## 2 Materials and Methods
### Feature Extraction
#### 2.3.2 The Features of EMD-Derived IMFs  
The features extracted from the EMD-derived IMFs were as follows: the area of analytical signal representation (\(ASR_{area}\)), the second-order difference plot area (\(SODP_{area}\)), and the central tendency measure of the second-order difference plot (\(SODP_{CTM}\)). Furthermore, the features extracted after power spectral density estimation of the IMFs were: peak amplitude (\(PSD_{pkamp}\)), band power (\(PSD_{ppow}\)), and mean frequency (\(PSD_{mfreq}\)). The specifics of how these features were obtained are explained below. * The area of analytical signal representation (\(ASR_{area}\))  
The IMF analytic signal s(n) can be plotted as the imaginary part, Im{s(n)}, against the real part, R{s(n)}. This IFM analytical signal representation (ASR) plot shows a circular pattern with a unique center of rotation [59]. Thus, a feature of the area can be estimated. One of the standard methods to summarize graph information is the central tendency measure (CTM). The CTM is computed by selecting a circular region of radius \(r\) around the origin, counting the number of points within the radius, and dividing by the total number of points N, as in [60]. This procedure is demonstrated in Equations (5) and (6):  
\[\text{CTM}=\frac{\sum_{n=1}^{N}\text{D}(n)}{N} \tag{5}\]  
\[\text{D}(n)=1\text{ if }(\left[\text{R}\{\text{s(n)}\}\right]^{2}+\left[\text{ Im}\{\text{s(n)}\}\right]^{2})^{0.5}<\text{r} \tag{6}\]  
The radius of the plot is computed using the CTM. In this work, a radius of 95% CTM was chosen to calculate the ASR area. Finally, the \(ASR_{area}\) is computed as in Equation (7). \[ASR_{area}=\pi\pi^{2} \tag{7}\]
* The second-order difference plots (\(SODP_{area}\) and \(SODP_{CTM}\))  
The second-order difference plots (SODPs) are centered around the origin representing the variability rate; they help model biological systems, such as heart rate variations, to characterize the degree of theoretical chaos [60]. The SODPs extract the rate of the data variability (e.g., they assess the variability present in the IMFs of the RR interval signals) and provide a graphical representation of successive differences in the same series when plotted against each other [29,43]. The SODP graph of a given signal x(n) can be obtained by plotting X(n) versus Y(n), as defined in Equations (8) and (9), respectively:  
\[\text{X}(n)=\text{IMF}_{\text{i}}(n+1)-\text{IMF}_{\text{i}}(n) \tag{8}\]  
\[\text{Y}(n)=\text{IMF}_{\text{i}}(n+2)-\text{IMF}_{\text{i}}(n+1) \tag{9}\]  
Once the SODP is obtained, we used the 95% confidence ellipse area of the SODP graph to obtain the \(SODP_{area}\). The procedure to calculate the ellipse area is given as the following process [26,61]. Compute \(S_{x}\), \(S_{y}\), and \(S_{xy}\), according to Equations (10)-(12):  
\[S_{x}=\sqrt{\frac{1}{N}}\ \sum_{n=0}^{N-1}\text{X}(n)^{2} \tag{10}\]  
\[S_{y}=\sqrt{\frac{1}{N}}\ \sum_{n=0}^{N-1}\text{Y}(n)^{2} \tag{11}\]  
\[S_{xy}=\frac{1}{N}\sum_{n=0}^{N}\text{X}(n)\ \text{Y}(n) \tag{12}\]Compute D, a, and b parameters according to Equations (13)-(15):  
\[\text{D}=\sqrt{\left(S_{x}^{2}+S_{y}^{2}\right)-4\left(S_{x}^{2}S_{y}^{2}-S_{xy} ^{2}\right)} \tag{13}\]  
\[\text{a}=1.7321\sqrt{S_{x}^{2}+S_{y}^{2}+D)} \tag{14}\]  
\[\text{D}(n)=1\text{ if }(\left[\text{R}\{\text{s(n)}\}\right]^{2}+\left[\text{ Im}\{\text{s(n)}\}\right]^{2})^{0.5}<\text{r} \tag{6}\]0  
From parameters a and b, the \(SODP_{area}\) is calculated as in Equation (16):  
\[\text{D}(n)=1\text{ if }(\left[\text{R}\{\text{s(n)}\}\right]^{2}+\left[\text{ Im}\{\text{s(n)}\}\right]^{2})^{0.5}<\text{r} \tag{6}\]1  
In addition, the CTM method was applied to the SODP graphs. The CTM was calculated for each IMF's fixed circular region around the SODP origin point. Therefore, a low-variable plot will have points clustered around the origin. Additionally, the diseased state of the heart exhibits a greater degree of chaos [60]. The \(SODP_{CTM}\) is defined as the ratio between the number of points within the fixed radius and the total number of points.