## 2 Materials and Methods
### Description of Variables and Signal Processing
The corrected (2) is then used as the physical coherence formula:  
\[\text{PhyCoh}=\frac{P_{\text{Peak}}}{P_{\text{Total}}} \tag{2}\]  
In our case, the coherence number was calculated for all three continuous signals, that is, HRV, PPT, and RES, using the same methodology mentioned above. Another variable used, derived from physiological coherence, is the global coherence index (GCI), which is calculated using (5). This equation is composed of three main values: the combined value of the physiological coherences of the three signals, the phase synchronization index (PSI), obtained by weighting the phase values of the signals, and the cross-correlations of the three signals. As shown in (3) and (4), the PSI term \(\varphi_{k}\) represents the phase of the signal \(k\), which is calculated using the Hilbert transform to find the analytic version of the signal and then determine the phase \(\varphi\). On the other hand, for the cross-correlation \(r_{xy}(k)\), the relationship between each pair of the three signals was found, with \(x\) and \(y\) representing the different signals (TTP, RES, or HRV). Finally, the Global Coherence Index (GCI) in (5) represents the relationship among the special summation of these terms [31]. \[\text{PSI}_{k}=\overline{\cos^{2}(\varphi_{k})}+\overline{\sin^{2}(\varphi_{k})} \tag{3}\]  
\[r_{xy}(k)=\frac{1}{N}\sum_{n=1}^{N}y(n)x(n+k) \tag{4}\]  
\[\text{GCI}=\frac{3}{5}\,\overline{\text{PhyCoh}}+\frac{1}{4}\overline{r_{xy} }+\frac{3}{20}\,\overline{\text{PSI}} \tag{5}\]  
Following the other variables found, the averages of each vector of values from the continuous subsignals were calculated. In this way, the general averages of PTT, RES, and HRV were obtained. The standard deviations of PTT and HRV were also calculated. Equation (6) shows how the standard deviation \(\sigma\) was found, where \(\overline{x}\) is the mean value of a certain time series signal, \(x_{i}\) is each value of the signal vector, and \(N\) is the total length of the time series vector:  
\[\sigma=\sqrt{\frac{\sum_{i}(x_{i}-\overline{x})^{2}}{N}} \tag{6}\]  
Another variable calculated was the variation coefficient for both the main PPG signals and the ECG, it was measured as illustrated in (7), where \(\sigma\) is the standard deviation of the signal and \(\overline{x}\) is the mean signal:  
\[\text{CV}=\frac{\sigma}{\overline{x}} \tag{7}\]  
Other types of variables that were calculated from the subsignals included Shannon entropy, frequency centroid, and zero crossings. The zero crossing was determined by programming a threshold detector with conditionals that incremented a counter each time the signal crossed the zero line. For the spectral centroid, as stated in (8), \(f_{i}\) represents a frequency component of the frequency spectrum and \(|X(f_{i})|\) corresponds to the magnitude of the signal at the frequency \(f_{i}\). The variable \(N\) represents the number of frequencies evaluated in the experiment. Finally, the spectral centroid is the average of the weighted frequencies [32]:  
\[C_{i}=\frac{\sum_{i=1}^{N}f_{i}\cdot|X(f_{i})|}{\sum_{i=1}^{N}|X(f_{i})|} \tag{8}\]  
Finally, for the Shannon entropy, as shown in (9), the entropy is calculated using the probability \(p(x_{i})\), which is related to the number of occurrences of each unique value in the signal vector. In this context, the probability is used as follows [30; 33]:  
\[S(x)=\sum_{i=1}^{N}p(x_{i})\log_{2}(p(x_{i})) \tag{9}\]  
In summary, the final variables were all applied to the three subsignals. All of the calculations mentioned above are the features that were extracted for the classification exercise. Table 1 explains each variable and subsignal extracted from the ECG and PPG signals. The preponderance of the selected features is intricately linked to the physiological coherence metrics. The principal signals acquired were used to compute the Global Coherence Index, using PPG and ECG to evaluate coherence among three additional physiological parameters: respiratory signal, pulse transit time, and heart rate variability.