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Mon Jul 7 22:34:52 2008 UTC (11 years, 1 month ago) by mmaechler
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Mon Jul 7 22:34:52 2008 UTC (11 years, 1 month ago) by mmaechler
File size: 18424 byte(s)
M[...] <- "sparseVector" etc, see ChangeLog
\documentclass{article} % \usepackage{myVignette} \usepackage{fullpage}% save trees ;-) \usepackage[authoryear,round]{natbib} \bibliographystyle{plainnat} \newcommand{\noFootnote}[1]{{\small (\textit{#1})}} \newcommand{\myOp}[1]{{$\left\langle\ensuremath{#1}\right\rangle$}} % %%\VignetteIndexEntry{2nd Introduction to the Matrix Package} %%\VignetteDepends{Matrix} \SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,width=7,height=4,strip.white=TRUE,keep.source=TRUE} % ^^^^^^^^^^^^^^^^ \title{2nd Introduction to the Matrix package} \author{Martin Maechler and Douglas Bates\\ R Core Development Team \\\email{maechler@stat.math.ethz.ch}, \email{bates@r-project.org}} \date{September 2006 ({\tiny typeset on \tiny\today})} % \begin{document} \maketitle \begin{abstract} % \emph{\Large Why should you want to work with this package and what % does it do for you?} Linear algebra is at the core of many areas of statistical computing and from its inception the \Slang{} has supported numerical linear algebra via a matrix data type and several functions and operators, such as \code{\%*\%}, \code{qr}, \code{chol}, and \code{solve}. However, these data types and functions do not provide direct access to all of the facilities for efficient manipulation of dense matrices, as provided by the Lapack subroutines, and they do not provide for manipulation of sparse matrices. The \pkg{Matrix} package provides a set of S4 classes for dense and sparse matrices that extend the basic matrix data type. Methods for a wide variety of functions and operators applied to objects from these classes provide efficient access to BLAS (Basic Linear Algebra Subroutines), Lapack (dense matrix), CHOLMOD including AMD and COLAMD and \code{Csparse} (sparse matrix) routines. One notable characteristic of the package is that whenever a matrix is factored, the factorization is stored as part of the original matrix so that further operations on the matrix can reuse this factorization. \end{abstract} %% Note: These are explained in '?RweaveLatex' : <<preliminaries, echo=FALSE>>= options(width=75) @ \section{Introduction} \label{sec:Intro} The most automatic way to use the \pkg{Matrix} package is via the \Rfun{Matrix} function which is very similar to the standard \RR\ function \Rfun{matrix}, @ <<ex1>>= library(Matrix) M <- Matrix(10 + 1:28, 4, 7) M tM <- t(M) @ %def Such a matrix can be appended to (using \Rfun{cBind} or \Rfun{rBind} with capital ``B'') or indexed, <<ex2>>= (M2 <- cBind(-1, M)) M[2, 1] M[4, ] @ where the last two statements show customary matrix indexing, returning a simple numeric vector each\footnote{because there's an additional default argument to indexing, \code{drop = TRUE}. If you add \hbox{``\code{\ ,\ drop = FALSE}\ ''} you will get submatrices instead of simple vectors.}. We assign 0 to some columns and rows to ``sparsify'' it, and some \code{NA}s (typically ``missing values'' in data analysis) in order to demonstrate how they are dealt with; note how we can \emph{``subassign''} as usual, for classical \RR{} matrices (i.e., single entries or whole slices at once), @ <<set0>>= M2[, c(2,4:6)] <- 0 M2[2, ] <- 0 M2 <- rBind(0, M2, 0) M2[1:2,2] <- M2[3,4:5] <- NA @ and then coerce it to a sparse matrix, @ <<asSparse>>= sM <- as(M2, "sparseMatrix") 10 * sM identical(sM * 2, sM + sM) is(sM / 10 + M2 %/% 2, "sparseMatrix") @ %def where the last three calls show that multiplication by a scalar keeps sparcity, as does other arithmetic, but addition to a ``dense'' object does not, as you might have expected after some thought about ``sensible'' behavior: @ <<add1>>= sM + 10 @ %def Operations on our classed matrices include (componentwise) arithmetic ($+$, $-$, $*$, $/$, etc) as partly seen above, comparison ($>$, $\le$, etc), e.g., <<Comp1>>= Mg2 <- (sM > 2) Mg2 @ returning a logical sparse matrix. When interested in the internal \textbf{str}ucture, \Rfun{str} comes handy, and we have been using it ourselves more regulary than \Rfun{print}ing (or \Rfun{show}ing as it happens) our matrices; alternatively, \Rfun{summary} gives output similar to Matlab's printing of sparse matrices. @ <<str_mat>>= str(Mg2) summary(Mg2) @ As you see from both of these, \code{Mg2} contains ``extra zero'' (here \code{FALSE}) entries; such sparse matrices may be created for different reasons, and you can