\chapter{Measures of Semantic Relatedness}

Measure of Semantic Relatedness (MSR) is a way for measuring a kind of semantic similarity between two lemmas . This relatedness is extracted in unsupervised way from the corpora on the basis of words co-occurrence.

\section{Notation}

The following notation is used throughout the chapter:

$M$ --- a matrix 

$r$ --- number of rows in the matrix

$c$ --- number of columns in the matrix

$M_{i,j}$ --- value of matrix for i-th row and j-th column

$TF_{i,*}$ --- global term frequency for i-th row

$TF_{*,j}$ --- global term frequency for j-th column

$tf_{i,*}$ --- term frequency for i-th row defined as $\sum_{j = 0}^{c} M_{i, j}$

$tf_{*,j}$ --- term frequency for j-th column defined as $\sum_{i = 0}^{r} M_{i, j}$

$nz_{i,*}$ --- number of non zero features for i-th row

$nz_{*,j}$ --- number of non zero features for j-th column

$weight_{i,j}$ --- weighted value of specified matrix cell

\section{Classical Measures}

\subsection{TF.IDF}

Term frequency, inversed document frequency \citep{ManRagSch:08}. Assuming that rows represent documents and columns
represent terms:

\begin{eqnarray}
tfidf_{i,j} &=& M[i,j] \times idf_j  \\
idf_j &=& \log{ \frac{r}{nz_{*,j}}} \\
\end{eqnarray}

\subsection{Logent}

Measure used for weighting in Latent Semantic Analysis (LSA) \citep{LanDum:97}

\begin{eqnarray}
logent_{i,j} &=& \frac{\log{M_{i,j}}} { entropy_{i,*} }\\
 &=& \frac{\log{M_{i,j}}} {\sum_{j=0}^{c} { p\times \log{p} }}
\end{eqnarray}


\section{Lin's Measure}


\section{Relative Frequency Focus}

RFF measure was described in \citep{gefdag:06, dagan2009bootstrapping}. Basically it is a weighting scheme based in
(Lin's) similarity space. First standard MI-based weighting scheme is used. In \citep{dagan2009bootstrapping} the
following formula is given:

\begin{eqnarray}
weight_{MI} &=& \log_{2} { \frac{ count(w,f) \times nrels } {count(w) \times count(f)} } \\
MI_{i,j} &=& \log_{2} { \frac{ M_{i,j} \times nrels } { tf_{i,*} \times tf_{*,j} } } \\
nrels &=& \sum_{i,j}{M_{i,j}}
\end{eqnarray}

$WS(f)$ --- set of words, where $f$ is \emph{active feature}, i.e., words $i$ where $MI_{i, f} > t_1$ and $tf_{*,f} > t_2$.
In original work, $t_1 = 4$ and $ t_2 = 10$.

$N(w)$ --- neighbourhood of word $w$, i.e., a set of words $v$, where $sim_{Lin}(w, v) > t_3$. In original work 
$t_3 = 0.04$.

Weighting is performed as follows:

\begin{eqnarray}
RFF_{w,f} = \sum_{v \in WS(f) \cap N(w) } { sim_{Lin}(w,v)}
\end{eqnarray}

After that the feature space is reduced to $t_4$ dimensions for every feature.\footnote{So basically it will be much
larger then $t_4$, because different words will select different features.}. Reduction is performed in straightforward
manner, i.e., top $t_4$ features are retained.

The last step of RFF is to compute similarity using Lin98 formula.

\section{Co-occurrence Retrieval Models}

\section{Rank Weight Functions}


