\documentclass{article}
%
\usepackage{fullpage}
\usepackage{myVignette}
\usepackage[authoryear,round]{natbib}
\bibliographystyle{plainnat}
\newcommand{\noFootnote}[1]{{\small (\textit{#1})}}
\newcommand{\myOp}[1]{{$\left\langle\ensuremath{#1}\right\rangle$}}
%%\VignetteIndexEntry{Sparse Model Matrices}
%%\VignetteDepends{Matrix,MASS}
\title{Sparse Model Matrices}
\author{Martin Maechler\\ R Core Development Team
\\\email{maechler@R-project.org}}
\date{July 2007, 2008 ({\tiny typeset on \tiny\today})}
%
\begin{document}
\maketitle
\SweaveOpts{engine=R, keep.source=TRUE}
\SweaveOpts{eps=FALSE, pdf=TRUE, width=8, height=5.5, strip.white=true}
\setkeys{Gin}{width=\textwidth}
% \begin{abstract}
% ............................ FIXME
% \end{abstract}
%% Note: These are explained in '?RweaveLatex' :
<>=
options(width=75)
@
\section*{Introduction}
Model matrices in the very widely used (generalized) linear models
of statistics, (typically fit via \Rfun{lm} or \Rfun{glm} in \RR) are often
practically sparse --- whenever categorical predictors, \code{factor}s in
\RR, are used.
%% FIXME: Introduce lm.fit.sparse() or not ?
We show for a few classes of such linear models how to construct sparse
model matrices using sparse matrix (S4) objects from the \pkg{Matrix}
package, and typically \emph{without} using dense matrices in intermediate
steps. %% only the latter is really novel, since "SparseM" (and others)
%% have used the equivalent of
%% as( model.matrix(.....), "sparseMatrix")
\section{One factor: \texttt{y $\sim$ f1}}
Let's start with an artifical small example:
<>=
(ff <- factor(strsplit("statistics_is_a_task", "")[[1]], levels=c("_",letters)))
factor(ff) # drops the levels that do not occur
f1 <- ff[, drop=TRUE] # the same, more transparently
@
and now assume a model $$y_i = \mu + \alpha_{j(i)} + E_i,$$ for $i=1,\dots,n
=$~\code{length(f1)}$= 20$, and
$\alpha_{j(i)}$ with a constraint such as $\sum_j \alpha_j = 0$ (``sum'')
or $\alpha_1 = 0$ (``treatment'') and $j(i) =$\code{as.numeric(f1[i])}
being the level number of the $i$-th observation. For such a ``design'',
the model is only estimable if the levels \code{c} and \code{k} are merged,
and
<>=
levels(f1)[match(c("c","k"), levels(f1))] <- "ck"
library(Matrix)
Matrix(contrasts(f1)) # "treatment" contrasts by default -- level "_" = baseline
Matrix(contrasts(C(f1, sum)))
Matrix(contrasts(C(f1, helmert)), sparse=TRUE) # S-plus default; much less sparse
@
where \Rfun{contrasts} is (conceptually) just one major ingredient in the
well-known \Rfun{model.matrix} function
to build the linear model matrix $\mathbf{X}$ of so-called ``dummy variables''.
%%
Since 2007, the \pkg{Matrix}
package has been providing coercion from a \code{factor} object to a
\code{sparseMatrix} one to produce the transpose of the model matrix
corresponding to a model with that factor as predictor (and no intercept):
<>=
as(f1, "sparseMatrix")
@
which is really almost the transpose of using the above sparsification of
\Rfun{contrasts} (and arranging for nice printing),
<>=
printSpMatrix( t( Matrix(contrasts(f1))[as.character(f1) ,] ),
col.names=TRUE)
@
and that is the same as the ``sparsification'' of \Rfun{model.matrix},
apart from the column names (here transposed),
<>=
t( Matrix(model.matrix(~ 0+ f1))) # model with*OUT* intercept
@
A more realistic small example is the \code{chickwts} data set,
<>=
str(chickwts)# a standard R data set, 71 x 2
x.feed <- as(chickwts$feed, "sparseMatrix")
x.feed[ , (1:72)[c(TRUE,FALSE,FALSE)]] ## every 3rd column:
@
% FIXME: Move this to ../../../MatrixModels/inst/doc/ ???
