# Online Algorithm for Deriving Heart Rate Variability Components and Their Time-Frequency Analysis
## 3 Results  
### Validation of Methods  
The efficiency of the newly proposed online algorithm VMDon and modified AVMD was assessed based on the known properties of the synthetic signal and additionally compared to those obtained with the previously used offline procedures: standard VMD, CWT, and WPD (Figures 3-5 and Table 2).  
Figure 5: Results of frequency modulation (FM) of HRV components extracted using VMDon (blue), AVMD (orange), classic VMD (purple), CWT (green), and WPD (red) algorithms compared to the synthetic signal (ground truth) components (black).  
Figures 3-5 show that the results of decomposition by classic VMD and WPD are clearly inferior to the other algorithms. VMDon and CWT perform best with lower-frequency components, where the extracted signals almost match the true values, although  
Figure 4: Results of amplitude modulation (AM) of HRV components extracted using VMDon (blue), AVMD (orange), classic VMD (purple), CWT (green), and WPD (red) algorithms compared to the synthetic signal (ground truth) components (black).  
there are temporary deviations visible in ULF AM. At higher frequencies, more noise and fluctuations are observed, but the overall reproduction of the signals is proper.  
All the numerical results of the validation of the proposed methods against those previously used in the literature are summarized in Table 2. They confirm the inferiority of the standard VMD method in comparison to its online and adaptive versions. Similarly, WPD also turns out to be worse than them. Nevertheless, the CWT proves to be a method giving similar results or even slightly better in terms of the VLF component. AVMD is superior in the derivation of HF and LF components, but VMDon yields slightly better results in case of VLF and ULF. In general, these three algorithms (VMDon, AVMD, and CWT) present good alignment to the synthetic signal, with the minor exception of HF FM and LF FM in the case of VMDon. The result of the Wilcoxon test for ULF AM is not obvious, as it indicates a significantly smaller median for VMDon where \(|\widetilde{\epsilon_{c}}|\) is larger than for AVMD. It is also worth noting that VMDon, AVMD, CWT, and WPD always have lower \(|\widetilde{\epsilon_{c}}|\) values than VMD, also according to the Wilcoxon test.  
In the next step, the best performing three algorithms, VMDon, AVMD and CWT, were tested on real overnight HRVs from the two datasets. The correlation coefficients (see Supplementary Material for details) show overall consistency between the methods, with most results having \(r_{s}\) greater than 0.8. Nevertheless, there is noticeable discrepancy in the frequency modulation of HF, LF, and VLF, where \(r_{s}\) is mostly below 0.5, and the values are more diverse among subjects.  
Results of VMDon, AVMD, and CWT were also compared by visualizing the extracted components, as well as their AM and FM (representative results for Subject #2 are shown in Figure 6). The results correspond well, although VMDon tends to slightly underestimate the magnitude range of FM. It is also visible that, while most of the signals propagate in some range of magnitude, there are well-rapid spikes out of the usual boundary, especially in case of HF, and these occurrences are detected regardless of the derivation method.