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\usepackage[%
baseurl={http://www.stamats.de},%
pdftitle={RandVar: Implementation of random variables by means of S4 classes and methods},%
pdfauthor={Matthias Kohl},%
pdfsubject={RandVar},%
pdfkeywords={random variable, S4 classes, S4 methods},%
pagebackref,bookmarks,colorlinks,linkcolor=darkblue,citecolor=darkblue,%
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\markboth{\sl Packages ``{\tt RandVar}''}{\sl Packages ``{\tt RandVar}''}
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\begin{document}
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\title{RandVar: Implementation of random variables by means
    of {\tt S4} classes and methods}
\author{Matthias Kohl\\
e-Mail: {\tt matthias.kohl@stamats.de}\medskip\\
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\maketitle
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\begin{abstract}
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In this package we implement random variables by means of {\tt S4} classes
and methods. This vignette corresponds to Appendix D.2 in Kohl~(2005)~\cite{MK:05}.
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\end{abstract}
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\tableofcontents
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\clearpage
% -----------------------------------------------------------------------
\section{{\tt S4} Classes}
% -----------------------------------------------------------------------
The {\tt S4} class {\tt RandVariable} (cf.\ Figure~\ref{ap.Rpack.RandVar.dia1})
has the slots {\tt Map}, {\tt Domain} and {\tt Range} where {\tt Map}
contains a list of functions which are measurable maps from {\tt Domain}
to {\tt Range}. The elements contained in the list {\tt Map} must be functions in 
one argument named {\tt x}. We do not allow further parameters for these 
functions as this would lead to inconsistent objects. Strictly speaking, an object of 
class {\tt RandVariable} would represent not only one random variable but a whole 
set of random variables depending on these parameters.
% -----------------------------------------------------------------------
\par
The slots {\tt Domain} and {\tt Range} are filled with an object of
class {\tt OptionalrSpace}; i.e., they contain {\tt NULL} or an object of class 
{\tt rSpace} (see package {\tt distr}~\cite{distr}). In case of
{\tt EuclRandVariable} and {\tt RealRandVariable} the slot {\tt Range} is filled 
with an object of class {\tt Euclideanspace} and {\tt Reals}, respectively. The 
class {\tt EuclRandMatrix} additionally has the slot {\tt Dim} which is a vector 
of integers and contains the dimensions of the Euclidean random matrix.
% -----------------------------------------------------------------------
\par
Using these {\tt S4} classes there are two possibilities to implement a ${\rm R}^k$
valued random variable. First, we could define a {\tt EuclRandVariable}
whose slot {\tt Map} contains a list with one function which maps to ${\rm R}^k$;
i.e., the slot {\tt Range} is a $k$-dimensional Euclidean space. Second, we could
define a {\tt EuclRandVariable} whose slot {\tt Map} contains a list with $n$
functions (projections) which map to ${\rm R}^m$ where $k = m*n$. Now, the slot
{\tt Range} is an $m$-dimensional Euclidean space.
Since it is sometimes convenient to regard a ${\rm R}^k$ valued random variable
as measurable map consisting of ${\rm R}^{k_i}$ valued maps where $\sum k_i = k$,
we introduce a further class called {\tt EuclRandVarList}.
With this class we can now define ${\rm R}^k$ valued random variables as a list
of ${\rm R}^{k_i}$ valued random variables with compatible domains. More precisely,
the elements of a {\tt EuclRandVarList} may even have very different ranges
(not necessarily Euclidean spaces) they only need to have compatible
domains which is checked via the generic function {\tt compatibleDomains}.
% -----------------------------------------------------------------------
\begin{figure}[!ht]
\begin{center}
  \ifpdf
    \includegraphics[scale=1.0, viewport = 14 15 244 275]{RandVariable.pdf}
  \else
    \includegraphics[scale=1.0]{RandVariable.eps}
  \fi
  \caption[Class {\tt RandVariable} and Subclasses]{Class
    {\tt RandVariable} and subclasses.}
  \label{ap.Rpack.RandVar.dia1}
\end{center}
\end{figure}
% -----------------------------------------------------------------------
\newpage
\section{Functions and Methods}
% -----------------------------------------------------------------------
As in case of package {\tt distrEx}~\cite{distr}, we follow the advices of 
Section~7.3 of \cite{Cham:98} and \cite{Gent:03}. That is, we introduce 
generating functions as well as accessor and replacement functions. A short
description of the implemented generating functions is given in
Table~\ref{ap.Rpack.RandVar.tab.gen}.
% -----------------------------------------------------------------------
\begin{table}[!ht]
\begin{center}
\begin{tabular}{p{4cm}|p{7.5cm}}
  \textbf{Generating Function} & \textbf{Description} \\ \hline\hline
  {\tt EuclRandMatrix} & Generates an object of class {\tt EuclRandMatrix}\\ \hline
  {\tt EuclRandVariable} & Generates an object of class {\tt EuclRandVariable}\\ \hline
  {\tt EuclRandVarList}   & Generates an object of class {\tt EuclRandVarList}\\ \hline
  {\tt RandVariable} & Generates an object of class {\tt RandVariable}\\ \hline
  {\tt RealRandVariable} & Generates an object of class {\tt RealRandVariable}
\end{tabular}
\caption[Generating Functions of Package {\tt RandVar}]{Generating
functions of package {\tt RandVar}.}\label{ap.Rpack.RandVar.tab.gen}%
\end{center}
\end{table}
