\begin{table}[httt] \vspace{-0.5cm} \scriptsize \caption{Metrics used to estimate the relevance of a candidate tag $c$ as a recommendation for an object $o$ [Belém et al. 2011; Lipczak and Milios 2011]% \cite{belem_sigir2011,lipczak11}. } \label{tab:attrib} \centering \begin{tabular}{|c|p{1.55cm}|p{12.5cm}|} \hline & \textbf{Name} & \textbf{Equation/Description} \\ \hline \hline \multirow{4}{*}{\rotatebox{90}{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Tag Co-occurrence \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }} & $Sum$ & Let $X$ be a set of tags and $c$ a candidate tag. $X \rightarrow c$ is an association rule and $\theta (X \rightarrow c)$ is its \textit{confidence}. $Sum$ is defined as: \vspace{-0.3cm} \begin{equation} \label{eq:soma} Sum(c, I_o, \ell) = \sum_{X \subseteq I_o} \theta (X \rightarrow c), \quad (X \rightarrow c) \in \mathcal{R}, |X| \le \ell, \end{equation} \noindent where $\mathcal{R}$ is a set of association rules computed offline over the training set $\mathcal{D}$, %given thresholds $\sigma_{min}$ and $\theta_{min}$, and $\ell$ is the size limit for the antecedent $X$. As in \cite{belem_sigir2011}, we use the $LATRE$ algorithm to generate these rules. \\ \cline{2-3} & $Sum^+$ & \begin{equation} \label{eq:sum+} \begin{array}{l} Sum^+(c, I_o, k_x, k_c, k_r) = \sum_{x \in I_o} \theta(x \rightarrow c) \times Stab(x, k_x) \times Stab(c, k_c) \times Rank(c, x, k_r), %|X| \leq \ell \end{array} \end{equation} \noindent where $Stab(x, k_x)$ is defined in Eq. (10), and $k_x$, $k_c$ and $k_r$ are tuning parameters. $Rank(c, x, k_r)$ is equal to $k_r/(k_r + p(c, x))$, where $p(c, x)$ is the position of $c$ in the ranking of candidates according to the confidence of the corresponding association rule (whose antecedent is $x$). \\ \cline{2-3} \vspace{-0.3cm} & $Vote$ & \begin{equation} \label{eq:vote} \begin{array}{l} Vote(c, I_o) = \sum_{x \in I_o} j, \mbox{ where } j = \left\lbrace \begin{array}{lr} 1 & \mbox{if } (x \rightarrow c) \in \mathcal{R} \\ 0 & otherwise \end{array} \right. % \mbox{, and } |X| \le \ell. \end{array} \end{equation} \\ \cline{2-3} \vspace{-0.3cm} & $Vote^+$ & \begin{equation} \label{eq:vote+} \begin{array}{l} Vote^{+}(c, I_o, k_x, k_c, k_r) = \sum_{x \in I_o} j \times Stab(x, k_x) \times Stab(c, k_c) \times Rank(c, x, k_r), \\ \mbox{ where } j = \left\lbrace \begin{array}{lr} 1, & if \quad x \rightarrow c \in \mathcal{R} \\ 0, & otherwise \end{array} \right. %\mbox{, } |X| \le \ell, \end{array} \end{equation} \\ \hline \hline \multirow{4}{*}{\rotatebox{90}{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Descriptive Power \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }} & \textit{Term \hspace{1cm} Spread (TS)} & \vspace{-0.3cm} \begin{equation}\label{eq:spread} TS(c, o) = \sum_{F_o^i \in F_o} j, \mbox{ where } j = \left\{ \begin{array}{l l} 1 & \quad \mbox{if $c \in F_o^i$}\\ 0 & \quad \mbox{otherwise}\\ \end{array} \right. \end{equation} \\ \cline{2-3} & \textit{Term \hspace{1cm} Frequency (TF)} & \vspace{-0.2cm} \begin{equation}\label{eq:tf} TF(c, o) = \sum_{F_o^i \in F_o} tf(c, F^i_o), \end{equation} \noindent where $tf(c, F^i_o)$ is the number of occurrences of $c$ in textual feature $F^i_o$ of object $o$. \\ \cline{2-3} & \textit{Weighted Term \hspace{1cm} Spread (wTS)} & Let the {\it Feature Instance Spread} of a feature $F_o^i$ associated with object $o$, $FIS(F_o^i)$, be the average $TS$ over all terms in $F_o^i$. We define the \textit{Average Feature Spread} $AFS(F^i)$ as the average $FIS(F_o^i)$ over all instances of $F^i$ associated with objects in the training set $\mathcal{D}$. The $wTS$ is defined as: \vspace{-0.3cm} \begin{equation}\label{eq:sfs} wTS(c, o) = \sum_{F_o^i \in F_o} j, \mbox{ where } j = \left\{ \begin{array}{l l} AFS(F^i) & \quad \mbox{if $c \in F_o^i$}\\ 0 & \quad \mbox{otherwise}\\ \end{array} \right. \end{equation} \\ \cline{2-3} & \textit{Weighted Term \hspace{1cm} Frequency (wTF)} & \vspace{-0.3cm} \begin{equation}\label{eq:tfxfs} wTF(c, o) = \sum_{F_o^i \in F_o} tf(c, F^i_o) \times AFS(F^i) \end{equation} \\ \hline %\end{tabular} %\end{table} %\begin{table} %\scriptsize %\vspace{-0.5cm} %\caption{Metrics used to estimate the relevance of a candidate tag $c$ as a recommendation for an object $o$ %\cite{belem_sigir2011,lipczak11} %[Belém et al. 2011; Lipczak and Milios 2011] (cont.)} %\label{tab:attrib2} %\centering %\begin{tabular}{|c|p{1.6cm}|p{12.5cm}|} %\hline % & \textbf{Name} & \textbf{Equation/Description} \\ %\hline %\hline \multirow{2}{*}{\rotatebox{90}{ \ \ Discriminative Power \ \ }} & \textit{Inverse Feature \hspace{1cm} Frequency (IFF)} & \vspace{-0.3cm} \begin{equation} \label{eq:ifa} IFF(c) = \mathit{log} \frac{|\mathcal{D}| + 1}{f^{tag}_{c} + 1}, \end{equation} \noindent where $f^{tag}_{c}$ is the number of objects % (objects for object-centered recommendation, or pairs object-user for personalized recommendation) in the training set $\mathcal{D}$ that contain $c$ {\it associated as a tag}. \\ \cline{2-3} & \textit{Stability (Stab)} & \vspace{-0.3cm} \begin{equation} \label{eq:stab} Stab(c, k_s) = \frac{k_s}{k_s + | k_s - \mathit{log}(f^{tag}_{c}) |}, \end{equation} \noindent where the tuning parameter $k_s$ represents the ``ideal frequency'' of a term in the data collection. \\ \hline \hline \multirow{4}{*}{\rotatebox{90}{ \ \ \ \ Term Predictability \ \ \ \ }} & \textit{Entropy ($H^{tags}$)} & \vspace{-0.3cm} \begin{equation} \label{eq:h} H^{tags}(c) = - \sum_{(c \rightarrow i) \in \mathcal{R}} \theta(c \rightarrow i) \mathit{ log } \theta(c \rightarrow i) \end{equation} \\ \cline{2-3} & \textit{Predictability \hspace{1cm} (Pred)} & \vspace{-0.3cm} \begin{equation} \label{eq:pred} Pred(c) = \frac{f^{tag, F}_{c}}{f^{F}_{c}}, \end{equation} \noindent where $f^{tag, F}_{c}$ is the number of objects in the training set $\mathcal{D}$ in which $c$ appears both as a tag and as a term in {\it any} other textual feature, and $f^{F}_{c}$ is the number of objects in which $c$ is a term associated with any of its textual features (except tags). \\ \hline \end{tabular} \end{table}