The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multivariate Multiscale Entropy <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> BIB001 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multivariate Multiscale Entropy <s> This work generalizes the recently introduced univariate multiscale entropy (MSE) analysis to the multivariate case. This is achieved by introducing multivariate sample entropy (MSampEn) in a rigorous way, in order to account for both within- and cross-channel dependencies in multiple data channels, and by evaluating it over multiple temporal scales. The multivariate MSE (MMSE) method is shown to provide an assessment of the underlying dynamical richness of multichannel observations, and more degrees of freedom in the analysis than standard MSE. The benefits of the proposed approach are illustrated by simulations on complexity analysis of multivariate stochastic processes and on real-world multichannel physiological and environmental data. <s> BIB002 </s> The Multiscale Entropy Algorithm and Its Variants: A Review <s> Multivariate Multiscale Entropy <s> Multivariate physical and biological recordings are common and their simultaneous analysis is a prerequisite for the understanding of the complexity of underlying signal generating mechanisms. Traditional entropy measures are maximized for random processes and fail to quantify inherent long-range dependencies in real world data, a key feature of complex systems. The recently introduced multiscale entropy (MSE) is a univariate method capable of detecting intrinsic correlations and has been used to measure complexity of single channel physiological signals. To generalize this method for multichannel data, we first introduce multivariate sample entropy (MSampEn) and evaluate it over multiple time scales to perform the multivariate multiscale entropy (MMSE) analysis. This makes it possible to assess structural complexity of multivariate physical or physiological systems, together with more degrees of freedom and enhanced rigor in the analysis. Simulations on both multivariate synthetic data and real world postural sway analysis support the approach. <s> BIB003
In order to be able to process multivariate data, multivariate MSE has been proposed BIB002 BIB003 . For this purpose, a multivariate sample entropy version has been introduced for considering both within and cross-channel dependencies in multichannel data. Multivariate MSE relies on the same steps as MSE BIB002 BIB003 : (I) a coarse-graining procedure; (II) a sample entropy computation for each coarse-grained time series. However, due to the multivariate nature of the data processed by multivariate MSE, these two steps are adapted to multivariate signals. Thus, for the coarse-graining procedure, temporal scales are defined by averaging a p-variate time series {x l,i } N i=1 (l = 1, . . . , p is the channel index and N is the number of samples in every channel) over non overlapping time segments of increasing length. Thus, for a scale factor τ, a coarse-grained multivariate time series is computed as y (τ) l,j = 1 τ jτ i=(j−1)τ +1 x l,i where 1 ≤ j ≤ N/τ, and the channel index l goes from 1 to p. For the entropy computation, the multivariate sample entropy is used for each coarse-grained multivariate. The multivariate sample entropy algorithm is an extension of the (univariate) sample entropy BIB001 . For a tolerance level r, multivariate sample entropy is calculated as the negative of the natural logarithm of the conditional probability that two composite delay vectors close to each other in a m dimensional space will also be close to each other when the dimensionality is increased by one. The detailed multivariate sample entropy algorithm can be found in BIB002 BIB003 . In the multivariate MSE algorithm, multivariate sample entropy is evaluated over multiple time scales.