% -----------------------------------------------------------------------
\par\noindent
While there are accessor functions for all slots of the newly defined
{\tt S4} classes, replacement functions are only implemented for those
slots a user should modify.
% -----------------------------------------------------------------------
\par
Our next goal was that one can use these classes of random variables like
ordinary numeric vectors or matrices. Hence, we overloaded the {\tt S4}
group generic functions {\tt Arith} and {\tt Math} as well as matrix
multiplication {\tt \%*\%}. For the matrix multiplication of {\tt EuclRandVarList}s 
we additionally introduced the operator {\tt \%m\%}.
Now, if we have random variables {\tt X} and {\tt Y}, a numerical
vector {\tt v} and a numerical matrix {\tt M} (with compatible dimensions)
we can for instance generate
<<exa1, eval = TRUE>>=
library(RandVar)
distroptions("withSweave" = TRUE) ## only for use in Sweave - document; set to FALSE else!
(X <- RealRandVariable(Map = list(function(x){x}, function(x){x^2}), Domain = Reals(), Range = Reals()))
Map(X)
evalRandVar(X, 2)
evalRandVar(X, as.matrix(seq(2, 10, 2)))
R1 <- exp(X-1)
Map(R1)
R2 <- exp(X-1:2)
Map(R2)
(Y <- RealRandVariable(Map = list(function(x){sin(x)}, function(x){cos(x)}), Domain = Reals(), Range = Reals()))
Map(Y)
R3 <- X %*% Y
dimension(R3)
#evalRandVar(R3, 2)
2*sin(2) + 2^2*cos(2)
(R4 <- X %*% t(Y))
dimension(R4)
#evalRandVar(R4, 2)
(M <- matrix(c(2*sin(2), 2^2*sin(2), 2*cos(2), 2^2*cos(2)), ncol = 2))
(R5 <- M %*% R4)
@
We also implemented {\tt S4} methods for the generic function {\tt E} of 
package {\tt distrEx}~\cite{distr}. That is, given some distribution {\tt D}, 
respectively some conditional distribution {\tt CD} and some random variable {\tt X}
we can compute the (conditional) expectation of {\tt X} under {\tt D}, respectively 
{\tt CD} simply by
<<exa2, eval = TRUE>>=
D <- Norm()
E(object = D, fun = X)
E(D)
var(D)
(CD <- LMCondDistribution(theta = 1))
E(object = CD, fun = X, cond = 2)
E(Norm(mean = 2))
E(Norm(mean = 2), fun = function(x){x^2})
@
for some given condition {\tt cond}.
% -----------------------------------------------------------------------
\par
In addition, we define methods for the generic function {\tt show} for the classes
{\tt RandVariable}, {\tt EuclRandMatrix} and {\tt EuclRandVarList}. There are 
also methods for the generic functions {\tt dimension} (see package 
{\tt distr}~\cite{distr}), {\tt length}, {\tt ncol}, {\tt nrow}, {\tt t} and 
{\tt [} (cf.\ package {\tt base}). For more details we refer to the corresponding 
help pages.
% -----------------------------------------------------------------------
\par
Finally, we introduce several new generic functions. A brief description
of these functions is given in Table~\ref{ap.Rpack.RandVar.tab}.
% -----------------------------------------------------------------------
\begin{table}[!ht]
\begin{center}
\begin{tabular}{p{3.5cm}|p{8cm}}
  \textbf{Generic Function} & \textbf{Description} \\ \hline\hline
  \texttt{\%m\%} & matrix multiplication for {\tt EuclRandVarList}s \\ \hline
  \texttt{compatibleDomains} & test if the domains of two random variables
                     are compatible \\ \hline
  \texttt{evalRandVar} & evaluation of random variables\\ \hline
  \texttt{imageDistr} & image distribution of some distribution under some
               random variable \\ \hline
  \texttt{numberOfMaps} & number of functions contained in the slots {\tt Map}
                 of the members of a {\tt EuclRandVarList}
\end{tabular}
\caption[New Generic Functions of Package {\tt RandVar}]
{New generic functions of package {\tt RandVar} (without accessor
and replacement functions).}\label{ap.Rpack.RandVar.tab}%
\end{center}
\end{table}
% -----------------------------------------------------------------------
\par\noindent
For more details about the full functionality of package {\tt RandVar}
we refer to the source code and the corresponding help pages, respectively.
% -----------------------------------------------------------------------
\section{Odds and Ends}
% -----------------------------------------------------------------------
The main issue is to reduce the computation time for methods using
objects of class {\tt RandVariable} and its subclasses as these classes
play an important role in the computation of optimally robust
estimators; confer Kohl~(2005)~\cite{MK:05}.
In particular, we are looking for ways to increase the computation speed
of {\tt evalRandVar} and {\tt E}.
%-------------------------------------------------------------------------------
\begin{thebibliography}{8}

\bibitem{Cham:98}
Chambers J.M. 
\newblock {\em {Programming with data. A guide to the S language}\/}.
\newblock {Springer}.
\newblock http://cm.bell-labs.com/stat/Sbook/index.html

\bibitem{Gent:03}
Gentleman R.
\newblock {\em Object Orientated Programming. Slides of a Short Course held in Auckland\/}.
\newblock http://www.stat.auckland.ac.nz/S-Workshop/Gentleman/Methods.pdf

\bibitem{MK:05}
Kohl M. 
\newblock {\em Numerical Contributions to the Asymptotic Theory of Robustness\/}. 
\newblock {Dissertation}, Universit\"at Bayreuth. 
\newblock See also http://stamats.de/ThesisMKohl.pdf

\bibitem{distr}
Ruckdeschel P., Kohl M., Stabla T., and Camphausen F. 
\newblock {S4 Classes for Distributions.} 
\newblock {\em R-News\/}, {\bf 6}(2): 10--13.
\newblock https://CRAN.R-project.org/doc/Rnews/Rnews\_2006-2.pdf
\newblock See also {http://www.uni-bayreuth.de/departments/math/org/mathe7/RUCKDESCHEL/pubs/distr.pdf}

\end{thebibliography}
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\end{document}
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