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%\VignetteKeywords{Bioinformatics, Machine learning, Organelle, Proteomics}
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\author{
  Laurent Gatto\thanks{\email{lg390@cam.ac.uk}}
}

\affil{
  Computational Proteomics Unit\\
  University of Cambridge
}


\begin{document}

\title{Machine learning techniques available in \Rpackage{pRoloc}}

\maketitle

% %% Abstract and keywords %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \vskip 0.3in minus 0.1in
% \hrule
% \begin{abstract}
% This document described the machine learning techniques that are made available in the \Rpackage{pRoloc}.
% \end{abstract}
% \textit{Keywords}: Bioinformatics, spatial/organelle proteomics, machine learning, visualisation
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\tableofcontents

<<env, include=FALSE, echo=FALSE, cache=FALSE>>=
library("knitr")
opts_chunk$set(fig.align = 'center', 
               fig.show = 'hold', 
               par = TRUE,
               prompt = TRUE,
               eval = TRUE,
               comment = NA)
options(replace.assign = TRUE, 
        width = 55)

suppressPackageStartupMessages(library("MSnbase"))
suppressWarnings(suppressPackageStartupMessages(library("pRoloc")))
suppressPackageStartupMessages(library("pRolocdata"))
## suppressPackageStartupMessages(library("class"))
suppressPackageStartupMessages(library("xtable"))
@ 
%%$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Section
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\newpage

\paragraph{NB} This vignette provides more general background about
machine learning (ML) and its application to the analysis of organelle
proteomics data. This document is currently under still construction;
please see the \Rpackage{pRoloc} tutorial for details about the
package itself.

\paragraph{TODO} Define nomenclature

\section{Introduction}\label{sec:intro} 

For a general practical introduction to \Rpackage{pRoloc}, readers are
referred to the tutorial, available using
\Rfunction{vignette("pRoloc-tutorial", package = "pRoloc")}. The
following document provides a overview of the algorithms available in
the package. The respective section describe unsupervised machine
learning (USML), supervised machine learning (SML), and
semi-supervised machine learning (SSML) as implemented in the novelty
detection algorithm.

\section{Data sets}

We provide \Sexpr{nrow(pRolocdata()$results)} test data sets in the
\Rpackage{pRolocdata} package%$
that can be readily used with \Rpackage{pRoloc}. The data set can be
listed with \Rfunction{pRolocdata} and loaded with the
\Rfunction{data} function. Each data set, including its origin, is
individually documented.

The data sets are distributed as \Robject{MSnSet} instances. Briefly,
these are dedicated containers for quantitation data as well as
feature and sample meta-data. More details about \Robject{MSnSet}s are
available in the \Rpackage{pRoloc} tutorial and in the
\Rpackage{MSnbase} package, that defined the class.

<<pRolocdata>>=
library("pRolocdata")
data(tan2009r1)
tan2009r1
@

\section{Unsupervised machine learning}\label{sec:usml}

Unsupervised machine learning refers to clustering, i.e. finding
structure in a quantitative, generally multi-dimensional data set of
unlabelled data.

Currently, unsupervised clustering facilities are available through
the \Rfunction{plot2D} function and the \Rpackage{MLInterfaces}
package \cite{MLInterfaces}. The former takes an \Robject{MSnSet}
instance and represents the data on a scatter plot along the first two
principal components. Arbitrary feature meta-data can be represented
using different colours and point characters. The reader is referred
to the manual page available through \Rfunction{?plot2D} for more
details and examples.

\Rpackage{pRoloc} also implements a \Rfunction{MLean} method for
\Robject{MSnSet} instances, allowing to use the relevant
infrastructure with the organelle proteomics framework. Although
provides a common interface to unsupervised and numerous supervised
algorithms, we refer to the \Rpackage{pRoloc} tutorial for its usage
to several clustering algorithms.

\paragraph{Note} Current development efforts in terms of clustering
are described on the \textit{Clustering infrastructure} wiki
page\footnote{\url{https://github.com/lgatto/pRoloc/wiki/Clustering-infrastructure}}
and will be incorporated in future version of the package.

