# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
## 3 Models and Problem Statement
### Tri-State
Consequently, we have \(\Sigma_{a_{i}}=\Sigma_{a_{j}}\), and \(\Sigma=\{\Sigma_{a}=(\Gamma_{a},\Lambda_{a},\Omega_{a})|a\in A\}\) which can be denoted by \(\Sigma=(\Gamma,\Lambda,\Omega)\). More choices are available for \(\alpha\) and \(\beta\) with a greater \(g\). Moreover, if \(g=4\), then \(D=\{-0.67,0,0.67\}\). When the threshold \(\alpha\) is set to \(0.67\), \(\beta\) can be set to \(0.67\) or \(0\). However, the predicted tri-state is incomplete if no strong, medium, or weak symbols are found following the whole matrix neighbors. Namely, what the current method can guarantee is that each variable has at least one predicted symbol. Formally, given a tri-state \(P_{m+1,*}\) at \(t_{m+1}\), \(\forall i\in[1,n]\), we have:  
\[(p_{i}^{\Gamma},p_{i}^{\Lambda},p_{i}^{\Gamma})^{\Gamma}\
eq(\phi,\phi,\phi)^{ \Gamma}.\]  
With Table 3, let \(\alpha=1.07\) and \(\beta=0.57\), \(\Sigma_{a_{1}}=\Sigma_{a_{2}}=\Sigma_{a_{3}}=(\Gamma,\Lambda,\Omega)\), where \(\Gamma=\{a,g\}\), \(\Lambda=\{b,f\}\) and \(\Omega=\{c,d,e\}\). Based on the four labels of Example 9, i.e., \(\mathbf{f_{6,*}}=(g,f,g)\), \(\mathbf{f_{5,*}}=(g,f,f)\), \(\mathbf{f_{6,*}}=(c,d,c)\), and \(\mathbf{f_{7,*}}=(e,f,f)\), the predicted tri-state at \(t_{11}\) is \(P_{11,*}=\left( \begin{array}{ccc}g,&\phi,&g\\ \phi,&f,&f\\ c/e,&d,&c\end{array} \right)\). Note that \(\phi\) means the symbol of the current position is temporally unknown. More specifically, the strong symbol of \(a_{2}\) and the medium symbol of \(a_{1}\) are unknown. Here, \"\(c\) / \(e\)\" indicates that the final predicted symbol was randomly selected from them. For brevity, the symbol \(c\) was selected. Moreover, the precision for the incomplete tri-state is calculated as follows:  
\[\mathcal{P}_{m+1}^{{}^{\prime}}=\frac{\Sigma_{1\leq i\leq n}I(p_{m+1,i}^{ \Gamma}==f_{m+1,i}\text{ or }p_{m+1,i}^{\Lambda}==f_{m+1,i}\text{ or }p_{m+1,i}^{\Omega}==f_{m+1,i})}{\Sigma_{i\in[1,n]} \Sigma_{j\in[1,3]}I(P_{m+1,*}\
eq\phi)} \tag{20}\]  
In order to remedy this defect, i.e., to obtain a completed tri-state, we propose two simplified and effective filling strategies called the individual and related ones, respectively. For each attribute, if one or two symbols are missing, the individual filling strategy (IFS) predicts them with the most frequent ones in its own history data. Then, \(\forall i\in[1,n]\), the IFS can be formally described as follows:  
\[ \begin{cases}p_{i}^{\Gamma}=\arg_{\gamma\in\Gamma}\max\text{IFS-Count}(\gamma),& \text{if }p_{i}^{\Gamma}=\phi;\\ p_{i}^{\Lambda}=\arg_{\gamma\in\Lambda}\max\text{IFS-Count}(\gamma),&\text{if }p_{i}^{\Lambda}=\phi;\\ p_{i}^{\Omega}=\arg_{\gamma\in\Omega}\max\text{IFS-Count}(\gamma),&\text{if }p_{i}^{ \Omega}=\phi,\end{cases}  \tag{21}\]  
where:  
\[\text{IFS-Count}(\gamma)=\frac{\Sigma_{j=1}^{m}\text{Index}(f_{j,i}=\gamma)}{m}.\]  
**Example 13**. : _According to Example 12 and Table 3, for variable \(a_{1}\), \(p_{1}^{\Lambda}=b\). This is because IFS-Count(b) = \(\frac{3}{10}>\) IFS-Count(f) = 0. For variable \(a_{2}\), \(p_{2}^{\Gamma}=a\). This is because IFS-Count(a) = \(\frac{2}{10}\) > IFS-Count(g) = \(\frac{1}{10}\). Hence, the tri-state filled by the IFS is \(P_{11,*}= \begin{pmatrix}g,&a,&g\\ b,&f,&f\\ c,&d,&c\end{pmatrix} .\)_  
The related filling strategy (RFS) predicts the missing symbols by considering the association relationships between any pair of variables. Given two variables \(a_{i}\) and \(a_{j}\) (\(i,j\in[1,n],i\
eq j\)), \(a_{j}\) is the most linear related variable of \(a_{i}\). Namely, \(a_{j}=\arg_{a_{j}\in\Lambda\setminus\{a_{1}\}}\max\text{Pearson}(a_{i},a_{j})\). Hence, their predicted vectors are \((p_{i}^{\Gamma},p_{i}^{\Lambda},p_{i}^{\Omega})^{\Gamma}\) and \((p_{j}^{\Gamma},p_{j}^{\Lambda},p_{i}^{\Omega})^{\Gamma}\). Then, the RFS can be formally described as follows:  
\[