The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Abstract ::: Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts. <s> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Entropy, as it relates to dynamical systems, is the rate of information production. Methods for estimation of the entropy of a system represented by a time series are not, however, well suited to analysis of the short and noisy data sets encountered in cardiovascular and other biological studies. Pincus introduced approximate entropy (ApEn), a set of measures of system complexity closely related to entropy, which is easily applied to clinical cardiovascular and other time series. ApEn statistics, however, lead to inconsistent results. We have developed a new and related complexity measure, sample entropy (SampEn), and have compared ApEn and SampEn by using them to analyze sets of random numbers with known probabilistic character. We have also evaluated cross-ApEn and cross-SampEn, which use cardiovascular data sets to measure the similarity of two distinct time series. SampEn agreed with theory much more closely than ApEn over a broad range of conditions. The improved accuracy of SampEn statistics should make them useful in the study of experimental clinical cardiovascular and other biological time series. <s> BIB002 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> There has been considerable interest in quantifying the complexity of physiologic time series, such as heart rate. However, traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long-range correlations. This paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics. We introduce a method to calculate multiscale entropy (MSE) for complex time series. We find that MSE robustly separates healthy and pathologic groups and consistently yields higher values for simulated long-range correlated noise compared to uncorrelated noise. <s> BIB003 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Traditional approaches to measuring the complexity of biological signals fail to account for the multiple time scales inherent in such time series. These algorithms have yielded contradictory findings when applied to real-world datasets obtained in health and disease states. We describe in detail the basis and implementation of the multiscale entropy (MSE) method. We extend and elaborate previous findings showing its applicability to the fluctuations of the human heartbeat under physiologic and pathologic conditions. The method consistently indicates a loss of complexity with aging, with an erratic cardiac arrhythmia (atrial fibrillation), and with a life-threatening syndrome (congestive heart failure). Further, these different conditions have distinct MSE curve profiles, suggesting diagnostic uses. The results support a general "complexity-loss" theory of aging and disease. We also apply the method to the analysis of coding and noncoding DNA sequences and find that the latter have higher multiscale entropy, consistent with the emerging view that so-called "junk DNA" sequences contain important biological information. <s> BIB004 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Abstract. In this work we use the multiscale entropy method to analyse the variability of geo-electric time series monitored in two sites located in Mexico. In our analysis we consider a period of time from January 1995 to December 1995. We systematically calculate the sample entropy of electroseismic time series. Important differences in the entropy profile for several time scales are observed in records from the same station. In particular, a complex behaviour is observed in the vicinity of a M =7.4 EQ occurred on 14 September 1995. Besides, we also compare the changes in the entropy of the original data with their corresponding shuffled version. <s> BIB005 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <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> BIB006 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Processes regulating the cardiovascular system (CVS) are numerous. Each possesses several temporal scales. Their interactions lead to interdependences across multiple scales. For the CVS analysis, different multiscale studies have been proposed, mostly performed on heart rate variability signals (HRV) reflecting the central CVS; only few were dedicated to data from the peripheral CVS, such as laser Doppler flowmetry (LDF) signals. Very recently, a study implemented the first computation of multiscale entropy for LDF signals. A nonmonotonic evolution of multiscale entropy with two distinctive scales was reported, leading to a markedly different behavior from the one of HRV. Our goal herein is to confirm these results and to go forward in the investigations on origins of this behavior. For this purpose, 12 LDF signals recorded simultaneously on the two forearms of six healthy subjects are processed. This is performed before and after application of physiological scales-based filters aiming at isolating previously found frequency bands linked to physiological activities. The results obtained with signals recorded simultaneously on two different sites of each subject show a probable central origin for the nonmonotonic behavior. The filtering results lead to the suggestion that origins of the distinctive scales could be dominated by the cardiac activity. <s> BIB007 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Bearing fault diagnosis has attracted significant attention over the past few decades. It consists of two major parts: vibration signal feature extraction and condition classification for the extracted features. In this paper, multiscale permutation entropy (MPE) was introduced for feature extraction from faulty bearing vibration signals. After extracting feature vectors by MPE, the support vector machine (SVM) was applied to automate the fault diagnosis procedure. Simulation results demonstrated that the proposed method is a very powerful algorithm for bearing fault diagnosis and has much better performance than the methods based on single scale permutation entropy (PE) and multiscale entropy (MSE). <s> BIB008 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <s> Laser speckle contrast imaging (LSCI) is a noninvasive full-field optical imaging technique that gives a 2-D microcirculatory blood flow map of tissue. Due to novelty of commercial laser speckle contrast imagers, image processing of LSCI data is new. By opposition, the numerous signal processing works of laser Doppler flowmetry (LDF) data-that give a 1-D view of microvascular blood flow-have led to interesting physiological information. Recently, analysis of multiscale entropy (MSE) of LDF signals has been proposed. A nonmonotonic evolution of MSE with two distinctive scales-probably dominated by the cardiac activity-has been reported. We herein analyze MSE of LSCI data. We compare LSCI results with the ones of LDF signals obtained during the same experiment. We show that when time evolution of LSCI single pixels is studied, MSE presents a monotonic decreasing pattern, similar to the one of Gaussian white noises. By opposition, when the mean of LSCI pixel values is computed in a region of interest (ROI) and followed with time, MSE pattern becomes close to the one of LDF data, for ROI large enough. LSCI is gaining increased interest for blood flow monitoring. The physiological implications of our results require future study. <s> BIB009 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <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> BIB010 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Introduction <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> BIB011
Complexity of time series can be studied through several measures, e.g., approximate entropy BIB001 and sample entropy BIB002 . However, some of these measures do not take into account the multiple time scales in physical systems. In the 2000s, the multiscale entropy (MSE) approach has been proposed by Costa et al. to represent the complexity of a signal BIB003 . MSE relies on the computation of the sample entropy over a range of scales: in the MSE algorithm, coarse-grained time series-that represent the system dynamics on different scales-are analyzed with the sample entropy algorithm. Since its introduction, MSE has become a prevailing method to quantify the complexity of signals. It has been used successfully in different research fields, including biomedical time series BIB004 BIB007 BIB009 , electroseismic time series BIB005 , vibration of rotary machine BIB008 , financial time series BIB011 . However, the sample entropy algorithm may give an imprecise estimation of entropy or even induce undefined entropy values for short time series (see, e.g., BIB010 ). Moreover, it has been reported that the coarse-graining procedure (that eliminates the fast temporal scales) exploits a filter with a frequency response that cannot prevent aliasing and which is therefore suboptimal, especially in the presence of fast oscillations BIB006 . Finally, some authors mentioned that the entropy-based complexity is partially linked to the reduction of variance generated by the elimination of the fast temporal scales BIB006 . This is why, other algorithms-based on MSE-have been developed to try to improve the original MSE algorithm. We herein propose to present a state-of-the art of these algorithms that have been conceived to overcome some limitations of the original MSE.