\section{Supervised machine learning}\label{sec:sml}

Supervised machine learning refers to a broad family of classification
algorithms. The algorithms learns from a modest set of labelled data
points called the training data. Each training data example consists
of a pair of inputs: the actual data, generally represented as a
vector of numbers and a class label, representing the membership to
exactly 1 of multiple possible classes. When there are only two
possible classes, on refers to binary classification. The training
data is used to construct a model that can be used to classifier new,
unlabelled examples. The model takes the numeric vectors of the
unlabelled data points and return, for each of these inputs, the
corresponding mapped class.

\subsection{Algorithms used}\label{sec:algo}

\paragraph{k-nearest neighbour (KNN)} Function \Rfunction{knn} from package
\Rpackage{class}. For each row of the test set, the $k$ nearest (in
Euclidean distance) training set vectors are found, and the
classification is decided by majority vote over the $k$ classes, with
ties broken at random. This is a simple algorithm that is often used
as baseline classifier.
%% If there are ties for the $k^{th}$ nearest vector, all candidates are included in the vote.

\paragraph{Partial least square DA (PLS-DA)} Function \Rfunction{plsda} from
package \Rpackage{caret}. Partial least square discriminant analysis
is used to fit a standard PLS model for classification.

\paragraph{Support vector machine (SVM)} 
A support vector machine constructs a hyperplane (or set of
hyperplanes for multiple-class problem), which are then used for
classification. The best separation is defined as the hyperplane that
has the largest distance (the margin) to the nearest data points in
any class, which also reduces the classification generalisation
error. To assure liner separation of the classes, the data is
transformed using a \textit{kernel function} into a high-dimensional
space, permitting liner separation of the classes.

\Rpackage{pRoloc} makes use of the functions \Rfunction{svm} from
package \Rpackage{e1071} and \Rfunction{ksvm} from \Rpackage{kernlab}.

\paragraph{Artificial neural network (ANN)} Function \Rfunction{nnet}
from package \Rpackage{nnet}. Fits a single-hidden-layer neural
network, possibly with skip-layer connections.

\paragraph{Naive Bayes (NB)} Function \Rfunction{naiveBayes} from
package \Rpackage{e1071}. Naive Bayes classifier that computes the
conditional a-posterior probabilities of a categorical class variable
given independent predictor variables using the Bayes rule. Assumes
independence of the predictor variables, and Gaussian distribution
(given the target class) of metric predictors.

\paragraph{Random Forest (RF)} Function \Rfunction{randomForest} from
package \Rpackage{randomForest}.

\paragraph{Chi-square ($\chi^2$)} Assignment based on squared
differences between a labelled marker and a new feature to be
classified. Canonical protein correlation profile method (PCP) uses
squared differences between a labelled marker and new features. In
\citep{Andersen2003}, $\chi^2$ is defined as \emph{the [summed]
  squared deviation of the normalized profile [from the marker] for
  all peptides divided by the number of data points}, i.e. $\chi^{2} =
\frac{\sum_{i=1}^{n} (x_i - m_i)^{2}}{n}$, whereas \citep{Wiese2007}
divide by the value the squared value by the value of the reference
feature in each fraction, i.e. $\chi^{2} = \sum_{i=1}^{n}\frac{(x_i -
  m_i)^{2}}{m_i}$, where $x_i$ is normalised intensity of feature $x$
in fraction $i$, $m_i$ is the normalised intensity of marker $m$ in
fraction $i$ and $n$ is the number of fractions available. We will use
the former definition.

\paragraph{PerTurbo}
From \cite{perturbo}: PerTurbo, an original, non-parametric and
efficient classification method is presented here. In our framework,
the manifold of each class is characterised by its Laplace-Beltrami
operator, which is evaluated with classical methods involving the
graph Laplacian. The classification criterion is established thanks to
a measure of the magnitude of the spectrum perturbation of this
operator. The first experiments show good performances against
classical algorithms of the state-of-the-art. Moreover, from this
measure is derived an efficient policy to design sampling queries in a
context of active learning. Performances collected over toy examples
and real world datasets assess the qualities of this strategy.

