Fig 1.
Kalman filtering on sample entropy time series of different power noise.
The first row (a-c): the generated waveforms of power noise with different parameters, β = 0 (white noise), 0.5 (pink noise), and 1 (1/f noise). Segments of 2 seconds were shown for clarity. The second row (d-f): theoretical variances of sample entropy of the corresponding waveforms (a-c), which were calculated from Eq 10. The third row (g-i): the original and smoothed sample entropy time series of the corresponding waveforms (a-c) by Kalman filtering.
Fig 2.
Kalman filtering on sample entropy time series of different logistic map signals.
The first row (a-c): the generated waveforms of logistic map with different parameters, r = 3.57, 3.77, 3.9. The second row (d-f): the theoretical variance of sample entropy time series of the corresponding waveforms (a-c) calculated from Eq 10. The third row (g-i): the original and smoothed sample entropy time seires of the corresponding waveforms (a-c) by Kalman filtering.
Fig 3.
Kalman filtering on sample entropy time series of different Rössler system signals.
The first row (a-c): the generated waveforms of logistic map with different parameters, c = 2.5, 4, 5.7. The second row (d-f): the theoretical variance of sample entropy time series of the corresponding waveforms (a-c) calculated from Eq 10. The third row (g-i): the original and smoothed sample entropy time seires of the corresponding waveforms (a-c) by Kalman filtering.
Fig 4.
Kalman filtering on sample entropy time series of sleep signals.
(a-d) demonstrate the smoothing effect on S1, S2, S3 and W stages for subjects SC4001E0, SC4011E0, SC4021E0, and SC4032E0. First column: EEG waveforms; second column: theoretical variance computed by Eq 10; third row: original sample entropy time series and the sample entropy time series after smoothing by Kalman filtering. S1: sleep stage 1; S2: sleep stage 2; S3: sleep stage 3; W: wake stage. From the third column, it is visually clear that Kalman filtering reduces the variance of sample entropy measures; detailed values, see Table 1.
Table 1.
Comparison of VRR by different methods for sleep data.
Fig 5.
Visual comparison of three smoothing methods for sleep data.
(a): subject SC4001E0, (b): subject SC4011E0, (c): subject SC4021E0, (d): subject SC4032E0. VRRs are shown in Table 1.
Fig 6.
Comparison of cumulative computational time for sleep EEG data.
Three smoothing methods for different subjects, as a function of iteration number. (a) subject SC4001E0, (b) subject SC4011E0, (c) subject SC4021E0, (d) subject SC4032E0.
Fig 7.
Kalman filtering on sample entropy time series of epilepsy signals.
First row: waveforms of EEG recordings. Second row: theoretical variance of sample entropy calculated according to Eq 10. Third row: original and Kalman-filtering-smoothed sample entropy time series. (a): channel FP1-F3; (b): channel F3-C3; (c): channel C3-P3; (d): channel P3-O1.
Fig 8.
Comparison of the three smoothing methods for the epilepsy data.
First row: original sample entropy time series and those smoothed by moving average, EWMA and Kalman filtering. Second row: cumulative computational time needed as a function of iteration number for channel FP1-F3 (a,e), channel F3-C3 (b,f), channel C3-P3 (c,g) and channel P3-O1 (d,h).
Table 2.
Comparison of VRR by different methods in for epilepsy data.
Fig 9.
Kalman filtering applied to different entropy measures of sleep signals.
Data are sleep signals from subjects SC4001E0 (a,b,c), SC4011E0 (d,e,f), SC4021E0 (g,h,i) and SC4032E0 (j,k,l). Kalman filtering is applied to approximate entropy (first row) and NNetEn (second row).
Fig 10.
Kalman filtering applied to different entropy measures of epilepsy signals.
Data are epilepsy signals from channel FP1-F3 (a,b,c), F3-C3 (d,e,f), C3-P3 (g,h,l) and P3-O1 (j,k,l). Kalman filtering is applied to approximate entropy (first row) and NNetEn entropy (second row).
Fig 11.
The effects of embedding dimension m and threshold r on VRR smoothed by Kalman filtering on sample entropy time series.
(a)m = 2. (b)m = 3. The sleep EEG time series (10 subjects; data length of each subject: 50000 datapoints) were used for the sample entropy calculation. Error bars indicate standard errors. σ is the standard deviation of the EEG time series.
Fig 12.
The effects of parameters Q and P on VRR smoothed by Kalman filtering on sample entropy time series.
(a) VRR as a function of R values when Q is fixed to 0.1. (b) VRR as a function of Q values when R is fixed to 0.5. The data segment is the same as in Fig 11. Error bars indicate standard errors. N = 10. The sign * indicates significance level p<0.05.