# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
## 3 Models and Problem Statement
### Tri-State
\begin{cases}p_{i}^{\Gamma}=\arg_{\gamma\in\Gamma}\max\text{ RFS-Count}(\gamma),&\text{if }p_{i}^{\Gamma}=\phi;\\ p_{i}^{\Lambda}=\arg_{\gamma\in\Lambda}\max\text{ RFS-Count}(\gamma),&\text{if }p_{i}^{\Lambda}=\phi;\\ p_{i}^{\Omega}=\arg_{\gamma\in\Omega}\max\text{ RFS-Count}(\gamma),&\text{if }p_{i}^{\Omega}=\phi,\end{cases}  \tag{22}\]  
where:  
\[\text{RFS-Count}(\gamma)=\frac{\Sigma_{\gamma^{{}^{\prime}}\in\{p_{i}^{ \Gamma},p_{i}^{\Lambda},p_{i}^{\Omega}\}\Sigma_{l=1}^{m}\text{Index}(f_{l,i}= \gamma\text{ and }f_{l,j}=\gamma^{{}^{\prime}}\text{ and }f_{l,j}\
eq\phi)}}{\Sigma_{ \gamma^{{}^{\prime}}\in\{p_{i}^{\Gamma},p_{i}^{\Lambda},p_{i}^{\Omega}\}\text {Index}(\gamma^{{}^{\prime}}\
eq\phi)\times m}}.\]  
**Example 14**. : _Based on Example 12 and Table 3, the Pearson correlations among the three variables are listed as follows. Pearson\((a_{1},a_{2})=0.892\), Pearson\((a_{1},a_{3})=0.919\), and Pearson\((a_{2},a_{3})=0.839\). Hence, for the variable \(a_{1}\), the most related one is \(a_{3}\). Then, when (\(a_{3},g\)) happens, the happening symbols set of \(a_{1}\) is {\(g\)}. When (\(a_{3},f\)) happens, the happening symbols set of \(a_{1}\) is {\(g\)}. When (\(a_{3},e\)) happens, the happening symbols set of \(a_{1}\) is {\(g\)}. When (\(a_{3},e\)) happens, the happening symbols set of \(a_{1}\) is {\(c\)}. No medium symbol for \(p_{1}^{\Lambda}\) by the RFS is available. Therefore, the result is \(b\), which is predicted using the IFS._  
_Then, for the variable \(a_{2}\), the most related one is \(a_{1}\). Then, when (\(a_{1},g\)) happens, the happening symbols' set of \(a_{2}\) is {\(f\), \(f\)}. When (\(a_{1},b\)) happens, the happening symbols set of \(a_{2}\) is {\(a\), \(a\), \(c\)}. When (\(a_{1},c\)) happens, the happening symbols set of \(a_{2}\) is {\(d\), \(c\)}. Therefore, the result of \(p_{i}^{\Gamma}\) is \(a\)._  
_Accordingly, the tri-state filled by the RFS is \(P_{11,*}= \begin{pmatrix}g,&a,&g\\ b,&f,&f\\ c,&d,&c\end{pmatrix} .\) This result of the RFS is consistent with that of the IFS._  
Moreover, the precision for the completed tri-state (IFS- and RFS-ones) is calculated as follows:  
\[\mathcal{P}_{m+1}^{{}^{\prime\prime}}=\frac{\Sigma_{1\leq i\leq n}I(p_{m+1,i}^ {\Gamma}==f_{m+1,i}\text{ or }p_{m+1,i}^{\Lambda}==f_{m+1,i}\text{ or }p_{m+1,i}^{\Omega}==f_{m+1,i})}{3n} \tag{23}\]  
With the above Equations (15), (20) and (23):* \(0\leq\mathcal{P}_{m+1}^{{}^{\prime\prime}}\leq\frac{1}{3}\);
* \(\mathcal{P}_{m+1}^{{}^{\prime}}\leq\mathcal{P}_{m+1}\); and \(\mathcal{P}_{m+1}^{{}^{\prime\prime}}\leq\mathcal{P}_{m+1}\). Finally, with all of the above definitions, we can define the problem of three-way state prediction as follows:  
**Problem 2**. : _Tri-state prediction for MTS._  
_Input:_\(S^{{}^{\prime}}=(T,A,V^{{}^{\prime}},f^{{}^{\prime}})\), \(S=(T,A,V,f)\), \(w\), \(k\), \(\alpha\), and \(\beta\);  
_Output:_\(P_{m+1,\kappa}=(\mathbf{p}_{m+1,n},\mathbf{p}_{m+1,n},\ldots,\mathbf{p}_{m+1,n})= \begin{pmatrix}p_{m+1,1}^{{}^{\prime}},&\ldots,&p_{m+1,n}^{{}^{\prime}}\\ p_{m+1,1}^{{}^{\prime}},&\ldots,&p_{m+1,n}^{{}^{\prime}}\\ p_{m+1,1}^{{}^{\prime}},&\ldots,&p_{m+1,n}^{{}^{\prime}}\end{pmatrix} .\)  
Compared with Problem 1, Problem 2 has two more parameters \(\alpha\) and \(\beta\). The first process that generates \(\Sigma\) is required, but it has a polynomial time complexity \(\Theta(mn)\). The output is a matrix \(P\) with size \(3\times n\). Hence, we can obtain three of the most likely occurring symbols from the strong, medium, and weak regions, respectively. Note that Problem 1 obtains one predicted state at once, while Problem 2 can obtain up to \(3^{n}\) possible states. Excitedly, the time and space complexity of the two problems remain the same.