The PerTurbo implementation comes from the \Rpackage{pRoloc} packages.

\subsection{Estimating algorithm parameters}

It is essential when applying any of the above classification
algorithms, to wisely set the algorithm parameters, as these can have
an important effect on the classification. Such parameters are for
example the width $sigma$ of the Radial Basis Function (Gaussian
kernel) $exp(-\sigma \| x - x' \|^2 )$ and the $cost$ (slack)
parameter (controlling the tolerance to mis-classification) of our SVM
classifier. The number of neighbours $k$ of the KNN classifier is
equally important as will be discussed in this sections.

Figure \ref{fig:knnboundaries} illustrates the effect of different
choices of $k$ using organelle proteomics data from
\citep{Dunkley2006} (\Rfunction{dunkley2006} from
\Rpackage{pRolocdata}). As highlighted in the squared region, we can
see that using a low $k$ ($k = 1$ on the left) will result in very
specific classification boundaries that precisely follow the contour
or our marker set as opposed to a higher number of neighbours ($k = 8$
on the right). While one could be tempted to believe that
\textit{optimised} classification boundaries are preferable, it is
essential to remember that these boundaries are specific to the marker
set used to construct them, while there is absolutely no reason to
expect these regions to faithfully separate any new data points, in
particular proteins that we wish to classify to an organelle. In other
words, the highly specific $k = 1$ classification boundaries are
\textit{over-fitted} for the marker set or, in other words, lack
generalisation to new instances. We will demonstrate this using
simulated data taken from \cite{ISLwR} and show what detrimental
effect \textit{over-fitting} has on new data.

\begin{figure}[!hbt]
\centering
    \includegraphics[width=1\textwidth]{./Figures/knnboundaries.pdf}
    \caption{Classification boundaries using $k=1$ or $k=8$ on the
      \Robject{dunkley2006} data.}
\label{fig:knnboundaries}
\end{figure}

Figure \ref{fig:ISL1} uses 2 $\times$ 100 simulated data points
belonging to either of the orange or blue classes. The genuine classes
for all the points is known (solid lines) and the KNN algorithm has
been applied using $k=1$ (left) and $k=100$ (right) respectively
(purple dashed lines). As in our organelle proteomics examples, we
observe that when k = 1, the decision boundaries are overly flexible
and identify patterns in the data that do not reflect to correct
boundaries (in such cases, the classifier is said to have low bias but
very high variance). When a large $k$ is used, the classifier becomes
inflexible and produces approximate and nearly linear separation
boudaries (such a classifier is said to have low variance but high
bias). On this simulated data set, neither $k = 1$ nor $k = 100$ give
good predictions and have test error rates (i.e. the proportion of
wronly classified points ) of 0.1695 and 0.1925, respectively.

\begin{figure}[!hbt]
\centering
    \includegraphics[width=1\textwidth]{./Figures/ISL-2_16.pdf}
    \caption{The KNN classifier using $k = 1$ (left, solid
      classification boudaries) and $k = 100$ (right, solid
      classification boundaries) compared the Bayes decision boudaries
      (see original material for details). Reproduced with permission
      from \cite{ISLwR}.}
\label{fig:ISL1}
\end{figure}

To quantify the effect of flexibility and lack thereof in defining the
classification boundaries, \cite{ISLwR} calculate the classification
error rates using training data (training error rate) and testing data
(testing error rate). The latter is completely new data that was not
used to assess the model error rate when defining algorithm
parameters; one often says that the model used for classification has
not \textit{seen} this data. If the model performs well on new data,
it is said to generalise well. This is a quality that is required in
most cases, and in particular in our organelle proteomics experiments
where the training data corresponds to our marker sets. Figure
\ref{fig:ISL2} plots the respective training and testing error rates
as a function of $1/k$ which is a reflection of the
flexibility/complexity of our model; when $1/k = 1$, i.e. $k = 1$ (far
right), we have a very flexible model with the risk of
over-fitting. Greater values of $k$ (towards the left) characterise
less flexible models. As can be seen, high values of $k$ produce poor
performance for both training and testing data. However, while the
training error steadily decreases when the model complexity increases
(smaller $k$), the testing error rate displays a typical U-shape: at a
value around $k = 10$, the testing error rate reaches a minimum and
then starts to increase due to over-fitting to the training data and
lack of generalisation of the model.