use \Rfun{drop0} to remove (``drop'') these extra zeros. This should \emph{never} matter for functionality, and does not even show differently for logical sparse matrices, but the internal structure is more compact: <<drop0>>= Mg2 <- drop0(Mg2) str(Mg2@x) # length 13, was 16 @ For large sparse matrices, visualization (of the sparsity pattern) is important, and we provide \Rfun{image} methods for that, e.g., <<image,fig=TRUE>>= data(CAex) print(image(CAex, main = "image(CAex)")) # print(.) needed for Sweave @ \smallskip Further, i.e., in addition to the above implicitly mentioned \code{"Ops"} operators (\code{+}, \code{*},\dots, \code{<=},\code{>},\dots, \code{\&} which all work with our matrices, notably in conjunction with scalars and traditional matrices), the \code{"Math"}-operations (such as \Rfun{exp}, \Rfun{sin} or \Rfun{gamma}) and \code{"Math2"} (\Rfun{round} etc) and the \code{"Summary"} group of functions, \Rfun{min}, \Rfun{range}, \Rfun{sum}, all work on our matrices as they should. Note that all these are implemented via so called \emph{group methods}, see e.g., \code{?Arith} in \RR. The intention is that sparse matrices remain sparse whenever sensible, given the matrix \emph{classes} and operators involved, but not content specifically. E.g., <sparse> + <dense> gives <dense> even for the rare cases where it would be advantageous to get a <sparse> result. These classed matrices can be ``indexed'' (more technically ``subset'') as traditional \Slang{} (and hence \RR) matrices, as partly seen above. This also includes the idiom \code{M [ M \myOp{\mathit{op}} \myOp{\mathrm{num}}~]} which returns simple vectors, @ <<sub_logi>>= sM[sM > 2] sml <- sM[sM <= 2] sml @ %def and \emph{``subassign''}ment similarly works in the same generality as for traditional \Slang{} matrices. %% We have seen that already above! %% This was the 2005 - Introduction vignette's first section: \subsection{\pkg{Matrix} package for numerical linear algebra} \label{ssec:intro-linalg} Linear algebra is at the core of many statistical computing techniques and, from its inception, the \Slang{} has supported numerical linear algebra via a matrix data type and several functions and operators, such as \code{\%*\%}, \code{qr}, \code{chol}, and \code{solve}. %% Initially the numerical linear algebra functions in \RR{} called underlying Fortran routines from the Linpack~\citep{Linpack} and Eispack~\citep{Eispack} libraries but over the years most of these functions have been switched to use routines from the Lapack~\citep{Lapack} library which is the state-of-the-art implementation of numerical dense linear algebra. %% Furthermore, \RR{} can be configured to use accelerated BLAS (Basic Linear Algebra Subroutines), such as those from the Atlas~\citep{Atlas} project or other ones, see the \RR~manual ``Installation and Administration''. Lapack provides routines for operating on several special forms of matrices, such as triangular matrices and symmetric matrices. Furthermore, matrix decompositions like the QR decompositions produce multiple output components that should be regarded as parts of a single object. There is some support in \RR{} for operations on special forms of matrices (e.g. the \code{backsolve}, \code{forwardsolve} and \code{chol2inv} functions) and for special structures (e.g. a QR structure is implicitly defined as a list by the \code{qr}, \code{qr.qy}, \code{qr.qty}, and related functions) but it is not as fully developed as it could be. Also there is no direct support for sparse matrices in \RR{} although \citet{koen:ng:2003} have developed the \pkg{SparseM} package for sparse matrices based on SparseKit. The \pkg{Matrix} package provides S4 classes and methods for dense and sparse matrices. The methods for dense matrices use Lapack and BLAS. The sparse matrix methods use CHOLMOD~\citep{Cholmod}, CSparse~\citep{Csparse} and other parts (AMD, COLAMD) of Tim Davis' ``SuiteSparse'' collection of sparse matrix libraries, many of which also use BLAS. \TODO{\Rfun{triu}, \Rfun{tril}, \Rfun{diag}, ... and \command{as(.,.)} , but of course only when they've seen a few different ones.} \TODO{matrix operators include \code{\%*\%}, \Rfun{crossprod}, \Rfun{tcrossprod}, \Rfun{solve}} \TODO{\Rfun{expm} is the matrix exponential ... ...} \TODO{\Rfun{symmpart} and \Rfun{skewpart} compute the symmetric part, \code{(x + t(x))/2} and the skew-symmetric part, \code{(x - t(x))/2} of a matrix \code{x}.} \TODO{factorizations include \Rfun{Cholesky} (or \Rfun{chol}), \Rfun{lu}, \Rfun{qr} (not yet for dense)} \TODO{Although generally the result of an operation on dense matrices is a dgeMatrix, certain operations return matrices of special types.