the kernel matrix as the covariance matrix is a convenient way of
extending Bayesian modeling of linear estimators to nonlinear
situations.  Furthermore it represents the counterpart of the ``kernel
trick'' in methods minimizing the regularized risk.

For regression estimation we assume that rather than observing $t(x_i)$
we observe $y_i = t(x_i) + \xi_i$ where $\xi_i$ is assumed to be
independed Gaussian distributed noise with zero mean.  The posterior
distribution is given by
\begin{equation}
  p(\mathbf{y}\mid \mathbf{t}) =
  \left[ \prod_ip(y_i - t(x_i)) \right]
  \frac{1}{\sqrt{(2\pi)^m \det(K)}} 
  \exp \left(\frac{1}{2}\mathbf{t}^T K^{-1} \mathbf{t} \right)
\end{equation}
and after substituting $\mathbf{t} = K\mathbf{\alpha}$ and taking
logarithms
\begin{equation}
\ln{p(\mathbf{\alpha} \mid \mathbf{y})} = - \frac{1}{2\sigma^2}\| \mathbf{y} - K \mathbf{\alpha} \|^2 -\frac{1}{2}\mathbf{\alpha}^T K \mathbf{\alpha} +c
\end{equation}
and maximizing $\ln{p(\mathbf{\alpha} \mid \mathbf{y})}$ for
$\mathbf{\alpha}$ to obtain the maximum a posteriori approximation
yields
\begin{equation}
  \mathbf{\alpha} = (K + \sigma^2\mathbf{1})^{-1} \mathbf{y}
\end{equation}
Knowing $\mathbf{\alpha}$ allows for prediction of $y$ at a new location
$x$ through $y = K(x,x_i){\mathbf{\alpha}}$.  In similar fashion
Gaussian processes can be used for classification.

\code{gausspr} is the function in \pkg{kernlab} implementing Gaussian
processes for classification and regression.


\subsection{Ranking}

The success of Google has vividly demonstrated the value of a good
ranking algorithm in real world problems.  \pkg{kernlab} includes a
ranking algorithm based on work published in \citep{kernlab:Zhou:2003}.
This algorithm exploits the geometric structure of the data in contrast
to the more naive approach which uses the Euclidean distances or inner
products of the data.  Since real world data are usually highly
structured, this algorithm should perform better than a simpler approach
based on a Euclidean distance measure.

First, a weighted network is defined on the data and an authoritative
score is assigned to every point.  The query points act as source nodes
that continually pump their scores to the remaining points via the
weighted network, and the remaining points further spread the score to
their neighbors.  The spreading process is repeated until convergence
and the points are ranked according to the scores they received.

Suppose we are given a set of data points $X = {x_1, \dots, x_{s},
  x_{s+1}, \dots, x_{m}}$ in $\mathbf{R}^n$ where the first $s$ points
are the query points and the rest are the points to be ranked.  The
algorithm works by connecting the two nearest points iteratively until a
connected graph $G = (X, E)$ is obtained where $E$ is the set of edges.
The affinity matrix $K$ defined e.g.\ by $K_{ij} = \exp(-\sigma\|x_i -
x_j \|^2)$ if there is an edge $e(i,j) \in E$ and $0$ for the rest and
diagonal elements.  The matrix is normalized as $L = D^{-1/2}KD^{-1/2}$
where $D_{ii} = \sum_{j=1}^m K_{ij}$, and
 \begin{equation}
   f(t+1) = \alpha Lf(t) + (1 - \alpha)y
\end{equation}
is iterated until convergence, where $\alpha$ is a parameter in $[0,1)$.
The points are then ranked according to their final scores $f_{i}(t_f)$.