\begin{figure}[!hbt]
\centering
    \includegraphics[width=.8\textwidth]{./Figures/ISL-2_17.pdf}
    \caption{Effect of train and test error with respect to model
      complexity. The former decreases for lower values of $k$ while
      the test error reaches a minimum around $k = 10$ before
      increasing again. Reproduced with permission from \cite{ISLwR}.}
\label{fig:ISL2}
\end{figure}


These results show that adequate optimisation of the model parameters
are essential to avoid either too flexible models (that do not
generalise well to new data) or models that do not describe the
decision boundaries adequately. Such parameter selection is achieved
by cross validation, where the initial marker proteins are separated
into training data used to build classification models and independent
testing data used to assess the model on new data.

We recommend the book \textit{An Introduction to Statistical
  Learning}\footnote{\url{http://www-bcf.usc.edu/~gareth/ISL/}} by
\cite{ISLwR} for a more detailed introduction of machine learning.


\subsection{Default analysis scheme}

Below, we present a typical classification analysis using
\Rpackage{pRoloc}. The analysis typically consists of two steps. The
first one is to optimise the classifier parameters to be used for
training and testing (see above). A range of parameters are tested
using the labelled data, for which the labels are known. For each set
of parameters, we hide the labels of a subset of labelled data and use
the other part to train a model and apply in on the testing data with
hidden labels. The comparison of the estimated and expected labels
enables to assess the validity of the model and hence the adequacy if
the parameters. Once adequate parameters have been identified, they
are used to infer a model on the complete organelle marker set and
used to infer the subcellular location of the unlabelled examples.

\minisec{Parameter optimisation}

Algorithmic performance is estimated using a stratified 20/80
partitioning. The 80\% partitions are subjected to 5-fold
cross-validation in order to optimise free parameters via a grid
search, and these parameters are then applied to the remaining
20\%. The procedure is repeated $n = 100$ \texttt{times} to sample $n$
accuracy metrics (see below) values using $n$, possibly different,
optimised parameters for evaluation.

Models accuracy is evaluated using the F1 score, $F1 = 2 ~
\frac{precision \times recall}{precision + recall}$, calculated as the
harmonic mean of the precision ($precision = \frac{tp}{tp+fp}$, a
measure of \textit{exactness} -- returned output is a relevant result)
and recall ($recall=\frac{tp}{tp+fn}$, a measure of
\textit{completeness} -- indicating how much was missed from the
output). What we are aiming for are high generalisation accuracy, i.e
high $F1$, indicating that the marker proteins in the test data set
are consistently correctly assigned by the algorithms.

The results of the optimisation procedure are stored in an
\Robject{GenRegRes} object that can be inspected, plotted and best
parameter pairs can be extracted.

For a given algorithm \texttt{alg}, the corresponding parameter
optimisation function is names \Rfunction{algOptimisation} or,
equivalently, \Rfunction{algOptimization}. See table \ref{tab:algo}
for details. A description of each of the respective model parameters
is provided in the optimisation function manuals, available through
\Rfunction{?algOptimisation}.

<<svmParamOptim, cache = TRUE, warning = FALSE, message = FALSE>>=
params <- svmOptimisation(tan2009r1, times = 10, xval = 5, verbose = FALSE)
params
@ 

\minisec{Classification}

<<svmRes, warning=FALSE, tidy=FALSE, eval=TRUE>>=
tan2009r1 <- svmClassification(tan2009r1, params)
tan2009r1
@ 

\subsection{Customising model parameters}

Below we illustrate how to weight different classes according to the 
number of labelled instances, were large sets are down weighted. 
This strategy can help with imbalanced designs.