} \TODO{E.g. show the distinction between \code{t(mm) \%*\% mm} and \code{crossprod(mm)}.} % \bigskip % ... ... ... The following is the old \file{Introduction.Rnw} ... FIXME ... ... \bigskip \section{Matrix Classes} The \pkg{Matrix} package provides classes for real (stored as double precision), logical and so-called ``pattern'' (binary) dense and sparse matrices. There are provisions to also provide integer and complex (stored as double precision complex) matrices. Note that in \RR, \code{logical} means entries \code{TRUE}, \code{FALSE}, or \code{NA}. To store just the non-zero pattern for typical sparse matrix algorithms, the pattern matrices are \emph{binary}, i.e., conceptually just \code{TRUE} or \code{FALSE}. In \pkg{Matrix}, the pattern matrices all have class names starting with \code{"n"} (patter\textbf{n}). \subsection{Classes for dense matrices} \label{ssec:DenseClasses} For the sake of brevity, we restrict ourselves to the \emph{real} (\textbf{d}ouble) classes, but they are paralleled by \textbf{l}ogical and patter\textbf{n} matrices for all but the positive definite ones. \begin{description} \item[dgeMatrix] Real matrices in general storage mode \item[dsyMatrix] Symmetric real matrices in non-packed storage \item[dspMatrix] Symmetric real matrices in packed storage (one triangle only) \item[dtrMatrix] Triangular real matrices in non-packed storage \item[dtpMatrix] Triangular real matrices in packed storage (triangle only) \item[dpoMatrix] Positive semi-definite symmetric real matrices in non-packed storage \item[dppMatrix] \ \ ditto \ \ in packed storage \end{description} Methods for these classes include coercion between these classes, when appropriate, and coercion to the \code{matrix} class; methods for matrix multiplication (\code{\%*\%}); cross products (\code{crossprod}), matrix norm (\code{norm}); reciprocal condition number (\code{rcond}); LU factorization (\code{lu}) or, for the \code{poMatrix} class, the Cholesky decomposition (\code{chol}); and solutions of linear systems of equations (\code{solve}). %-- mentioned above already: % Further, group methods have been defined for the \code{Arith} (basic % arithmetic, including with scalar numbers) and the \code{Math} (basic % mathematical functions) group.. Whenever a factorization or a decomposition is calculated it is preserved as a (list) element in the \code{factors} slot of the original object. In this way a sequence of operations, such as determining the condition number of a matrix then solving a linear system based on the matrix, do not require multiple factorizations of the same matrix nor do they require the user to store the intermediate results. \subsection{Classes for sparse matrices} \label{sec:SparseClasses} Used for large matrices in which most of the elements are known to be zero. \TODO{E.g. model matrices created from factors with a large number of levels} \TODO{ or from spline basis functions (e.g. COBS, package \pkg{cobs}), etc.} \TODO{Other uses include representations of graphs. indeed; good you mentioned it! particularly since we still have the interface to the \pkg{graph} package. I think I'd like to draw one graph in that article --- maybe the undirected graph corresponding to a crossprod() result of dimension ca. $50^2$} \TODO{Specialized algorithms can give substantial savings in amount of storage used and execution time of operations.} \TODO{Our implementation is based on the CHOLMOD and CSparse libraries by Tim Davis.} \subsection{Representations of sparse matrices} \label{ssec:SparseReps} \subsubsection{Triplet representation (\class{TsparseMatrix})} Conceptually, the simplest representation of a sparse matrix is as a triplet of an integer vector \code{i} giving the row numbers, an integer vector \code{j} giving the column numbers, and a numeric vector \code{x} giving the non-zero values in the matrix.\footnote{For efficiency reasons, we use ``zero-based'' indexing in the \pkg{Matrix} package, i.e., the row indices \code{i} are in \code{0:(nrow(.)-1)} and the column indices \code{j} accordingly.} In \pkg{Matrix}, the \class{TsparseMatrix} class is the virtual class of all sparse matrices in triplet representation. Its main use is for easy input or transfer to other classes. As for the dense matrices, the class of the \code{x} slot may vary, and the subclasses may be triangular, symmetric or unspecified (``general''), such that the \class{TsparseMatrix} class has several\footnote{the $3 \times 3$ actual subclasses of \class{TsparseMatrix} are the three structural kinds, namely \textbf{t}riangular, \textbf{s}ymmetric and \textbf{g}eneral, times three entry classes, \textbf{d}ouble, \textbf{l}ogical, and patter\textbf{n}.