## 2 Fundamental Concepts of TDA  
This section outlines a simplified description of the mathematics behind Homology and Persistent Homology (PH). In TDA, cloud data are frequently seen as a simplicial complex, which is a set of points, lines, segments, triangles, and its n-dimensional counterparts. This allows one to use the methods from simplicial homology to quantify the shape of the data in terms of connections [28] and enables us to make a topological feature extraction. The process of topological feature extraction using PH can be summarized in the following steps:  
* Data Point Cloud \(\chi\in\mathbb{R}^{n}\) is employed as an input. * For each data point (or vertex) \(v_{i}\in\chi\), make B(\(v_{i}\)) a ball of radius \(\epsilon\) centered at each \(v_{i}\), where \(\epsilon\in\mathbb{R}^{+}\). * Raise the value of \(\epsilon\). * A simplicial complex is built for each \(\epsilon\) using Vietoris Rips and filtration. * Measure PH and take note of its appearance and disappearance. * Plot the \((\epsilon_{birth},\epsilon_{death})\) appearance and disappearance coordinates for each PH on an extended real plane \(\mathbb{R}^{2}\cup\{\pm\infty\}\). The Persistence Diagram comes as an output. * Lastly, the topological features are extracted. In terms of mathematics, the input to a PH are the Point Cloud data. In the case of ECG, the input data are the time series. [PERSON]'s Delay Embedding Theorem can be used in the conversion of time series data to point cloud data without losing topological properties. The approach consists of transforming a time series \(x_{t}\), where \(t\in\{1,2,\ldots,T\}\), into its phase space representation. A point cloud or a set of points is obtained according to the following equation where \(i=(1,2,\ldots,T+n\tau)\) and \(\tau\) is a delay parameter and \(n\) specifies the dimension of the point cloud [29]:  
\[v_{i}=x_{i},x_{i+\tau},\ldots,x_{i+n\tau} \tag{1}\]  
Simplicial complexes are essential in the extraction of topological features from point cloud data. A single data point may define 0-Simplex. A line between two points denotes 1-Simplex. A triangle is a 2-Simplex. Tetrahedra represent 3-simplices (see Figure 1). Finally, a combination of simplices gives way to a Simplicial Complex called Vietoris Rips Complex [30; 31; 32]. A simplicial complex can be taken from a dataset using the \(Viterois-Rips\) construction. Being \(X=(x_{1},\ldots,x_{n})\) a point cloud in an euclidean space \(\mathbb{R}_{n}\), for each distance \(\epsilon>0\), represented by \(VR(X;\epsilon)\), there is a simplicial complex with vertex set in X where \(x_{0},x_{1},\ldots,x_{k}\) spreads a \(k-simplex\) if the reciprocal distance between any pair of its varieties is smaller than \(\epsilon\), where \(d(x,x)\leq\epsilon\), for all \(0\leq i\), \(j\leq k\). When building a Simplicial Complex with Point Cloud data, it is needed to follow a set of rules. Firstly, a circle should be drawn with radius \(\epsilon\) for each point in a point cloud. Then, when two circles intersect with each other and the radius is increased, a line is drawn to link the two points, which can be seen in Figure 2.  
As \(\epsilon\) gets longer, the Vietoris-Rips complex of a Point Cloud does, too. This is a filtration of simplicial complexes, i.e., a nested sequence of simplicial complexes, where  
Figure 1: Gradual construction of various simplices (0-Simplex, 1-Simplex, 2-Simplex, \(\ldots\)) eventually gives way to a Simplicial Complex. Figure 2: Diagram illustrating circular intersections and linked point clouds required to build a Simplicial Complex. \(VR(X;\epsilon)\), \(\epsilon\geq 0\) satisfying \(VR(X;\epsilon_{1})\subseteq VR(X;\epsilon_{2})\) if \(\epsilon_{1}\leq\epsilon_{2}\). To represent the distance between them, balls are drawn around each point. If two balls with radius \(\epsilon\) intersect with each other, the two points are at a distance at most \(2\epsilon\). The Persistence Diagram representation (PDR) is a standard way to represent PH [33; 34].