<<weigths, eval=FALSE>>=
w <- table(fData(dunkley2006)$markers)
w <- 1/w[-5]
wpar <- svmOptimisation(dunkley2006, class.weights = w)
wres <- svmClassification(dunkley2006, pw, class.weights = w)
@ 


<<getmlfunction, echo=FALSE>>=
## Add chi^2.
tab <- data.frame('parameter optimisation' = grep("Optimisation", ls("package:pRoloc"), value = TRUE), 
                  'classification' = grep("Classification", ls("package:pRoloc"), value = TRUE))
tab$algorithm <- c("nearest neighbour", "support vector machine", "naive bayes",
                   "neural networks", "PerTurbo", "partial least square", 
                   "random forest", "support vector machine")
tab$package <- c("class", "kernlab", "e1071",
                 "nnet", "pRoloc", "caret",
                 "randomForest", "e1071")
colnames(tab)[1] <- c("parameter optimisation")
@ 


<<comptab, results='asis', echo=FALSE>>=
xt <- xtable(tab, label = "tab:algo",
             caption = "Supervised ML algorithm available in \\Rpackage{pRoloc}.")
print(xt, include.rownames = FALSE, size = "small")
@ 

\section{Comparison of different classifiers}

Several supervised machine learning algorithms have already been
applied to organelle proteomics data classification: partial least
square discriminant analysis in \cite{Dunkley2006, Tan2009}, support
vector machines (SVMs) in \cite{Trotter2010}, random forest in
\cite{Ohta2010}, neural networks in \cite{Tardif2012}, naive Bayes
\cite{Nikolovski2012}. In our HUPO 2011 poster (see
\Rfunction{vignette("HUPO\_2011\_poster", package = "pRoloc")}), we
show that different classification algorithms provide very similar
performance. We have extended this comparison on various datasets
distributed in the \Rpackage{pRolocdata package}. On figure
\ref{fig:f1box}, we illustrate how different algorithms reach very
similar performances on most of our test datasets.

\begin{figure}[!hbt]
\centering
    \includegraphics[width=0.7\textwidth]{./Figures/F1boxplots.pdf}
\caption{Comparison of classification performances of several contemporary classification algorithms on data from the \Rpackage{pRolocdata} package.}
\label{fig:f1box}
\end{figure}


\section{Semi-supervised machine learning and novelty detection}\label{sec:ssml}

The \emph{phenoDisco} algorithm is a semi-supervised novelty detection
method by \cite{Breckels2013} (figure \ref{fig:pd}). It uses the
labelled (i.e. markers, noted $D_L$) and unlabelled (i.e. proteins of
unknown localisation, noted $D_U$) sets of the input data. The
algorithm is repeated $N$ times (the \Robject{code} argument in the
\Rfunction{phenoDisco} function). At each iteration, each organelle
class $D_{L}^{i}$ and the unlabelled complement are clustered using
Gaussian mixture modelling. While unlabelled members that
systematically cluster with $D_{L}^{i}$ and pass outlier detection are
labelled as new putative members of class $i$, any example of $D_U$
which are not merged with any any of the $D_{L}^{i}$ and are
consistently clustered together throughout the $N$ iterations are
considered members of a new phenotype.

\begin{figure}[!hbt]
\centering
    \includegraphics[width=0.7\textwidth]{./Figures/phenodisco.pdf}
\caption{The PhenoDisco iterative algorithm.}
\label{fig:pd}
\end{figure}

\singlespacing

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Section
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\clearpage

\section*{Session information}\label{sec:sessionInfo} 

All software and respective versions used to produce this document are listed below.

<<sessioninfo, results='asis', echo=FALSE>>=
toLatex(sessionInfo())
@

\bibliographystyle{plainnat}
\bibliography{pRoloc}

\end{document}