% ## Provisional (hence unexported) sparse lm.fit():
% Matrix:::lm.fit.sparse(x = t(x.feed), y = chickwts[,1])
%- for emacs: $
\section{One factor, one continuous: \texttt{y $\sim$ f1 + x}}
To create the model matrix for the case of one factor and one continuous
predictor---called ``analysis of covariance'' in the historical literature---
we can adopt the following simple scheme.
%% Possible examples:
%% - Puromycin
%% - ToothGrowth
%--- FIXME ---
The final model matrix is the concatenation of:
1) create the sparse 0-1 matrix \code{m1} from the f1 main-effect
2) the single row/column 'x' == 'x' main-effect
3) replacing the values 1 in \code{m1@x} (the x-slot of the factor model matrix),
by the values of \code{x} (our continuous predictor).
\section{Two (or more) factors, main effects only: \texttt{y $\sim$ f1 + f2}}
%% FIXME: 'warpbreaks' is smaller and more natural as fixed effect model!
Let us consider the \code{warpbreaks} data set of 54 observations,
<>=
data(warpbreaks)# a standard R data set
str(warpbreaks) # 2 x 3 (x 9) balanced two-way with 9 replicates:
xtabs(~ wool + tension, data = warpbreaks)
@
%It is \emph{not} statistically sensible to assume that \code{Run} is a
%fixed effect, however the example is handy to depict how a model matrix
This example depicts how a model matrix
would be built for the model \code{breaks ~ wool + tension}.
Since this is a main effects model (no interactions), the desired model
matrix is simply the concatenation of the model matrices of the main
effects. There are two here, but the principle applies to general main
effects of factors.
The most sparse matrix is reached by \emph{not} using an intercept,
(which would give an all-1-column) but rather have one factor fully coded
(aka ``swallow'' the intercept), and all others being at \code{"treatment"}
contrast, i.e., here, the \emph{transposed} model matrix, \code{tmm}, is
<>=
tmm <- with(warpbreaks,
rBind(as(tension, "sparseMatrix"),
as(wool, "sparseMatrix")[-1,,drop=FALSE]))
print( image(tmm) ) # print(.) the lattice object
@
\\
The matrices are even sparser when the factors have more than just two or
three levels, e.g., for the morley data set,
<>=
data(morley) # a standard R data set
morley$Expt <- factor(morley$Expt)
morley$Run <- factor(morley$Run)
str(morley)
t.mm <- with(morley,
rBind(as(Expt, "sparseMatrix"),
as(Run, "sparseMatrix")[-1,]))
print( image(t.mm) ) # print(.) the lattice object
@
%% Also see Doug's E-mail to R-help
% From: "Douglas Bates"
% Subject: Re: [R] Large number of dummy variables
% Date: Mon, 21 Jul 2008 18:07:26 -0500
\section{Interactions of two (or more) factors,.....}
%% Of course, this is *the* interesting part
%% To form interactions, we would have to ``outer-multiply''
%% the single-factor model-matrices (after "[, -1]")
In situations with more than one factor, particularly with interactions,
the model matrix is currently not directly available via \pkg{Matrix}
functions --- but we still show to build them carefully.
The easiest---but not at memory resources efficient---way is to go via the
dense \Rfun{model.matrix} result:
<>=
data(npk, package="MASS")
npk.mf <- model.frame(yield ~ block + N*P*K, data = npk)
## str(npk.mf) # the data frame + "terms" attribute
m.npk <- model.matrix(attr(npk.mf, "terms"), data = npk)
class(M.npk <- Matrix(m.npk))
dim(M.npk)# 24 x 13 sparse Matrix
t(M.npk) # easier to display, column names readably displayed as row.names(t(.))
