# Online Algorithm for Deriving Heart Rate Variability Components and Their Time-Frequency Analysis
## 4 Discussion
In the case of VMDon, the efficiency of component extraction is lower at the beginning and end of a finite signal due to a reduced number of averaged samples. On the other hand, during online analysis, older samples already exist and the ECG recording continues. Another attribute of the online algorithm is that it operates with a certain delay--component extraction is delayed by the length of the associated window and the operating rule used. This delay is 6.5, 25, and 303 s for HF, LF and VLF, respectively, and 30 min for ULF. Such a window length for ULF was adopted despite the lack of verified information about its lower frequency limit; however, no difference in results was noticed for trials with windows as long as 90 min, while the 30 min window has shorter delay and much lower computational complexity. In addition, it should be noted that the 30 min delay applies only to ULF, which is inherently slow to change, while the delays of the most used HF or LF components are much shorter, which should enable, for example, the real-time detection of sought-after episodes related to the ANS. The Kolmogorov-Smirnov tests showed that the distributions of values in all analyzed data groups were strongly non-normal, justifying the use of [PERSON]'s correlation coefficient and the Wilcoxon test. In the case of AM of ULF, the indication of the test in Table 2 is inconsistent with the value of \(\mid\tilde{\epsilon}_{c}\mid\), where the median value of the residuals is smaller for AVMD, but the Wilcoxon test shows that VMDon has a statistically lower median in the population. With the Wilcoxon test, this can indeed happen when discrete and extremely skew data are analyzed [53; 54]. This is the case in this particular instance because the Wilcoxon test calculates the differences between the corresponding elements from both samples from AVMD and VMDon, and they are mostly positive for AM ULF, having also a positive median. The different nature of the procedures has resulted in a higher accuracy of AVMD in extracting HF and LF and an advantage of VMDon in extracting VLF and ULF; although for AM and FM of ULF, the results of both are comparable. Additionally, CWT proves to be the best for extracting the VLF component (Table 2). The importance of this observation is highlighted by the results of the real data decomposition analysis, in particular the lower correlation between the FM samples of components extracted for HF, LF, and VLF (see Supplementary Material). This effect is also seen in Figure 6.  
range of their variability (Figure 7). Similar conclusions emerge regarding the ranges of FM variability (Figure 8). This observation is in agreement with the power laws (scale invariance) observed in nature [55]. Figures 7 and 8, as well as Table 3, allow the comparison of results obtained with the same methods from two datasets of different origin, which allows for cross-dataset validation. The obtained outcomes show a high similarity between them and allow one to conclude about the reliability of components extracted with significantly different algorithms from two separate datasets. At the same time, the two-sample Kolmogorov-Smirnov tests performed for the corresponding results from these two datasets rejected the hypothesis that they come from the same distributions. This indicates, however, a statistical difference in the results obtained from HRVs originating from these different databases. Nevertheless, additional checks showed that the main reason for this outcome is the sensitivity of the test itself to small differences in the data; it was enough to remove data of 1 patient from a given set of 18 patients, and the test returned a statistical difference between these two cases. The above results, which validate the HRV component waveforms obtained using the proposed algorithms, are of particular importance for further analyses of the VLF and ULF components, which are currently the least understood.