\documentclass[a4paper]{article}

%\VignetteIndexEntry{Data with outliers}

\usepackage{hyperref}

\title{Handling of data containing outliers}
\author{Wolfram Stacklies and Henning Redestig\\
CAS-MPG Partner Institute for Computational Biology (PICB)\\
Shanghai, P.R. China \\
and\\
Max Planck Institute for Molecular Plant Physiology\\
Potsdam, Germany\\
\url{http://bioinformatics.mpimp-golm.mpg.de/}
}
\date{\today}

\begin{document}
\setkeys{Gin}{width=1.0\textwidth}
@
\maketitle

\section{PCA robust to outliers}
Away from often showing missing values, Microarray or Metabolite data are often
corrupted with extreme values (outliers). 
Standard SVD is highly susceptible to outliers. In
the extreme case, an individual data point, if sufficiently outlying, can draw even the
leading principal component toward itself.
This problem can be addressed by using 
a robust analysis method. Hereto we provide \texttt{robustSvd}, a singular value
decomposition robust to outliers. \texttt{robustPca} is a PCA implementation that
resembles the original \texttt{R} \texttt{prcomp} method, with the difference
that it uses \texttt{robustSvd} instead of the standard \texttt{svd} function.\\
Robust SVD and its application to microarray data were proposed in \cite{hawkins01} 
and \cite{liu03}. The algorithm is based on the idea to use a sequential estimation 
of the eigenvalues and left and right eigenvectors that
ignores missing values and is resistant to outliers. \\
The \texttt{robustSvd} script included here was contributed by Kevin Wright. Thanks
a lot to him! 

\section{Outliers and missing value imputation}
The problem of outliers is similar to the missing data problem in the sense that
extreme values provide no or wrong information. They are generally artifacts of
the experiment and provide no information about the underlying biological
processes. \\
Most of the PCA methods coming with the package were not designed to be robust to
outliers in the sense that they will converge to the standard PCA solution on a complete
data set.
Yet, an applicable solution is to remove obvious outliers from the data first (by setting
them NA) and to then estimate the PCA solution on the incomplete data. This is likely
to produce accurate results if the number of missing data does not exceed a certain
amount, less than 10\% should be a good number.

The following example illustrates the effect of outliers and the use of robust methods.
First, we attach the complete metabolite data set and create 5\% outliers. We
mean center the data before we create outliers because these large artificial
outliers will strongly shift the original means. This would not allow for objective comparison
between the differnt results obtained, e.g. when doing scatterplots.
<<results=hide>>=
library(pcaMethods)
@
<<>>=
data(metaboliteDataComplete)
mdc          <- scale(metaboliteDataComplete, center=TRUE, scale=FALSE)
cond         <- runif(length(mdc)) < 0.05;
mdcOut       <- mdc; mdcOut[cond] <- 10
@
Then we calculate a PCA solution using standard SVD and robust SVD.
<<results=hide>>=
resSvd       <- pca(mdc, method = "svd", nPcs = 5, center = FALSE)
resSvdOut    <- pca(mdcOut, method = "svd", nPcs = 5, center = FALSE)
resRobSvd    <- pca(mdcOut, method = "robustPca", nPcs = 5, center = FALSE)
@
Now we use \texttt{PPCA} to estimate the PCA solution, 
but set the outliers NA before.
<<results=hide>>=
mdcNa        <- mdc;
mdcNa[cond]  <- NA
resPPCA      <- pca(mdcNa, method = "ppca", nPcs = 5, center = FALSE)
@

To check the robustness to outliers we can just do a scatterplot comparing the
results to the optimal PCA solution for the complete data set (which is \texttt{resSvd}).
In Figure \ref{fig:svdPlot} we plot the estimated and original loadings against each other.
\begin{figure}[!ht]
  \centering
<<fig=true, width=16, height=16, results=hide, echo=false>>=
par(mar=c(5, 6, 4, 2))
par(mfrow=c(2,2))
plot(resSvd@loadings[,1], resSvdOut@loadings[,1], 
xlab = "Loading 1 SVD", ylab = "Loading 1 SVD with outliers",
cex.lab = 3, cex.axis = 2, cex = 3, lwd = 3)
plot(resSvd@loadings[,1], resRobSvd@loadings[,1],
xlab = "Loading 1 SVD", ylab = "Loading 1 robustSVD with outliers",
cex.lab = 3, cex.axis = 2, cex = 3, lwd = 3)
plot(resSvd@loadings[,1], resPPCA@loadings[,1],
xlab = "Loading 1 SVD", ylab = "Loading 1 PPCA with outliers = NA",
cex.lab = 3, cex.axis = 2, cex = 3, lwd = 3)
plot(resRobSvd@loadings[,1], resPPCA@loadings[,1],
xlab = "Loading 1 roubst SVD with outliers", ylab = "Loading 1 svdImpute with outliers = NA",
cex.lab = 3, cex.axis = 2, cex = 3, lwd = 3)
@
  \caption{Figures show (from left to right): \newline
  Original PCA solution vs. solution on data with outliers; \newline
  Original PCA solution vs. robust PCA solution on data with outliers; \newline
  Original PCA solution vs. PPCA solution on data where outliers = NA; \newline
  Robust PCA solution vs. PPCA solution on data with outliers / outliers = NA.
  \label{fig:svdPlot}
  }
\end{figure}

\begin{thebibliography}{2006}
\bibitem{hawkins01} Hawkins, D.M., Liu, L. and Young, S.S.
{\sl Robust Singular Value Decomposition.}
National Institute of Statistical Sciences, 2001, Tech Report 122.

\bibitem{liu03} Liu, L., Hawkins, D.M., Ghosh, S. and Young, S.S.
{\sl Robust singular value decomposition analysis of microarray data.}
PNAS, 2003;100:13167--13172.
\end{thebibliography}

\end{document}
