# Stochastic Modeling and Optimal Time-Frequency Estimation of Task-Related HRV
## 3 Results and Discussion
### Comparison between Time-Frequency Estimates  
The MSE optimal eigenvectors and eigenvalues, corresponding to multitapers and weights of the MSE optimal LSCP\({}_{c}\) spectral estimator, are shown in Figure 4. These will in general differ from the MSE optimal eigenvectors and eigenvalues based on the individually estimated parameters, used for computing  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{LSCP\({}_{i}\)} & \multicolumn{2}{c}{LSCP\({}_{c}\)} \\ \cline{2-6}
**Parameter** & **Median** & **Mean** & **sd** & \(c_{v}\) & \\ \hline \(m\) & 0.0003 & 0.0003 & 0.0001 & 0.3139 & 0.0003 \\ \(d\) & \(-396.36\) & \(-379.16\) & 68.71 & \(-0.1812\) & \(-402.62\) \\ \(a\) & 0.0111 & 0.0155 & 0.0162 & 1.0423 & 0.0126 \\ \(b\) & 0.0052 & 0.0146 & 0.0462 & 3.1709 & 0.0043 \\ \(c\) & 4.2023 & 3.9032 & 1.8951 & 0.4855 & 5.6208 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Median, mean, standard deviation and coefficient of variation \(c_{v}\) of the estimated LSCP\({}_{i}\) parameters, and the estimated LSCP\({}_{c}\) parameters for the 92 TPs.  
the MSE optimal LSCP\({}_{i}\) spectral estimator, an example of which can be seen in Figure 5. In particular, the number of eigenvalues significantly different from zero varies depending on the parameters.  
In Figure 6, we present examples of TF estimates for three TPs, obtained with the spectrogram with 64 samples Hanning window (16 s) S64, the spectrogram with 256 samples Hanning window (64 s) S256, the Wigner-Ville spectrum WVS, the MSE optimal LSCP\({}_{c}\) spectrum, and the MSE optimal LSCP\({}_{i}\) spectrum. The number of tapers \(K\) used for computing the LSCP\({}_{i}\) spectrum is dependent on the individual parameters. In the examples shown in Figure 6, \(K\) is equal to 10, 19, 3 for Ex. 1, Ex. 2, and Ex. 3 respectively.  
Figure 4: MSE optimal (**a**) eigenvalues and (**b**) eigenvectors, with \(K=4\), corresponding to weights and multitapers of the MSE optimal LSCP\({}_{c}\) spectral estimator.  
The longer window spectrogram S256 offers a lower temporal resolution, compared to the shorter window spectrogram S64, while the opposite holds for the frequency resolution. Additionally to the resolution trade-off, the spectrogram suffers from a large variance, which is a significant problem when the estimate is based on a single realization of the underlying process, see the panels in the first two rows of Figure 6. The typical cross-terms of the WVS, large oscillating terms located in between the actual signal components, are visible in the panels in the middle row of Figure 6. By construction, the MSE optimal spectral estimators minimize bias and variance of the spectral estimates for LSCPs. LSCP\({}_{i}\) is preferred for capturing features that vary between the TPs, see panels in the last row of Figure 6. On the other hand, the LSCP\({}_{c}\) parameters are more robust than the individually estimated ones. This must be taken into account when the spectral estimates are used for extracting further features, e.g., frequency bands content from the time-frequency marginals or other spectral measures. It is worth noticing that for non-stationary signals of this kind, spectral measures such as the power in the low- and high- frequency bands and their ratio have limited interest, since the spectral content of the chirp signal is found above 0.12 Hz, which is above the usual low-frequency band.  
Figure 5: An example of (**a**) eigenvalues and (**b**) eigenvectors, \(K=7\), corresponding to weights and multitapers of the MSE optimal LSCP\({}_{i}\) spectral estimator, based on individual parameter estimates.