Fig. 6: Evaluation of the predicted transition rates of time series with a low signal-to-noise ratio.

Two datasets of 2D-histograms were generated with the COCOC topology (Table 1, datasets No. 2,5). The underlying time series had a signal-to-noise ratio of SNR = 5 and SNR = 2, respectively. The regression NNs were trained with the data, and the rates of the models in the test dataset were estimated. The predictions were ranked according to the averaged error scores (RAE) for each model. A–C Excerpts of time series simulated using the ground truth and predictions of the best-predicted model (A), a selected model approximately below the median (B), and the worst-predicted model (C) are shown. They are accompanied by their respective 2D-histograms (2D_GT and 2D_Pr for the ground truth and prediction). The closed dwell-times are represented on the horizontal axis and the open dwell-times on the vertical axis, ranging from 0.01 ms to 100 ms. For computing the 2D-histograms, time series with a length of 10 million samples were used. Furthermore, using time series with a length of 1 million samples, the current distributions of ground truth and prediction are plotted together in the same graph. The red lines indicate the open (O) and closed (C) current amplitudes, spanning 2000 arbitrary units (AU), with SNR = 2. In addition (inset), the segment of the time series between the vertical dashed blue lines is displayed with its corresponding idealization (black on gray). 100 simulations are computed with each predicted Markov model. The time series are idealized, and the 2D-histograms are generated. According to Eqs. 6, 7, the mean volume deviation and mean reference volume are then calculated (\({\bar{V}}_{{{\rm{D}}}}({{\bf{G}}},{{{\bf{H}}}}_{1},\ldots ,{{{\bf{H}}}}_{100})\) and \({\bar{V}}_{{{\rm{R}}}}({{{\bf{H}}}}_{1},\ldots ,{{{\bf{H}}}}_{100})\), respectively). The volume differences are depicted in the 2D_GT and 2D_Pr histograms, respectively. Furthermore, distribution of the transition rates of the 100 re-predictions is displayed in (D–F). The horizontal dashed lines indicate the parameter range on which the NNs were trained. The diamond indicates the median as well as the 75 and 25 percentiles, while the whiskers denote the 10 and 90 percentiles. The orange dots connected with orange lines illustrate the ground truth as indicated in (A–C). G The predictions on all test datasets (10,000 for each SNR) are summarized as cumulative distributions of the error scores (RAE), for each k_ij. Solid lines indicate the results for the data with an SNR = 5 and the dashed lines for an SNR = 2. For comparison, the dashed pink line shows the cumulative distribution of error scores (RAE) from randomly drawn rates within the parameter space of the datasets.