The Multiscale Entropy Algorithm and Its Variants: A Review <s> Composite Multiscale Entropy <s> Multiscale entropy (MSE) was recently developed to evaluate the complexity of time series over different time scales. Although the MSE algorithm has been successfully applied in a number of different fields, it encounters a problem in that the statistical reliability of the sample entropy (SampEn) of a coarse-grained series is reduced as a time scale factor is increased. Therefore, in this paper, the concept of a composite multiscale entropy (CMSE) is introduced to overcome this difficulty. Simulation results on both white noise and 1/f noise show that the CMSE provides higher entropy reliablity than the MSE approach for large time scale factors. On real data analysis, both the MSE and CMSE are applied to extract features from fault bearing vibration signals. Experimental results demonstrate that the proposed CMSE-based feature extractor provides higher separability than the MSE-based feature extractor. <s> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> 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> BIB002 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Composite Multiscale Entropy <s> Abstract It is significant to study the complexity of financial time series since the financial market is a complex evolved dynamic system. Multiscale entropy is a prevailing method used to quantify the complexity of a time series. Due to its less reliability of entropy estimation for short-term time series at large time scales, a modification method, the composite multiscale entropy, is applied to the financial market. To qualify its effectiveness, its applications in the synthetic white noise and 1 / f noise with different data lengths are reproduced first in the present paper. Then it is introduced for the first time to make a reliability test with two Chinese stock indices. After conducting on short-time return series, the CMSE method shows the advantages in reducing deviations of entropy estimation and demonstrates more stable and reliable results when compared with the conventional MSE algorithm. Finally, the composite multiscale entropy of six important stock indices from the world financial markets is investigated, and some useful and interesting empirical results are obtained. <s> BIB003
In 2013, Wu et al. proposed the composite multiscale entropy (CMSE) in order to reduce the variance of estimated entropy values at large scales BIB001 . For this purpose, and for a discrete time series x = {x 1 , . . . , x i , . . . , x N }, the kth coarse-grained time series for a scale factor τ is defined as BIB001 y Then, at a scale factor τ the sample entropy values of all coarse-grained time series are computed and the CMSE is defined as the means of the τ entropy values BIB001  where S E (y k , m, r) corresponds to the sample entropy for the time series y k , which can also be written as BIB002  where n m k,τ represents the total number of m-dimensional matched vector pairs and is computed from the kth coarse-grained time series at a scale factor τ. In order to compare the original MSE and CMSE, Wu et al. applied the two algorithms on white and 1/f noises BIB001 . Moreover, different lengths for the data have been tested. For the white noise, they reported that the estimated MSE values (numerical solutions) are significantly different from the analytic solution, for short signals. For the 1/f noise, they reported differences between the numerical solution and the analytic solution, for all scales BIB001 . The authors also mentioned that the variance of the entropy estimator is improved by CMSE. However, the over estimation due to the short length of the time series still exists with CMSE on 1/f noises BIB001 . The authors report that CMSE shows better performances on short time series than the original MSE algorithm (the original MSE and CMSE give entropy values that are almost the same but CMSE can estimate entropy values more accurately than the original MSE: the standard deviation values of sample entropy are smaller with CMSE). They also mention that for white noise and 1/f noise, CMSE gives a more reliable estimation of entropy than the original MSE algorithm, as shown by the simulation results. CMSE has also been applied on real data (see, e.g., BIB003 BIB001 ).