The Multiscale Entropy Algorithm and Its Variants: A Review <s> Refined Composite Multiscale Entropy <s> Abstract Multiscale entropy (MSE) is an effective algorithm for measuring the complexity of a time series that has been applied in many fields successfully. However, MSE may yield an inaccurate estimation of entropy or induce undefined entropy because the coarse-graining procedure reduces the length of a time series considerably at large scales. Composite multiscale entropy (CMSE) was recently proposed to improve the accuracy of MSE, but it does not resolve undefined entropy. Here we propose a refined composite multiscale entropy (RCMSE) to improve CMSE. For short time series analyses, we demonstrate that RCMSE increases the accuracy of entropy estimation and reduces the probability of inducing undefined entropy. <s> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Refined Composite Multiscale Entropy <s> Multiscale entropy (MSE) is an effective method to measure the complexity of signals from complex systems, which has been applied to various fields successfully. However, MSE may yield an inaccurate estimate of entropy and induce undefined entropy as the coarse-graining procedure reduces the length of data considerably at large scales. Refined composite multiscale entropy (RCMSE) is then developed to solve this problem. However, trends superimposed in signals may significantly affect the complexity measurement. Thus we introduce an empirical mode decomposition based RCMSE, called EMD–RCMSE to first eliminate such effects of trends and then measure the complexity of signals. It is validated from simulated signals and has good estimation. In addition, this method is also applied to study the complexity of traffic signals and we obtain some interesting results: (1) Traffic signals are more complex than the results showing from RCMSE/MSE; (2) Weekday and weekend patterns (different combination of trends) greatly affect the results; (3) Complexity indices change with time at each day, due to the degree of human activities. <s> BIB002
In 2014, Wu et al. proposed the refined CMSE (RCMSE) to improve CMSE because CMSE does not resolve undefined entropy: CMSE estimates entropy more accurately but increases the probability of inducing undefined entropy BIB001 . From Equation (10), one can observe that in CMSE the logarithms for all τ coarse-grained time series are first computed and only then the average of the results is computed to give rise to the entropy value. As a result, CMSE leads to an undefined value when either n m+1 k,τ or n m k,τ is zero. Consequently, when short time series are analyzed, CMSE leads to much more undefined values than does the original MSE algorithm. To address this drawback, the RCMSE has been proposed, based on the following three steps BIB001 (1) the same coarse-graining procedure as in CMSE is used (see Equation (8)) (2) for each scale factor τ, and for all τ coarse-grained time series, the number of matched vector pairs n m+1 k,τ and n m k,τ is computed (3) RCMSE is then defined as BIB001 RCM SE(x, τ, m, r) = − ln From Equation (11), one can note that RCMSE leads to undefined entropy values only when all n m+1 k,τ or n m k,τ are zeros. The RCMSE algorithm therefore reduces the probability of having undefined entropy values, compared to CMSE BIB001 . Wu et al. also processed white and 1/f noises with the RCMSE algorithm BIB001 . They concluded that, regarding validity, the RCMSE algorithm is superior to both MSE and CMSE algorithms (the probability of obtaining undefined entropy was zero). Moreover, RCMSE leads to more consistent entropy values than CMSE or MSE (the standard deviations of the entropy values obtained with RCMSE were lower than those obtained with CMSE or MSE). Finally, Wu et al. reported that RCMSE is better than the CMSE and MSE algorithms regarding independence of the data length BIB001 . Nevertheless, the computational cost of RCMSE is worse than the one of MSE (but better than the one of CMSE) BIB001 . RCMSE has recently been applied (in conjunction with empirical mode decomposition) on real data BIB002 .