The Multiscale Entropy Algorithm and Its Variants: A Review <s> Refined Multiscale Entropy <s> Multiscale entropy (MSE) was proposed to characterize complexity as a function of the time-scale factor tau. Despite its broad use, this technique suffers from two limitations: 1) the artificial MSE reduction due to the coarse graining procedure and 2) the introduction of spurious MSE oscillations due to the suboptimal procedure for the elimination of the fast temporal scales. We propose a refined MSE (RMSE), and we apply it to simulations and to 24-h Holter recordings of heart rate variability (HRV) obtained from healthy and aortic stenosis (AS) groups. The study showed that the refinement relevant to the elimination of the fast temporal scales was more helpful at short scales (spanning the range of short-term HRV oscillations), while that relevant to the procedure of coarse graining was more useful at large scales. In healthy subjects, during daytime, RMSE was smaller at short scales (i.e., tau = 1-2) and larger at longer scales (i.e., tau = 4-20) than during nighttime. In AS population, RMSE was smaller during daytime both at short and long time scales (i.e., tau = 1 -11) than during nighttime. RMSE was larger in healthy group than in AS population during both daytime (i.e., tau = 2 -9) and nighttime (i.e., tau = 2). RMSE overcomes two limitations of MSE and confirms the complementary information that can be derived by observing complexity as a function of the temporal scale. <s> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Refined Multiscale Entropy <s> In this work, Refined Multiscale Entropy (RMSE) was applied to characterize risk of cardiac death in ischemic cardiomyopathy patients, analyzing heart rate variability (HRV) by means of RR series during daytime and nighttime. RMSE approach measures an entropy rate in different time scales of a series, giving a multiscale characterization of complexity of that series. RMSE showed statistically significant differences (p<0.05) during daytime and nighttime only in middle time scales (τ=4–15 and τ=3–16, respectively). For these scales, RMSE was higher in low risk (SV) than in high risk (CM) group of cardiac death, indicating a reduction of the entropy-based complexity in CM when it was compared with SV. No statistical differences between risk groups were presented at time scale τ=1 (unfiltered original RR series). It can be concluded that the dynamics in middle time scales should be considered to better describe the HRV of patients with cardiac death. <s> BIB002
In 2009, Valencia et al. proposed the refined MSE (RMSE) BIB001 . In the latter algorithm, the authors proposed a way to remove the fast temporal scales and used a coarse-graining that prevents the influence of the reduced variance on the complexity evaluation BIB001 . In order to improve the elimination of the fast temporal scales, the FIR filter is replaced by a low-pass Butterworth filter. The squared magnitude of the filter frequency response is chosen as BIB001 |H(e 2πjf ) where n is the filter order and f c is the cutoff frequency BIB001 . This filter has the advantage of presenting a flat magnitude for its frequency response in the passband. Moreover, it has no side lobes in the stopband and the roll-off is fast. It thus reduces aliasing when the filtered time series is downsampled BIB001 . Regarding the coarse-graining procedure, Valencia et al. proposed to reduce the dependence of the estimated entropy rate on the decrease of variance by updating the value of r as a percentage of the standard deviation of the filtered signal BIB001 . In order to analyze the performance of RMSE, Valencia et al. processed (among others) a fully unpredictable process (Gaussian white noise) and a signal with long-range correlation (1/f noise). They reported that RMSE is flat with scale factor τ for the Gaussian white noise and presents a slow but progressive increase with scale factor τ in the case of 1/f noise. By opposition, the original MSE presents a monotonic decrease with scale factor τ both for the Gaussian white noise and the 1/f noise BIB001 . Since aliasing is more significant at short time scales when the fast oscillations are dominant, the largest differences between RMSE and the original MSE algorithms are found in the presence of high frequency oscillations BIB001 . The RMSE algorithm has also been applied on experimental data (see, e.g., BIB001 BIB002 ).