The Multiscale Entropy Algorithm and Its Variants: A Review <s> Other Improvements <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> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Other Improvements <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> BIB002 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Other Improvements <s> Methods from nonlinear dynamics (NLD) have shown new insights into heart rate (HR) variability changes under various physiological and pathological conditions, providing additional prognostic information and complementing traditional time- and frequency-domain analyses. In this review, some of the most prominent indices of nonlinear and fractal dynamics are summarized and their algorithmic implementations and applications in clinical trials are discussed. Several of those indices have been proven to be of diagnostic relevance or have contributed to risk stratification. In particular, techniques based on mono- and multifractal analyses and symbolic dynamics have been successfully applied to clinical studies. Further advances in HR variability analysis are expected through multidimensional and multivariate assessments. Today, the question is no longer about whether or not methods from NLD should be applied; however, it is relevant to ask which of the methods should be selected and under which basic and standardized conditions should they be applied. <s> BIB003 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Other Improvements <s> This paper presents a novel framework to determine the number of resolution levels in the application of a wavelet transformation to a rainfall time series. The rainfall time series are decomposed using the a trous wavelet transform. Then, multi-scale entropy (MSE) analysis that helps to elucidate some hidden characteristics of the original rainfall time series is applied to the decomposed rainfall time series. The analysis shows that the Mann-Kendall (MK) rank correlation test of MSE curves of residuals at various resolution levels could determine the number of resolution levels in the wavelet decomposition. The complexity of rainfall time series at four stations on a multi-scale is compared. The results reveal that the suggested number of resolution levels can be obtained using MSE analysis and MK test. The complexity of rainfall time series at various locations can also be analyzed to provide a reference for water resource planning and application. <s> BIB004 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Other Improvements <s> Complexity in physiological outputs is believed to be a hallmark of healthy physiological control. How to accurately quantify the degree of complexity in physiological signals with outliers remains a major barrier for translating this novel concept of nonlinear dynamic theory to clinical practice. Here we propose a new approach to estimate the complexity in a signal by analyzing the irregularity of the sign time series of its coarse-grained time series at different time scales. Using surrogate data, we show that the method can reliably assess the complexity in noisy data while being highly resilient to outliers. We further apply this method to the analysis of human heartbeat recordings. Without removing any outliers due to ectopic beats, the method is able to detect a degradation of cardiac control in patients with congestive heart failure and a more degradation in critically ill patients whose life continuation relies on extracorporeal membrane oxygenator (ECMO). Moreover, the derived complexity measures can predict the mortality of ECMO patients. These results indicate that the proposed method may serve as a promising tool for monitoring cardiac function of patients in clinical settings. <s> BIB005
Other improvements than those presented above have been proposed for the original MSE algorithm. Thus, in 2011, Chou proposed the use of MSE on residuals (at various resolution levels) given by the à trous redundant wavelet transform BIB004 . Based on this, the authors proposed to determine the number of resolution levels in the wavelet transformation, according to the Mann-Kendall rank correlation test BIB004 . Regarding the selection of the parameter r value, and as mentioned in Section 2, Nikulin and Brismar reported that the coarse-graining procedure is similar to smoothing and decimation of the original sequences BIB001 . Due to the constant value of r with scales, in the original MSE algorithm, they concluded that the changes in MSE on each scale depend on both the regularity and variation of the coarse-grained time series BIB001 . To this point, Costa et al. argued that the degree of irregularity of a complex signal is a property measured by entropy that cannot be entirely captured by the standard deviation or correlation measures, individually or in combination BIB001 : they argue that, following the normalization, modifications in the variance generated by the coarse-graining procedure are due to the temporal structure of the original signal. Therefore, it should be accounted for by the entropy measure BIB002 . Regarding the application of MSE on noisy data: it has been reported that MSE is not reliable when applied on heartbeat signals containing outliers BIB002 BIB003 . Therefore, in order to quantify the complexity of noisy data (data with outliers), Lo et al. proposed to analyze the irregularity of the sign time series of the coarse-grained data and, for this purpose, proposed the multiscale symbolic entropy (MSSE) analysis BIB005 . MSSE uses the sign time series of each coarse-grained series at a given time scale (1 when the time series increases and 0 otherwise). The sign time series is divided into bit sequences of length m. Based on the same concept as the approximate/sample entropy, these bit sequences are divided into different categories based on their temporal patterns (see details in BIB005 ). Then, based on the distribution of the sequences in different categories, the Shannon entropy (which gives information richness) and the mean rank (which gives the degree of uncertainty of the fluctuations) are computed for the sign time series. Lo et al. used the MSSE on human heartbeat recordings. While the outliers due to ectopic beats were not removed, the authors were able to stress a modification in the cardiac control of patients with congestive heart failure, and even more BIB005 . As a result, they mentioned that MSSE enables the application of the complexity theory in clinical practice BIB005 . We can also report that Govindan et al. proposed a modification in the definition of sample entropy for a better characterization of the complexity: in the original sample entropy algorithm, the time series under study is divided into blocks of size m with a time gap of one (sample unit) between the successive components of the block  . Govindan et al. incorporated a time delay between the successive components of the block  . The delay is larger than one in the presence of dynamical correlations. Govindan et al. reported that this modification leads to a better characterization of the complexity than the original algorithm of sample entropy  .