\pkg{kernlab} includes an \proglang{S4} method implementing the ranking
algorithm.  The algorithm can be used both with an edge-graph where the
structure of the data is taken into account, and without which is
equivalent to ranking the data by their distance in the projected space.

\begin{figure}
\centering
<<ranking,echo =TRUE,fig=TRUE,width=12,height=7>>=
data(spirals)
ran <- spirals[rowSums(abs(spirals) < 0.55) == 2,]
ranked <- ranking(ran, 54, kernel = "rbfdot", kpar = list(sigma = 100), edgegraph = TRUE)
ranked[54, 2] <- max(ranked[-54, 2])
c<-1:86
op <- par(mfrow = c(1, 2),pty="s")
plot(ran)
plot(ran, cex=c[ranked[,3]]/40)
@
\caption{The points on the left are ranked according to their similarity
  to the upper most left point.  Points with a higher rank appear
  bigger. Instead of ranking the points on simple Euclidean distance the
  structure of the data is recognized and all points on the upper
  structure are given a higher rank although further away in distance
  than points in the lower structure.}
\label{fig:Ranking}
\end{figure}

\subsection{Online learning with kernels}

The \code{onlearn} function in \pkg{kernlab} implements the online kernel algorithms 
for classification, novelty detection and regression descibed in \citep{kernlab:Kivinen:2004}.
In batch learning, it is typically assumed that all the examples are immediately 
available and are drawn independently from some distribution $P$. One natural measure 
of quality for some $f$ in that case is the expected risk 
\begin{equation}
R[f,P] := E_{(x,y)~P}[l(f(x),y)]
\end{equation}  
Since usually $P$ is unknown a standard approach is to instead minimize the empirical risk 
\begin{equation}
R_{emp}[f,P] := \frac{1}{m}\sum_{t=1}^m l(f(x_t),y_t)
\end{equation}
Minimizing $R_{emp}[f]$ may lead to overfitting (complex functions that fit well on the training
data but do not generalize to unseen data). One way to avoid this is to penalize complex functions by 
instead minimizing the regularized risk.
\begin{equation}
R_{reg}[f,S] := R_{reg,\lambda}[f,S] := R_{emp}[f] = \frac{\lambda}{2}\|f\|_{H}^2
\end{equation}
where $\lambda > 0$ and $\|f\|_{H} = {\langle f,f \rangle}_{H}^{\frac{1}{2}}$ does indeed measure
the complexity of $f$ in a sensible way. The constant $\lambda$ needs to be chosen appropriately for each problem.
Since in online learning one is interested in dealing with one example at the time the definition 
of an instantaneous regularized risk on a single example is needed
\begin{equation}
R_inst[f,x,y] := R_{inst,\lambda}[f,x,y] := R_{reg,\lambda}[f,((x,y))]
\end{equation}

The implemented algorithms are classical stochastic gradient descent algorithms performing gradient
descent on the instantaneous risk. The general form of the update rule is :
\begin{equation}
f_{t+1} = f_t - \eta \partial_f R_{inst,\lambda}[f,x_t,y_t]|_{f=f_t}
\end{equation} 
where $f_i \in H$ and $\partial_f$< is short hand for $\partial \ \partial f$ 
(the gradient with respect to $f$) and $\eta_t > 0$ is the learning rate. 
Due to the learning taking place in a \textit{reproducing kernel Hilbert space} $H$ 
the kernel $k$ used has the property $\langle f,k(x,\cdotp)\rangle_H = f(x)$ 
and therefore 
\begin{equation}
\partial_f l(f(x_t)),y_t) = l'(f(x_t),y_t)k(x_t,\cdotp)
\end{equation} 
where $l'(z,y) := \partial_z l(z,y)$. Since $\partial_f\|f\|_H^2 = 2f$ the update becomes
\begin{equation}
f_{t+1} := (1 - \eta\lambda)f_t -\eta_t \lambda '( f_t(x_t),y_t)k(x_t,\cdotp)
\end{equation}

The \code{onlearn} function implements the online learning algorithm for regression, classification and novelty 
detection. The online nature of the algorithm requires a different approach to the use of the function. An object 
is used to store the state of the algorithm at each iteration $t$ this object is passed to the function as an 
argument and is returned at each iteration $t+1$ containing the model parameter state at this step. 
An empty object of class \code{onlearn} is initialized using the \code{inlearn} function. 
<<onlearn, echo = TRUE>>=
## create toy data set
x <- rbind(matrix(rnorm(90),,2),matrix(rnorm(90)+3,,2))
y <- matrix(c(rep(1,45),rep(-1,45)),,1)

