## 4 Results
### Accuracy of the Instantaneous Respiratory Frequency Estimate  
From the HRV TF-images, the instantaneous respiratory frequency (IRF) is estimated as the maximum value at each time instant. Figure 5 illustrates the IF estimates (red solid lines) obtained using the ADTFD, AOKTFD, the S-method and the spectrogram together with the GTRF (blue solid lines). Note that the MBD method is omitted as the appearance of the estimate was similar to the AOKTFD. Differences between methods can be seen, where the ADTFD is more precise in the estimation of the faster variations in the respiratory frequency than other methods, e.g., the AOKTFD, which seems to over-estimate the oscillations, see Figure 5c) between frequencies 0.12-0.14 Hz. The experiment is repeated for all TPs. The grand average MSE is calculated and presented in Table 4 for all methods. The smallest value is given from AOKTFD followed by the S-method, ADTFD and the spectrogram. However, the MSE is too sensitive to large deviations at few time points (outliers). Thus, few outliers can degrade the MSE performance for a method which is otherwise consistent in giving good results. In HRV spectra, a narrow band around the IF is important for estimating RSA related signal energy. Thus, a more reliable method for estimating the IF is one that generates a minimum number of outliers outside this band. In order to test the consistency of selected TEDs, we choose to apply an outlier limit (outlim) that is set to be 0.005 Hz. Figure 4: TF analysis of simulated HR signals based on the Amplitude Modulation-Frequency Modulation (AM-FM) model; (**a**) ADTFD (\(a=3\), \(b=8\)); (**b**) AOKTFD; (**c**) MBD (\(\eta=0.05\), lag window = 64); (**d**) S-method (STFT window length = 91, frequency axis convolution window length = 21); (**e**) Spectrogram (window length = 91). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**TFD** & **ADTFD** & **AOKTFD** & **MBD** & **S-method** & **Spectrogram** \\ \hline EC & \(3.7660\times 10^{3}\) & \(9.2798\times 10^{3}\) & \(1.0854\times 10^{4}\) & \(6.6109\times 10^{3}\) & \(1.0974\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of TEDs for the analysis of simulated HRV signal based on the Amplitude Modulation-Frequency Modulation model. The results of all 35 TPs indicate that the ADTFD is the most reliable method to estimate the IRF. The ADTFD based method was able to accurately estimate all frequency values inside the outlier (between the dashed red lines) in 25 out of total of 35 TPs (71.4%). The results of the other methods are not in this range (see Table 5) and, especially, the spectrogram gives unreliable results, as none of the subjects has all estimates inside the outlier limits. \begin{table}
\begin{tabular}{c c c c c c} \hline
**TFD** & **ADTFD** & **AOKFTD** & **MBD** & **S-method** & **Spectrogram** \\ \hline gr.av.MSE & \(0.0656\times 10^{-3}\) & \(0.0387\times 10^{-3}\) & \(0.111\times 10^{-3}\) & \(0.0633\times 10^{-3}\) & \(0.0668\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Grand average mean square error of all test participants and all instantaneous respiratory frequency estimates. Figure 5: The estimated HRV peak frequency from TP 11 and the corresponding ground truth respiratory frequency (GTRF). The dashed lines represents the outlier limit (outlim = 0.005 Hz); (**a**) ADTFD (\(a=3\), \(b=8\)); (**b**) AOKFTD; (**c**) S-method (STFT window length = 91, frequency axis convolution window length = 21); (**d**) spectrogram (window length = 91). \begin{table}
\begin{tabular}{c c c c c c} \hline
**TFD** & **ADTFD** & **AOKFTD** & **MBD** & **S-method** & **Spectrogram** \\ \hline No. of TPs (perc.) & 25 (71.4\%) & 10 (28.6\%) & 5 (14.3\%) & 15 (42.9\%) & 0 (0\%) \\ \hline \end{tabular}
\end{table} 
Table 5: Number and percentage of the TPs without any outlier IRFs. In Figure 6, histograms of the number of TPs that have a certain percentage of the IRFs outside the outlier limits are presented for all methods.