# Tri-Partition Alphabet-Based State Prediction for Multivariate Time-Series
## 3 Models and Problem Statement
### Data Model  
The PAA and SAX versions of MTS are defined on the basis of the original numerical MTS. Definition 1: An original numerical MTS is the quadruple:  
\[S^{\prime\prime}=(T^{\prime\prime},A,V^{\prime\prime}=\cup_{a\in A}V^{\prime \prime}_{a},f^{\prime\prime}), \tag{1}\]  
where \(T^{\prime\prime}=\{t_{1},t_{2},\ldots,t_{\mathrm{M}}\}\) is the finite set of time points, \(A=\{a_{1},a_{2},\ldots,a_{n}\}\) is the finite set of variables, \(V^{\prime\prime}_{a}\subset\{\text{real number}\}\) is the value ranges of variable \(a\), and \(f^{\prime\prime}:T^{\prime\prime}\times A\to V^{\prime\prime}\) is the mapping function. For brevity, \(f^{\prime\prime}(t_{i},a_{j})\) can be denoted by \(f^{\prime\prime}_{i,j}\). We further assume that \(t_{i+1}-t_{i}=t_{i}-t_{i-1}\) (\(2\leq i\leq n-1\)). \begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Descriptions** \\ \hline \(S^{\prime\prime}=(T^{\prime\prime},A,V^{\prime\prime}=\cup_{a\in A}V^{\prime \prime}_{a},f^{\prime\prime})\) & The original numerical MTS. \\ \(S^{\prime}=(T,A,V^{\prime}=\cup_{a\in A}V_{a},f^{\prime})\) & The PAA version of numerical \(S^{\prime\prime}\). \\ \(S=(T,A,V=\cup_{a\in A}V_{a},f)\) & The SAX version of numerical \(S^{\prime\prime}\). \\ \(m\) & The number of all time stamps, \(|T|\). \\ \(n\) & The number of all variables, \(|A|\). \\ \(g\) & The number of partitions; \(\forall a\in A\), \(|V_{a}|=g\). \\ \(D\) & The set of breakpoints for \(S\), \(|D|=g-1\). \\ \(\delta\) & \(\delta\in D\). \\ \(\Gamma\) & The strong region. \\ \(\Lambda\) & The medium region. \\ \(\Omega\) & The weak region. \\ \(\Sigma=(\Gamma,\Lambda,\Omega)\) & Tri-partition alphabet. \\ \(\beta\geq 0\) & The threshold for the weak region. \\ \(\alpha\geq\beta\) & The threshold for the strong region. \\ \(\mathbf{f}_{i,*}\) & A symbolic state occurring at time \(t_{i}\). \\ \(\mathbf{f}^{\prime}_{i,*}\) & A numerical state occurring at time \(t_{i}\). \\ \(\mathbf{p}_{m+1,*}\) & A prediction of state occurring at time \(t_{m+1}\). \\ \(w\) & The length of sliding window. \\ \(O\) & A matrix instance; \(|O|=w\times n\). \\ \(\Delta\) & The similarity of two matrix instances. \\ \(k\) & The number of nearest matrix neighbors. \\ \(\mathbf{N}\) & The set of \(k\)-nearest matrix neighbors. \\ \(P_{m+1,*}\) & The form of the tri-state with area \(3\times n\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. **Definition 2**. : _The PAA-MTS \(S^{{}^{\prime}}=(T,A,V^{{}^{\prime}}=\cup_{a\in A}V_{a}^{{}^{\prime}},f^{{}^{ \prime}})\) has similar forms to Definition 1. However, two differences are present, namely (i) \(T=\{t_{1},t_{2},\ldots,t_{m}\}\), \(m<M\), and (ii) \(\forall i\in[i,m],j\in[i,M]\):_  
\[f_{i,a}^{{}^{\prime}}=\frac{m}{M}\sum_{j=(i-1)\frac{M}{m}+1}^{i\frac{M}{m}}f_{ j,a}^{{}^{\prime}}. \tag{2}\]  
**Example 1**. : _Figure 1 shows an example of the transition of NO\({}_{2}\) from the original numerical MTS (\(S^{{}^{\prime\prime}}\)) to the PAA version of MTS (\(S^{{}^{\prime}}\)). Here, \(m=10\) and \(M=100\). This way, the dimension is reduced from 100 to 10._  
**Definition 3**. : _The SAX-MTS \(S=(T,A,V=\cup_{a\in A}V_{a},f)\) also has a similar form to Definition 2. The only difference is that the numerical value is transformed into a symbolic one. To produce symbols with equiprobability, a set of breakpoints \(D=\{\delta_{1},\delta_{2},\ldots,\delta_{g-1}\}\) dividing the area of under the probability distribution function (PDF) of \(a\in A\) is required. Therefore, let \(V_{a}=\{\gamma_{1},\gamma_{2},\ldots,\gamma_{g}\}\) containing \(g\) symbols; then, we have:_  
\[f_{i,a}=\gamma_{j},\text{ if }f_{i,a}^{{}^{\prime}}\in(\delta_{j-1},\delta_{j}), \tag{3}\]  
_where \(j\in[1,g]\), and \(\delta_{0}\), and \(\delta_{g}\) are defined as \(-\infty\) and \(+\infty\), respectively._  
**Example 2**. : _Table 2 shows a lookup table of breakpoints for the \(\mathcal{N}(0,1)\) distribution. In practice, \(g\) can be set as an integer that is not less than 2.