## 4 Self-Tuning Exploratory Data Analytics
### Joint-Approach
#### 4.1.1 Latent Semantic Analysis  
To make the cluster analysis problem more effectively tractable, ESCAPE includes a natural language process named LSA (latent semantic analysis) [33]. LSA allows a reduction in the dimensionality of the document-term matrix \(X\), which captures the latent semantic structure. Choosing the right dimensionality reduction, while avoiding the loss of significant information, is an open research issue and a very complex task. If there are not enough dimensions after the LSA process, the data representation will be poor, while if there are too many dimensions, it will lead to more noisy data. LSA maps both words and documents in a concept-space, wherein it is able to find the relationships between them. To find the hidden concepts, LSA applies the singular value decomposition (SVD). SVD is a matrix factorization method that decomposes the original matrix (document-term matrix) \(X\) into three matrices (\(U\); \(S\); \(V^{T}\)). To find the principal dimensions (\(K_{LSA}\)) in \(X\), ESCAPE includes an innovative algorithm named ST-DaRe. Given \(K_{LSA}\), ESCAPE uses only the highest singular \(K_{LSA}\) values in \(S\), setting the others to zero. The approximated matrix of \(X\), denoted as \(X_{K_{LSA}}=U_{K_{LSA}}S_{K_{LSA}}V^{T}_{K_{LSA}}\), is obtained through the reduction of all three decomposed matrices (\(U,S,V^{T}\)) to rank \(K_{LSA}\). In general, the low-rank approximation of X by \(X_{K_{LSA}}\) can be viewed as a constrained optimization problem, with respect to the constraint that \(X_{K_{LSA}}\) has a rank at most \(K_{LSA}\). When the document-term matrix is tightened down to a k-dimensional space, terms with alike co-occurences should be brought together by the SVD. This insight indicates that the dimensionality reduction could improve the results. **Self-Tuning Data-Reduction Algorithm. ** \begin{table}
\begin{tabular}{c c c} \hline \hline
**Weight** & **WId** & **Definition** \\ \hline \multirow{4}{*}{Local} & L1 & TF = tf\({}_{ij}\) \\ \cline{2-3}  & L2 & LogTF = log\({}_{2}\)(tf\({}_{ij}+1\)) \\ \cline{2-3}  & L3 & Boolean = \(\left\{\begin{array}{ll}0\text{ if }\text{tf}_{ij}=0\\ 1\text{ otherwise}\end{array}\right.\) \\ \hline \multirow{4}{*}{Global} & G1 & IDF = log\(\frac{|D|}{\text{df}_{i}}\) \\ \cline{2-3}  & G2 & Entropy = \(1+\sum_{i}\frac{P_{i}\log P_{i}}{\log|D|}\) \\ \cline{2-3}  & G3 & TF\({}_{glob}\) = g\({}_{j}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Local and global weight functions exploited in ESCAPE. The goal of the ST-DaRe (self-tuning data-reduction) algorithm in ESCAPE is to pick out a proper number of dimensions to take into account in the successive analytics steps, while avoiding the loss of relevant information, by identifying three reasonable values for the LSA parameter. The correct choice of the number of dimensions to be considered is an open research issue [41]. Selecting the maximum decrease point inside the singular-value curve is an easy approach, but if a local minimum is hit, the resulting choice would be inaccurate. The original ST-DaRe algorithm [1] needs three parameters that have been experimentally set. These parameters are the singular-value step and two thresholds. In this case, the singular values are plotted in descending order and, from the obtained curve, the singular values are analyzed in pairs, using the singular-value step set as a parameter. For each pair, the marginal decrease of the curve is calculated. If this decrease is comparable to one of the two parameters chosen as thresholds, or to their average, then the smallest singular value of the analyzed pair is chosen as one of three values. Different from this original approach, in ESCAPE, we propose a new strategy based on a single parameter \(T\), indicating the number of singular values to consider. In particular, after having ordered the singular values in descending order, for our analysis, we consider only the first \(T\) of them.