## 3 Results and Discussion
### The Comparison between the Matrix Norm and the Matrix Discriminant  
The three cardiac parameters (the \(JT\) wave, the duration of \(QRS\) complex and \(RR\) interval) represented by time series \((x_{n}:n=0,1,2,3\ldots)\), \((y_{n}:n=0,1,2,3\ldots)\), and \((z_{n}:n=0,1,2,3\ldots)\), are preprocessed. A threshold range is applied to ensure that each input value remains within physiologically plausible limits. For instance, the minimum and maximum limits are chosen for each of the time series: \(x_{(\min)}=100\) ms; \(x_{(\max)}=400\) ms; \(y_{(\min)}=800\) ms; \(y_{(\max)}=110\) ms; \(z_{(\min)}=600\) ms; and \(z_{(\max)}=1200\) ms, respectively. Any values falling below the lower bounds or exceeding the upper bounds are adjusted to the respective minimum or maximum values. The adjusted values of the time series \(x_{n},y_{n}\) and \(z_{n}\) are then normalized to a range between 0 and 1. For instance, the time series \(x_{n}\) is normalized as \(x_{normalized}=(x-100)/(400-100)\). Similarly, the time series \(y_{n}\) and \(z_{n}\) are processed through the same normalization technique, respectively. The same normalization techniques are also used in [32]. Once the input parameters are preprocessed and normalized, then, without losing the generality, the preprocessed and normalized time series \(x_{n},y_{n}\) and \(z_{n}\) are fitted to the three matrix architectures as represented by Equation (2), Equation (4) and Equation (5). Moreover, it is crucial to acknowledge that the degree of freedom allows for the choice of any variant specified in Equations (6)-(9). Thus, for this research, the variant mentioned in Equation (7) is chosen due to the reasons explained in [20]. To transform the third-order matrix into a scalar series via the mapping function \(\mathcal{F}:\mathbb{R}^{3\times 3}\longrightarrow\mathbb{R}^{1\times 3}\), twofold mapping techniques are employed. The large discriminant of the matrix is used as the mapping function for MA1 and MA2 architectures. The norm is used as the mappingfunction for the PMLD architecture following the criteria described in [20]. The internal and external smoothing techniques are applied with the smoothing radius adjusted to the values optimized in [18]. Finally, a statistical metric, such as the variance of the norm or the variance of the large discriminant, is calculated for each produced time sequence from the three original time series \(x_{n},y_{n}\) and \(z_{n}\). The resulting values are depicted in Tables 1 and 2. In Tables 1 and 2, the variance values for both the large discriminant and the norm of the matrix are presented for the healthy and unhealthy cohort, respectively. It is pertinent to note that while the variance of the norm provides meaningful analytical insights into the matrix architectures, the variance of the large discriminant appears to yield misleading information without a clear analytical significance appliable for classification purposes. It can be observed that the variance of the large discriminant exhibited a wide range of values, with minimum and maximum ranges varying significantly across both healthy and unhealthy cohorts. For instance, in the healthy cohort, the variance of the large discriminant ranges from 0.042 to 1618.7723, whereas in the unhealthy cohort this variance ranged from 0.0006 to 1358.7248. In contrast, the variance of the norm demonstrated more consistency and interpretable trends, with narrower minimum and maximum ranges. For example, the variance of the norm ranges from 0.0010 to 0.0216 in the healthy cohort. The variance of the norm ranges from 0.0004 to 0.0177 in the unhealthy cohort. This observation underscores the importance of employing the norm of the matrices as a useful and relevant statistical metric for further computations for the distribution and classification (seen in Figure 2).