Fig 1.
Combined trace plots for and
over 5000 iterations.
The red dashed lines indicate the true values of and
.
Table 1.
Posterior estimates for and
compared with the true values used in the simulation.
Fig 2.
Circular histograms of the two von Mises samples.
Sample 1 is centered around (blue), and Sample 2 is centered around
(red). The histograms clearly show the distinct mean directions of the two samples.
Fig 3.
Permutation distribution of Watson’s statistic.
The observed test statistic (dashed line) lies in the extreme tail of the permutation distribution, indicating a significant difference between the two samples.
Fig 4.
Circular Histogram of Simulated Rhythms.
The distribution of phases for each rhythm is shown, highlighting the different concentration parameters.
Fig 5.
Comparison of Estimated and True Mean Directions () for Each Rhythm.
Dashed lines represent the estimated values, and solid lines represent the true values.
Table 2.
Comparison of True and Estimated Parameters for Rhythms.
Fig 6.
Heatmap of Phase Locking Value (PLV) Between Rhythms.
Higher values indicate stronger phase synchronization between the rhythms.
Fig 7.
Heatmap of Mutual Information (MI) Between Rhythms.
Higher values indicate stronger dependencies between rhythms.
Fig 8.
Stochastic Temporal Evolution of Phases on a Circular Axis.
The plot displays the wrapped phase evolution of three rhythms over time, comparing true and estimated parameters shown in the Table 3.
Table 3.
Comparison of True and Estimated Parameters for Stochastic Temporal Evolution (Wrapped Phases).
Table 4.
Mean absolute estimation errors in and
under different noise levels and sample sizes. Results are averaged across all simulated rhythms.
Fig 9.
Boxplot of absolute estimation error in across rhythms.
Estimation improves with increasing N, and errors are more sensitive to noise at lower N.
Fig 10.
Boxplot of absolute estimation error in across rhythms.
Errors remain relatively low and consistent even under high noise.
Fig 11.
Stochastic Temporal Evolution of the ECG Phase.
The simulated evolution accounts for both deterministic drift and random fluctuations over time.
Fig 12.
(Left) Rose Plot of the ECG Phase Data () with the Posterior Distribution of
.
The red arrow represents the circular mean of the posterior for . (Right) Posterior distribution of the concentration parameter
.
Fig 13.
Q-Q Plot of Real ECG Phases vs. Fitted von Mises Distribution.
The red dashed line represents the 45-degree line for reference.
Fig 14.
Polar Histogram of the ECG Phase () and Simulated Rhythm (
).
The ECG phase is shown in blue, and the simulated rhythm is shown in green. Overlapping areas indicate similar phase distributions.
Fig 15.
Stochastic temporal evolution of Rhythm 1 and Rhythm 2 phases.
The phases evolve over time under the influence of deterministic drift and stochastic fluctuations.
Fig 16.
Stochastic Temporal Evolution of the Phases of Rhythm 1 and Rhythm 2 on a Circular Plot.
The radial axis represents time, and the angular axis represents the phase of each rhythm.
Table 5.
Estimated Parameters of the Stochastic Diffusion Process for Rhythm 1 and Rhythm 2.
Fig 17.
Rose Plot of Rhythm 1.
Fig 18.
Rose Plot of Rhythm 2.
Fig 19.
Rose Plot of Phase Difference between Rhythm 1 and Rhythm 2.
Fig 20.
Heatmap representing the joint distribution of the phases of Rhythm 1 and Rhythm 2, used for calculating Mutual Information.
Table 6.
Comparison of Methods (Quality Metrics), for a threshold (3 radians) for the circular SDE model.
Table 7.
Comparison of Methods (Quality Metrics), a threshold (2 radians) for the circular SDE model.
Fig 21.
ECG Segment with Abnormal Beat Detections.
Red circles denote detections by the AR model, blue crosses by the Fourier-based approach, and green triangles by the Bayesian circular SDE model. Note that our circular model is specifically designed to capture subtle phase anomalies.
Fig 22.
Rose Plot of Extracted Phases.
The plot demonstrates that the phase data are intrinsically circular, which underpins the need for circular statistical methods.
Fig 23.
Short-Term Phase Prediction via Bayesian SDE.
The Fig illustrates the forward simulation of the phase over a 0.5-second prediction horizon. This capability is crucial for anticipatory diagnostics in arrhythmia detection.