@
%% printSpMatrix(M.npk, col.names = "abb1")
Another example was reported by a user on R-help (July 15, 2008,
{\small \url{https://stat.ethz.ch/pipermail/r-help/2008-July/167772.html}})
about an ``aov error with large data set''.
\begin{citation} % RAS: in my PDF, I don't see the first character I
I'm looking to analyze a large data set: a within-Ss 2*2*1500 design
with 20 Ss. However, aov() gives me an error. %, reproducible as follows:
\end{citation}
And gave the following code example (slightly edited):
<>=
id <- factor(1:20)
a <- factor(1:2)
b <- factor(1:2)
d <- factor(1:1500)
aDat <- expand.grid(id=id, a=a, b=b, d=d)
aDat$y <- rnorm(length(aDat[, 1])) # generate some random DV data
dim(aDat) # 120'000 x 5 (120'000 = 2*2*1500 * 20 = 6000 * 20)
@
%% ^^^^^^^ MM: "fix" and generate much more interesting data
and then continued with
\begin{Sinput}
m.aov <- aov(y ~ a*b*d + Error(id/(a*b*d)), data=aDat)
\end{Sinput}
\begin{citation}\sffamily
which yields the following error:\\
\ttfamily
Error in model.matrix.default(mt, mf, contrasts) :\\
allocMatrix: too many elements specified\\
\end{citation}
to which he got the explanation by Peter Dalgaard that the formal model matrix involved
was much too large in this case, and that PD assumed, \pkg{lme4} would be
able to solve the problem.
However, currently there would still be a big problem with using \pkg{lme4},
because of the many levels of \emph{fixed} effects:
Specifically\footnote{the following is not run in \RR\ on purpose, rather just
displayed here},
\begin{Sinput}
dim(model.matrix( ~ a*b*d, data = aDat)) # 120'000 x 6000
\end{Sinput}
where we note that $120'000 \times 6000 = 720 \textrm{mio}$, which is
$720'000'000 * 8 / 2^{20} \approx 5500$ Megabytes.
\emph{Unfortunately} \pkg{lme4} does \emph{not} use a sparse $X$-matrix for
the fixed effects (yet), it just uses sparse matrices for the $Z$-matrix of
random effects and sparse matrix operations for computations related to $Z$.
Let us use a smaller factor \code{d} in order to investigate how sparse the
$X$ matrix would be:
<>=
d2 <- factor(1:150) # 10 times smaller
tmp2 <- expand.grid(id=id, a=a, b=b, d=d2)
dim(tmp2)
dim(mm <- model.matrix( ~ a*b*d, data=tmp2))
## is 100 times smaller than original example
class(smm <- Matrix(mm)) # automatically coerced to sparse
round(object.size(mm) / object.size(smm), 1)
@
shows that even for the small \code{d} here, the memory reduction would be
more than an order of magnitude.
\\
%% Reasons to fake here:
%% 1) print() is needed for lattice -- but looks ugly,
%% 2) the resulting pdf file is too large -- use png instead:
<>=
image(t(smm), aspect = 1/3, lwd=0, col.regions = "red")
<>=
png("sparseModels-X-sparse-image.png", width=6, height=3,
units='in', res=150)
print(
<>
)
dev.off()
@
%%--NB: 'keep.source=FALSE' above is workaround-a-bug-in-R-devel-(2.13.x)---
\par\vspace*{-1ex}
\centerline{%
\includegraphics[width=1.1\textwidth]{sparseModels-X-sparse-image.png}}
and working with the sparse instead of the dense model matrix is
considerably faster as well,
<>=
x <- 1:600
system.time(y <- smm %*% x) ## sparse is much faster
system.time(y. <- mm %*% x) ## than dense
identical(as.matrix(y), y.) ## TRUE
@
<>=
toLatex(sessionInfo())
@
\end{document}