## initialize onlearn object
on <- inlearn(2,kernel="rbfdot",kpar=list(sigma=0.2),type="classification")
ind <- sample(1:90,90)
## learn one data point at the time
for(i in ind)
on <- onlearn(on,x[i,],y[i],nu=0.03,lambda=0.1)
sign(predict(on,x))
@ 

\subsection{Spectral clustering}

Spectral clustering \citep{kernlab:Ng:2001} is a recently emerged promising alternative to
common clustering algorithms. In this method one
uses the top eigenvectors of a matrix created by some similarity measure
to cluster the data.  Similarly to the ranking algorithm, an affinity
matrix is created out from the data as
\begin{equation}
  K_{ij}=\exp(-\sigma\|x_i - x_j \|^2) 
\end{equation}
and normalized as $L = D^{-1/2}KD^{-1/2}$ where $D_{ii} = \sum_{j=1}^m
K_{ij}$.  Then the top $k$ eigenvectors (where $k$ is the number of
clusters to be found) of the affinity matrix are used to form an $n
\times k$ matrix $Y$ where each column is normalized again to unit
length.  Treating each row of this matrix as a data point,
\code{kmeans} is finally used to cluster the points.

\pkg{kernlab} includes an \proglang{S4} method called \code{specc}
implementing this algorithm which can be used through an formula
interface or a matrix interface.  The \proglang{S4} object returned by the
method extends the class ``vector'' and contains the assigned cluster
for each point along with information on the centers size and
within-cluster sum of squares for each cluster.  In case a Gaussian RBF
kernel is being used a model selection process can be used to determine
the optimal value of the $\sigma$ hyper-parameter.  For a good value of
$\sigma$ the values of $Y$ tend to cluster tightly and it turns out that
the within cluster sum of squares is a good indicator for the
``quality'' of the sigma parameter found.  We then iterate through the
sigma values to find an optimal value for $\sigma$.

\begin{figure}
\centering
<<fig=TRUE,echo=TRUE,width=7,height=7>>=
data(spirals)
sc <- specc(spirals, centers=2)
plot(spirals, pch=(23 - 2*sc))
@  
\caption{Clustering the two spirals data set with \code{specc}}
\label{fig:Spectral Clustering}
\end{figure}

\subsection{Kernel principal components analysis}

Principal component analysis (PCA) is a powerful technique for
extracting structure from possibly high-dimensional datasets.  PCA is an
orthogonal transformation of the coordinate system in which we describe
the data.  The new coordinates by which we represent the data are called
principal components.  Kernel PCA \citep{kernlab:Schoelkopf:1998}
performs a nonlinear transformation of the coordinate system by finding
principal components which are nonlinearly related to the input
variables.  Given a set of centered observations $x_k$, $k=1,\dots,M$,
$x_k \in \mathbf{R}^N$, PCA diagonalizes the covariance matrix $C =
\frac{1}{M}\sum_{j=1}^Mx_jx_{j}^T$ by solving the eigenvalue problem
$\lambda\mathbf{v}=C\mathbf{v}$.  The same computation can be done in a
dot product space $F$ which is related to the input space by a possibly
nonlinear map $\Phi:\mathbf{R}^N \rightarrow F$, $x \mapsto \mathbf{X}$.
Assuming that we deal with centered data and use the covariance matrix
in $F$,
\begin{equation}
\hat{C}=\frac{1}{C}\sum_{j=1}^N \Phi(x_j)\Phi(x_j)^T
\end{equation}
the kernel principal components are then computed by taking the
eigenvectors of the centered kernel matrix $K_{ij} = \langle
\Phi(x_j),\Phi(x_j) \rangle$.

