\name{robustSvd}
\alias{robustSvd}
\title{Alternating L1 Singular Value Decomposition}
\usage{robustSvd(x)}
\description{A robust approximation to the singular value decomposition of a
rectangular matrix is computed using an alternating L1 norm
(instead of the more usual least squares L2 norm). As the SVD is
a least-squares procedure, it is highly susceptible to outliers
and in the extreme case, an individual cell (if sufficiently
outlying) can draw even the leading principal component toward
itself.}
\details{See Hawkins et al (2001) for details on the robust SVD algorithm.
Briefly, the idea is to sequentially estimate the left and right
eigenvectors using an L1 (absolute value) norm minimization.

Note that the robust SVD is able to accomodate missing values in
the matrix \code{x}, unlike the usual \code{svd} function.

Also note that the eigenvectors returned by the robust SVD
algorithm are NOT (in general) orthogonal and the eigenvalues need
not be descending in order.}
\value{The robust SVD of the matrix is x = u d v'. \item{d}{A
vector containing the singular values of \code{x}.} \item{u}{A
matrix whose columns are the left singular vectors of \code{x}.}
\item{v}{A matrix whose columns are the right singular vectors of
\code{x}.}}
\note{Two differences from the usual SVD may be noted. One relates
to orthogonality. In the conventional SVD, all the eigenvectors
are orthogonal even if not explicitly imposed.  Those returned by
the AL1 algorithm (used here) are (in general) not orthogonal.
Another difference is that, in the L2 analysis of the conventional
SVD, the successive eigen triples (eigenvalue, left eigenvector,
right eigenvector) are found in descending order of
eigenvalue. This is not necessarily the case with the AL1
algorithm.  Hawkins et al (2001) note that a larger eigen value
may follow a smaller one.}
\references{Hawkins, Douglas M, Li Liu, and S Stanley Young (2001)
Robust Singular Value Decomposition, National Institute of
Statistical Sciences, Technical Report Number
122. \url{http://www.niss.org/technicalreports/tr122.pdf}}
\author{Kevin Wright, modifications by Wolfram Stacklies}
\seealso{\code{\link{svd}}, \code{\link[ade4:nipals]{nipals}} for
an alternating L2 norm method that also accommodates missing data.}
\keyword{algebra}
\arguments{\item{x}{A matrix whose SVD decomposition is to be
computed. Missing values are allowed.}}
\examples{## Load a complete sample metabolite data set and mean center the data
data(metaboliteDataComplete)
mdc <- prep(metaboliteDataComplete, center=TRUE, scale="none")
## Now create 5% of outliers.
cond   <- runif(length(mdc)) < 0.05;
mdcOut <- mdc
mdcOut[cond] <- 10
## Now we do a conventional SVD and a robustSvd on both, the original and the 
## data with outliers.
resSvd <- svd(mdc)
resSvdOut <- svd(mdcOut)
resRobSvd <- robustSvd(mdc)
resRobSvdOut <- robustSvd(mdcOut)
## Now we plot the results for the original data against those with outliers
## We can see that robustSvd is hardly affected by the outliers.
plot(resSvd$v[,1], resSvdOut$v[,1])
plot(resRobSvd$v[,1], resRobSvdOut$v[,1])}