## 3 Method
### Multivariate VMD for VCG Signal Analysis  
In this study, we have used MVMD to evaluate the modes of VCG beat along each orthogonal lead. The MVMD is the extension of VMD algorithm used to decompose multi-channel signals into modes [26]. The VCG beat is given as \(v_{m}(n)\), with \(n=1,2,...N\). \(N\) is the number of samples in the VCG beat. The parameter \(m\) is denoted as the \(m\)th orthogonal lead of VCG beat. The VCG beat synthesized from its modes is given as follows: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Stage 1: MI detection**} \\ \hline Class & NSR & \multicolumn{2}{c}{MI} \\ \hline VCG beats & \multicolumn{2}{c}{9874} & \multicolumn{2}{c}{13982} \\ \hline \multicolumn{6}{c}{**Stage 2: MI localization**} \\ \hline Class & AMI & ALMI & ASMI & IMI & ILMI & IPLMI \\ \hline VCG beats & 1664 & 1778 & 3057 & 2807 & 3049 & 1981 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of VCG beats used for MI detection and localization. Figure 1: (**a**) Stage1 1: MI detection block using VCG signals. (**b**) Stage 2: MI localization using MI detected multi-scale VCG tensor. \[v_{m}(n)=\sum_{i=1}^{k}u_{m}^{i}(n) \tag{1}\]  
where, \(u_{m}^{i}(n)\) is the \(i\)th mode of \(m\)th lead VCG signal \(u_{m}^{i}(n)=\left[u_{1}^{i}(n),u_{2}^{i}(n),u_{3}^{i}(n)\right]\) is also interpreted as the multivariate modulated oscillations of VCG signal with \(i=1,2,.....k\), and \(k\) is the total number of modes [26]. The vector analytic representation of \(i\)th mode of \(m\)th lead VCG is written as follows [26]:  
\[\hat{u}_{m}^{i}(n)=u_{m}^{i}(n)+j\mathrm{H}\left(u_{m}^{i}(n)\right) \tag{2}\]  
where, \(\mathrm{H}\left(u_{m}^{i}(n)\right)\) is the Hilbert transform of \(i\)th mode of \(m\)th lead VCG signal [26]. In MVMD, the objective is to evaluate the modes of VCG signal based on the criteria as (a) the sum of bandwidth of components or modes of VCG should be minimum and (b) sum of all modes should recover the VCG signal along each lead [26]. The optimization problem of MVMD for the decomposition of VCG signal is formulated as follows [26]:  
\[ \begin{split}\min_{u_{m}^{i}(n),u^{i}}&\quad \left\{\sum_{i=1}^{k}\sum_{m=1}^{M}\left\|\frac{\partial}{\partial n}\left[ \hat{u}_{m}^{i}(n)e^{-jw^{i}n}\right]\right\|_{F}^{2}\right\}\\ \mathrm{s.t.}&\quad\sum_{i=1}^{k}u_{m}^{i}(n)=v_{m} (n),\quad m=1,2,\text{and }3\end{split}  \tag{3}\]  
where, \(\left\|\bullet\right\|_{F}\) is the representation of Frobenious norm [26]. The optimization problem in Equation (3) can be reformulated using augmented Lagrangian and it is given as follows:  
\[L\left\{u_{m}^{i}(n),w^{i},\eta_{m}(n)\right\}=\beta\sum_{i=1}^{k}\sum_{m=1}^{ M}\left\|\frac{\partial}{\partial n}\left[u_{m}^{i}(n)e^{-jw^{i}n}\right] \right\|_{F}^{2}+\sum_{m=1}^{M}\left\|v_{m}(n)-\sum_{i=1}^{k}u_{m}^{i}(n) \right\|_{F}^{2}+\sum_{m=1}^{M}\left\langle\eta_{m}(n),v_{m}(n)-\sum_{i=1}^{ k}u_{m}^{i}(n)\right\rangle \tag{4}\]  
where, \(\eta_{m}(n)\) is the Lagrangian multiplier for \(m\)th lead VCG beat, and \(\beta\) is interpreted as the penalty factor for MVMD. The modes of VCG beat along each lead is iteratively evaluated based on the solution of Equation (4) using alternating direction method of multipliers (ADMM) [26]. The complete algorithm of MVMD for the extraction of modes from the non-stationary signals has been given in [26]. In this study, we have evaluated five modes from the VCG beat along each orthogonal lead. The multi-scale VCG tensor is formulated using the modes of VCG beat and the size of multi-scale VCG tensor is \(651\times 3\times 5\). For NSR class, the V\({}_{\mathrm{x}}\), V\({}_{\mathrm{y}}\), and V\({}_{\mathrm{z}}\) lead VCG beat are shown in Figure 2a,g,m, respectively. The modes of V\({}_{\mathrm{x}}\), V\({}_{\mathrm{y}}\), and V\({}_{\mathrm{z}}\) lead VCG beats evaluated using MVMD are shown Figure 2b-f,h-l,n-r, respectively. Similarly, the V\({}_{\mathrm{x}}\), V\({}_{\mathrm{y}}\), and V\({}_{\mathrm{z}}\) channel VCG beats for IPLMI class are shown in Figure 3a,g,m, respectively.