\code{kpca}, the the function implementing KPCA in \pkg{kernlab}, can
be used both with a formula and a matrix interface, and returns an
\proglang{S4} object of class \code{kpca} containing the principal
components the corresponding eigenvalues along with the projection of
the training data on the new coordinate system.  Furthermore, the
\code{predict} function can be used to embed new data points into the
new coordinate system.

\begin{figure}
\centering
<<kpca,echo =TRUE,fig=TRUE,width=7,height=7>>=
data(spam)
train <- sample(1:dim(spam)[1],400)
kpc <- kpca(~.,data=spam[train,-58],kernel="rbfdot",kpar=list(sigma=0.001),features=2)
kpcv <- pcv(kpc)
plot(rotated(kpc),col=as.integer(spam[train,58]),xlab="1st Principal Component",ylab="2nd Principal Component")
@
\caption{Projection of the spam data on two kernel principal components
  using an RBF kernel}
\label{fig:KPCA}
\end{figure}

\subsection{Kernel feature analysis}

Whilst KPCA leads to very good results there are nevertheless some issues to be addressed. 
First the computational complexity of the standard version of KPCA, the algorithm scales
$O(m^3)$ and secondly the resulting feature extractors are given as a dense expansion in terms
of the of the training patterns. 
Sparse solutions are often achieven in supervised learning settings by using an $l_1$ penalty on the 
expansion coefficients. An algorithm can be derived using the same approach in feature extraction 
requiring only $n$ basis functions to compute the first $n$ feature.
Kernel feature analysis \citep{kernlab:Olvi:2000} is computationaly simple and scales approximately
 one order of magnitute better on large data sets than standard KPCA.    
Choosing $\Omega [f] = \sum_{i=1}^m |\alpha_i |$ 
this yields 
\begin{equation}
F_{LP} =  \{  \mathbf{w} \vert  \mathbf{w} = \sum_{i=1}^m \alpha_i \Phi(x_i) \mathrm{with} \sum_{i=1}^m |\alpha_i | \leq 1 \}
\end{equation}

This setting leads to the first ``principal vector'' in the $l_1$ context
\begin{equation}
\mathbf{\nu}^1 = \mathrm{argmax}_{\mathbf{\nu} \in F_{LP}} \frac{1}{m} \sum_{i=1}^m \langle \mathbf{\nu},\mathbf{\Phi}(x_i) - \frac{1}{m}\sum_{j=1}^m\mathbf{\Phi}(x_i)  \rangle^2 
\end{equation}

Subsequent ``principal vectors'' can be defined by enforcing optimality with respect to the remaining 
orthogonal subspaces. Due to the $l_1$ constrain the solution has the favorable property of beeing 
sparse in terms of the coefficients $\alpha_i$.

The function \code{kfa} in \pkg{kernlab} implements Kernel Feature Analysis by using a projection 
pursuit technique on a sample of the data. Results are then returned in an \proglang{S4} object. 

\begin{figure}
\centering
<<kfa, echo = TRUE,fig=TRUE,width=7,height=7>>=
data(promotergene)
f <- kfa(~.,data=promotergene,features=2,kernel="rbfdot",kpar=list(sigma=0.013))
plot(predict(f,promotergene),col=as.numeric(promotergene[,1]),xlab="1st Feature",ylab="2nd Feature")
@ 
\caption{Projection of the spam data on two features using an RBF kernel}
\label{fig:KFA}
\end{figure}

\subsection{Kernel canonical correlation analysis}

Canonical correlation analysis (CCA) is concerned with describing the
linear relations between variables.  If we have two data sets $x_1$ and
$x_2$, then the classical CCA attempts to find linear combination of the
variables which give the maximum correlation between the combinations.
I.e., if
\begin{eqnarray*}
  && y_1 = \mathbf{w_1}\mathbf{x_1} = \sum_j w_1 x_{1j} \\
  && y_2 = \mathbf{w_2}\mathbf{x_2} = \sum_j w_2 x_{2j}
\end{eqnarray*}
one wishes to find those values of $\mathbf{w_1}$ and $\mathbf{w_2}$
which maximize the correlation between $y_1$ and $y_2$.  Similar to the
KPCA algorithm, CCA can be extended and used in a dot product space~$F$
which is related to the input space by a possibly nonlinear map
$\Phi:\mathbf{R}^N \rightarrow F$, $x \mapsto \mathbf{X}$ as
\begin{eqnarray*}
  && y_1 = \mathbf{w_1}\mathbf{\Phi(x_1)} = \sum_j w_1 \Phi(x_{1j}) \\
  && y_2 = \mathbf{w_2}\mathbf{\Phi(x_2)} = \sum_j w_2 \Phi(x_{2j})
\end{eqnarray*}

