## 3 Materials and Methods
### Feature Extraction and Reduction
\begin{table}
\begin{tabular}{l l} \hline \hline
**Representation or Domain** & **Parameter** \\ \hline Time Domain & Std (Standard deviation along the specified axis) [48] \\  & Permutation entropy [49] \\  & Spectral entropy [50] \\  & Singular Value decomposition entropy [51] \\  & Aproximate entropy [52] \\  & Sample entropy [53] \\  & Lempel-Ziv complexity [54] \\  & Shannon entropy [55] \\  & Petrosian fractal dimension [56] \\  & Katz fractal dimension [57] \\  & Higuchi fractal dimension [58] \\  & Detrended fluctuation analysis [59] \\ \hline Representation of Point Cloud (RPC) & Std (Standard deviation along the specified axis) [53] \\  & Tsem (trimmed standar error of the mean) [60] \\  & Nanmean [61] \\  & Tvar (Tail value at risk) [62] \\ \hline Persistence Diagram Representation (PDR) & Persistence Weighted Gaussian Kernel [63] \\  & Approximate PWG kernel [63] \\  & Persistence Scale Spacekerne [64] \\  & Approximate PSS kernel [64] \\  & Sliced Wasserstein Distance [27] \\  & Sliced Wasserstein Kernel [27] \\ \hline Landscape Representation (LR) & Tsem (Trimmed standard error of the mean) [60] \\  & Tstd (Trimmed sample standard deviation) [53] \\  & Wasserstein distance [65] \\  & Heat kernel distance bottleneck [53] \\ \hline Silhouette Representation (SR) & RMS (Root Mean Square) [66] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: List of extracted features using TDA and other time-domain parameters used as an input vector to the classifier. time windows. The only classifier used in this work was the k-Nearest Neighbor (kNN) algorithm, which is commonly used in the bibliography [12; 67]. And the used distance is the Euclidean distance. In addition, two different input sets are evaluated. The first input set corresponds to the features described in Table 3, named the TDA method. The second input set is based on the Gaussian KDE bidimensional representation, named Persistence Diagram Image - PDI. It has been used as a repeated random sub-sampling validation technique. Thus, for each class, 67% of data was randomly selected for training and the remaining 33% for testing. The kNN training process was done, and then the testing dataset was used to evaluate the classification performance by measuring the Sensibility (Sen), Specificity (Spe), and Accuracy (Acc). This cross-validation approach was repeated five times at random, and the performance of the classifiers was evaluated overall by taking this five iterations average. This number of iterations was chosen after some trials as it showed the lowest generalization error. Standard statistical parameters were followed to assess the performance in accurate classification of the ECG signal into the _VF_, _VT_, _Normal_, or _Others_ classes. These include the Sensitivity (Sen), Specificity (Spe), and Accuracy (Acc), which are calculated using the following equations where _TP_, _TN_, _FP_, and _FN_ represent the number of true positives, true negatives, false positives, and false negatives, respectively:  
\[Sensitivity=\frac{TP}{TP+FN}\times 100\% \tag{6}\]  
\[Specificity=\frac{TN}{TN+FP}\times 100\% \tag{7}\]  
\[Accuracy=\frac{TP+TN}{TP+FP+TN+FN}\times 100\% \tag{8}\]