} `actual'' subclasses, the most typical (numeric, general) is \class{dgTMatrix}: <<Tsparse-class>>= getClass("TsparseMatrix") # (i,j, Dim, Dimnames) slots are common to all getClass("dgTMatrix") @ Note that the \emph{order} of the entries in the \code{(i,j,x)} vectors does not matter; consequently, such matrices are not unique in their representation. \footnote{ Furthermore, there can be \emph{repeated} \code{(i,j)} entries with the customary convention that the corresponding \code{x} entries are \emph{added} to form the matrix element $m_{ij}$. } %% The triplet representation is row-oriented if elements in the same row %% were adjacent and column-oriented if elements in the same column were %% adjacent. \subsubsection{Compressed representations: \class{CsparseMatrix} } % (and \class{RsparseMatrix})} -- but don't advertize them For most sparse operations we use the compressed column-oriented representation (virtual class \class{CsparseMatrix}) (also known as ``csc'', ``compressed sparse column''). Here, instead of storing all column indices \code{j}, only the \emph{start} index of every column is stored. Analogously, there is also a compressed sparse row (csr) representation, which e.g. is used in in the \pkg{SparseM} package, and we provide the \class{RsparseMatrix} for compatibility and completeness purposes, in addition to basic coercion (\code({as(., \textit{<cl>})} between the classes. %% (column-oriented triplet) except that \code{i} (\code{j}) just stores %% the index of the first element in the row (column). (There are a %% couple of other details but that is the gist of it.) These compressed representations remove the redundant row (column) indices and provide faster access to a given location in the matrix because you only need to check one row (column). There are certain advantages \footnote{routines can make use of high-level (``level-3'') BLAS in certain sparse matrix computations} to csc in systems like \RR{}, Octave and Matlab where dense matrices are stored in column-major order, therefore it is used in sparse matrix libraries such as CHOLMOD or CSparse of which we make use. For this reason, the \class{CsparseMatrix} class and subclasses are the principal classes for sparse matrices in the \pkg{Matrix} package. The Matrix package provides the following classes for sparse matrices \FIXME{many more --- maybe explain naming scheme?} \begin{description} \item[dgTMatrix] general, numeric, sparse matrices in (a possibly redundant) triplet form. This can be a convenient form in which to construct sparse matrices. \item[dgCMatrix] general, numeric, sparse matrices in the (sorted) compressed sparse column format. \item[dsCMatrix] symmetric, real, sparse matrices in the (sorted) compressed sparse column format. Only the upper or the lower triangle is stored. Although there is provision for both forms, the lower triangle form works best with TAUCS. \item[dtCMatrix] triangular, real, sparse matrices in the (sorted) compressed sparse column format. \end{description} \TODO{Can also read and write the Matrix Market and read the Harwell-Boeing representations.} \TODO{Can convert from a dense matrix to a sparse matrix (or use the Matrix function) but going through an intermediate dense matrix may cause problems with the amount of memory required.} \TODO{similar range of operations as for the dense matrix classes.} \section{More detailed examples of ``Matrix'' operations} Have seen \texttt{drop0()} above, %(p.3); only with logical showe a nice double example (where you see ``.'' and ``0''). Show the use of \code{dim<-} for \emph{resizing} a (sparse) matrix. Maybe mention \Rfun{nearPD}. \TODO{Solve a sparse least squares problem and demonstrate memory / speed gain} \TODO{mention \code{lme4} and \Rfun{lmer}, maybe use one example to show the matrix sizes.} \section{Notes about S4 classes and methods implementation} Maybe we could % even here (for R News, not only for JSS) give some glimpses of implementations at least on the \RR{} level ones? \TODO{The class hierarchy: a non-trivial tree where only the leaves are ``actual'' classes.} \TODO{The main advantage of the multi-level hierarchy is that methods can often be defined on a higher (virtual class) level which ensures consistency [and saves from ``cut \& paste'' and forgetting things]} \TODO{Using Group Methods} \section{Session Info} <<sessionInfo, results=tex>>= toLatex(sessionInfo()) @ \bibliography{Matrix} \end{document}
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