# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
## 3 Models and Problem Statement
### Tri-State  
To enrich the semantics of predictions, we extend each component of \(\mathbf{p}_{m}\) to a column vector with length 3. For each vector, different components have various semantics. This way, the form of prediction is changed from a \(1\times n\) vector into a \(3\times n\) matrix. First, we introduce the definition of the tri-partition alphabet as follows. **Definition 9**. : _Given an SAX-MTS \(S=(T,A,V,f)\), \(\forall a\in A\),_  
\[\Sigma_{a}=(\Gamma_{a},\Lambda_{a},\Omega_{a}) \tag{16}\]  
_is called a tri-partition alphabet of a if_  
* \(\Gamma_{a}\cup\Lambda_{a}\cup\Omega_{a}=\Sigma_{a}=V_{a}\)_; and_
* \(\Gamma_{a}\cap\Lambda_{a}=\Gamma_{a}\cap\Omega_{a}=\Omega_{a}\cap\Lambda_{a}=\emptyset\)_._  
_Additionally, we call \(\Gamma_{a}\), \(\Lambda_{a}\), and \(\Omega_{a}\) the strong, medium, and weak regions of attribute \(a\in A\), respectively._  
**Example 11**. : _With Table 3, the range of values for variable NO\({}_{2}\) (\(a_{2}\)) is {a, b, c, d, e, f, g}. Let \(\Gamma_{a_{2}}=\{a,g\}\), \(\Lambda_{a_{2}}=\{b,f\}\), and \(\Omega_{a_{2}}=\{c,d,e\}\), \(\Sigma_{a_{2}}\) is called a tri-partition alphabet of NO\({}_{2}\)._  
**Definition 10**. : _Given a SAX-MTS \(S=(T,A,V=\cup_{a\in A}V_{a},f)\): \(\Sigma=\cup_{a\in A}\Sigma_{a}\) and \(O_{m}\), a tri-state at time stamp \(t_{m+1}\) is:_  
\[P_{m+1,*}= \begin{pmatrix}P_{m+1,1}^{\Gamma_{a_{1}}}&\ldots&P_{m+1,n}^{\Gamma_ {a_{m}}}\\ P_{m+1,1}^{\Lambda_{a_{1}}}&\ldots&P_{m+1,n}^{\Lambda_{a_{1}}}\\ P_{m+1,1}^{\Lambda_{a_{1}}}&\ldots&P_{m+1,n}^{\Omega_{a_{m}}}\\ \end{pmatrix}  \tag{17}\]  
_where \(\forall i\in[1,n]\), \(p_{m+1,i}^{\Gamma_{a_{i}}}\in\Gamma_{i}\), \(p_{m+1,i}^{\Lambda_{a_{i}}}\in\Lambda_{i}\), and \(p_{m+1,i}^{\Omega_{a_{i}}}\in\Omega_{i}\)._Compared with the state in Definition 4, we replace \(p_{m+1,i}\) with \(\mathbf{p}_{m+1,i}=(p_{m+1,i}^{\Gamma_{a_{i}}}\)\(p_{m+1,i^{\prime}}^{\Lambda_{a_{i}}}\)\(p_{m+1,i^{\prime}}^{\Lambda_{a_{i}}}\)\(p_{m+1,i}^{\Gamma_{a_{i}}}\)\(p_{m+1,i}^{\Gamma_{a_{i}}}\)\(p_{m+1,i}^{\Gamma_{a_{i}}}\)\(p_{m+1,i}^{\Gamma_{a_{i}}}\)\(\forall i\in[1,n]\). Moreover, this predicted vector can be interpreted as the most probable symbol from the strong, medium, and weak regions, respectively. Note that the three-way state is useless for historical data. Therefore, we present the voting strategy for the three-way state prediction as follows. Given \(\forall i\in[1,n]\):  
\[ \begin{cases}p_{m+1,i}^{\Gamma_{a_{i}}}=\arg\max_{v_{a_{i}}\in\Gamma_{a_{i}}} vote(v_{a_{i}});\\ p_{m+1,i}^{\Lambda_{a_{i}}}=\arg\max_{v_{a_{i}}\in\Lambda_{a_{i}}} vote(v_{a_{i}});\\ p_{m+1,i}^{\Omega_{a_{i}}}=\arg\max_{v_{a_{i}}\in\Omega_{a_{i}}} vote(v_{a_{i}}).\end{cases}  \tag{18}\]  
Practically, regions \(\Gamma_{a}\), \(\Lambda_{a}\), and \(\Omega_{a}\) can be obtained using various partition strategies and have meaningful explanations. Here, we partition the range of symbolic values for each attribute using the following strategy. Based on Equations (2) and (3), we can evaluate the level deviating from the mean for each symbol. For each attribute \(a_{i}(i\in[1,n])\), given a set of thresholds pair \(\{(a_{i},\beta_{i})\}\) with cardinality \(n\), where \(a_{i},\beta_{i}\in D=\{\delta_{1},\delta_{2},\ldots,\delta_{g-1}\}\), \(a_{i}\geq\beta_{i}>0\). \(\forall j\in[1,g-1]\), the tri-partition strategy is formally described as  
\[ \begin{cases}\gamma_{i,j}\in\Gamma_{a_{i}},&\text{if }|\delta_{j}|\geq\alpha_{i};\\ \gamma_{i,j}\in\Lambda_{a_{i}},&\text{if }\alpha_{i}>|\delta_{j}|\geq\beta_{i};\\ \gamma_{i,j}\in\Omega_{a_{i}},&\text{if }|\delta_{j}|<\beta_{i}.\end{cases}  \tag{19}\]  
The combination of PAA-MTS \(S^{{}^{\prime}}=(T,A,V^{{}^{\prime}},f^{{}^{\prime}})\) and SAX-MTS \(S=(T,A,V,f)\) is first used here. The breakpoint \(\delta_{g}\) is \(+\infty\), and \(\delta_{g}>\alpha\) always holds. Hence, \(\gamma_{g}\) always belongs to \(\Gamma_{a_{i}}\). However, up to \(2n\) thresholds need to be specified. Therefore, we assume that \(\forall i,j\in[1,n],i\
eq j\), \(a_{i}=\alpha_{j}\), and \(\beta_{i}=\beta_{j}\) for brevity.