# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
## 3 Models and Problem Statement
### State
In other words, the pairs of vectors in Equations (9) and (10) are either PAA-MTS or SAX-MTS. Namely, the case that \(\mathbf{h}_{s,l}^{i}\) is PAA-MTS while \(\mathbf{h}_{s,l}^{j}\) is SAX-MTS is not permitted. Table 4 presents the availability of similarities and distances for Equation (11). Two things need to be further explained. One is the availability of the metrics. Given any two indices \(r\) and \(c\) (\(r,c\in\) IDs), PAA(\(r\)) = True or PAA(\(c\)) = True indicates that the \(r\)-th or the \(c\)-th metric fits the numerical data. Similarly, SAX(\(r\)) = True or SAX(\(c\)) = False means that the \(r\)-th or the \(c\)-th metric fits the symbolic data. For example, PAA(0) = True indicates that the Euclidean distance fits PAA-MTS but not SAX-MTS. The other is the transformation from the distance to similarity. As similarity and distance metrics are simultaneously used here, the distance needs be transformed into the similarity. Therefore, given two vectors \(\mathbf{h}^{i}\) and \(\mathbf{h}^{j}\), the transformation from distance to similarity is: \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**IDs**} & \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Type**} & \multicolumn{2}{c}{**Availability**} \\ \cline{3-5}  & & & **PAA** & **SAX** \\ \hline
0 & Euclidean & distance & True & False \\
1 & Manhattan & distance & True & False \\
2 & LCSubstring [53] & distance & False & True \\
3 & Levenshtein [54] & distance & False & True \\
4 & Cosine & similarity & True & False \\
5 & Pearson & similarity & True & False \\
6 & Tanimoto [55] & similarity & True & False \\
7 & Triangle [56] & similarity & True & False \\
8 & Jaccard & similarity & False & True \\
9 & Jaro & similarity & False & True \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The availability of similarities and distances. \[s(\mathbf{h}^{i},\mathbf{h}^{j})=\frac{1}{d(\mathbf{h}^{i},\mathbf{h}^{j})+1}, \tag{12}\]  
where \(d\) denotes the distance between \(\mathbf{h}^{i}\) and \(\mathbf{h}^{j}\). This way, 100 combinations of distances and similarities exist. Their performances are discussed in Section 5. **Example 8**. : _With Tables 3 and 4, let \(r=8\) (Jaccard similarity, PAA\((8)=\)True), \(c=1\) (Manhattan distance, SAX\((1)=\)True), and \(w=2\), and the along-across similarity between \(O_{6}\) and \(O_{7}\) (i.e., \(\Delta(O_{6},O_{7})\)) is illustrated as follows. First, the SAX-MTS and PAA-MTS of \(O_{6}\) is:_  
\[O_{6}= \begin{pmatrix}g&f&f\\ g&f&g\end{pmatrix} ,and \begin{pmatrix}1.173&1.007&0.922\\ 1.496&0.842&1.490\end{pmatrix} ,respectively.\]  
_Those of \(O_{7}\) is:_  
\[O_{7}= \begin{pmatrix}g&f&g\\ e&f&f\end{pmatrix} ,and \begin{pmatrix}1.496&0.842&1.490\\ 0.272&0.609&0.959\end{pmatrix} ,respectively.\]  
Second, the row vector similarity \(R_{6,7}=\frac{\frac{3}{2}+\frac{1}{2}}{2}=  
Fourth, the results of voting can hence be obtained as \(p_{11,1}=g\), \(p_{11,2}=f\), \(p_{11,3}=f\). Namely, \(\mathbf{p}_{11,*}=(g,f,f)\). More specially, in terms of \(a_{1}\), \(vote(g)=2>vote(c)=1=vote(e)\). Sixth, the prediction performance is better with less difference between the \(\mathbf{p}_{m+1,*}\) and \(\mathbf{f}_{m+1,*}\) in general. The measures of prediction performance such as the precision and recall are introduced here. \(\forall i\in[1,n]\), and the precision and recall of the state \(\mathbf{f}_{m+1,*}\) have the same form, namely:  
\[\mathcal{P}_{m+1}=\frac{\Sigma_{1\leq i\leq n}I(p_{m+1,i}==f_{m+1,i})}{n} \tag{15}\]  
Finally, with the above definitions, the problem of state prediction is proposed as follows. **Problem 1**. : _kNMN-based state prediction for MTS:_  
_Input:_: \(S^{{}^{\prime}}=(T,A,V^{{}^{\prime}},f^{{}^{\prime}})\), \(S=(T,A,V,f)\), \(w\) and \(k\);  
_Output:_: \(\mathbf{p}_{m+1,*}=(p_{m+1,1},p_{m+1,2},\ldots,p_{m+1,n})\). Although two types of datasets, i.e., the PAA- and SAX-MTSs, are both used here, the space complexity remains the same. The time complexity is closely related to the size of the matrix instance and similarity metrics for vectors. **Example 10**.