Following \citep{kernlab:kuss:2003}, the \pkg{kernlab} implementation of
a KCCA projects the data vectors on a new coordinate system using KPCA
and uses linear CCA to retrieve the correlation coefficients.  The
\code{kcca} method in \pkg{kernlab} returns an \proglang{S4} object
containing the correlation coefficients for each data set and the
corresponding correlation along with the kernel used.
 

\subsection{Interior point code quadratic optimizer}

In many kernel based algorithms, learning implies the minimization of
some risk function.  Typically we have to deal with quadratic or general
convex problems for support vector machines of the type
\begin{equation}
  \begin{array}{ll}
    \mathrm{minimize} & f(x) \\
    \mbox{subject to~} & c_i(x) \leq 0 \mbox{~for all~} i \in [n].
  \end{array}
\end{equation}
$f$ and $c_i$ are convex functions and $n \in \mathbf{N}$.
\pkg{kernlab} provides the \proglang{S4} method \code{ipop} implementing
an optimizer of the interior point family \citep{kernlab:Vanderbei:1999}
which solves the quadratic programming problem
\begin{equation}
  \begin{array}{ll}
    \mathrm{minimize} & c^\top x+\frac{1}{2}x^\top H x \\
    \mbox{subject to~} & b \leq Ax \leq b + r\\
    & l \leq x \leq u \\
  \end{array}
\end{equation}

This optimizer can be used in regression, classification, and novelty
detection in SVMs.

\subsection{Incomplete cholesky decomposition}

When dealing with kernel based algorithms, calculating a full kernel
matrix should be avoided since it is already a $O(N^2)$ operation.
Fortunately, the fact that kernel matrices are positive semidefinite is
a strong constraint and good approximations can be found with small
computational cost.  The Cholesky decomposition factorizes a positive
semidefinite $N \times N$ matrix $K$ as $K=ZZ^T$, where $Z$ is an upper
triangular $N \times N$ matrix.  Exploiting the fact that kernel
matrices are usually of low rank, an \emph{incomplete Cholesky
  decomposition} \citep{kernlab:Wright:1999} finds a matrix $\tilde{Z}$
of size $N \times M$ where $M\ll N$ such that the norm of
$K-\tilde{Z}\tilde{Z}^T$ is smaller than a given tolerance $\theta$.
The main difference of incomplete Cholesky decomposition to the standard
Cholesky decomposition is that pivots which are below a certain
threshold are simply skipped.  If $L$ is the number of skipped pivots,
we obtain a $\tilde{Z}$ with only $M = N - L$ columns.  The algorithm
works by picking a column from $K$ to be added by maximizing a lower
bound on the reduction of the error of the approximation.  \pkg{kernlab}
has an implementation of an incomplete Cholesky factorization called
\code{inc.chol} which computes the decomposed matrix $\tilde{Z}$ from
the original data for any given kernel without the need to compute a
full kernel matrix beforehand.  This has the advantage that no full
kernel matrix has to be stored in memory.

\section{Conclusions}

In this paper we described \pkg{kernlab}, a flexible and extensible
kernel methods package for \proglang{R} with existing modern kernel
algorithms along with tools for constructing new kernel based
algorithms.  It provides a unified framework for using and creating
kernel-based algorithms in \proglang{R} while using all of \proglang{R}'s
modern facilities, like \proglang{S4} classes and namespaces.  Our aim for
the future is to extend the package and add more kernel-based methods as
well as kernel relevant tools. Sources and binaries for 
the latest version of \pkg{kernlab} are available at CRAN\footnote{\url{http://CRAN.R-project.org}} 
under the GNU Public License.


\bibliography{jss}

\end{document}
