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.

Abstract-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.Index Terms-Multivariate embedding, long term correlation, multivariate multiscale entropy (MMSE), multivariate sample entropy (MSampEn), multivariate system complexity.

Abstract:The recently introduced multivariate multiscale entropy (MMSE) has been successfully used to quantify structural complexity in terms of nonlinear within-and cross-channel correlations as well as to reveal complex dynamical couplings and various degrees of synchronization over multiple scales in real-world multichannel data. However, the applicability of MMSE is limited by the coarse-graining process which defines scales, as it successively reduces the data length for each scale and thus yields inaccurate and undefined entropy estimates at higher scales and for short length data. To that cause, we propose the multivariate multiscale fuzzy entropy (MMFE) algorithm and demonstrate its superiority over the MMSE on both synthetic as well as real-world uterine electromyography (EMG) short duration signals. Based on MMFE features, an improvement in the classification accuracy of term-preterm deliveries was achieved, with a maximum area under the curve (AUC) value of 0.99.

Abstract. Established complexity measures typically operate at a single scale and thus fail to quantify inherent long-range correlations in real-world data, a key feature of complex systems. The recently introduced multiscale entropy (MSE) method has the ability to detect fractal correlations and has been used successfully to assess the complexity of univariate data. However, multivariate observations are common in many real-world scenarios and a simultaneous analysis of their structural complexity is a prerequisite for the understanding of the underlying signal-generating mechanism. For this purpose, based on the notion of multivariate sample entropy, the standard MSE method is extended to the multivariate case, whereby for rigor, the intrinsic multivariate scales of the input data are generated adaptively via the multivariate empirical mode decomposition (MEMD) algorithm. This allows us to gain better understanding of the complexity of the underlying multivariate real-world process, together with more degrees of freedom and physical interpretation in the analysis. Simulations on both synthetic and real-world biological multivariate data sets support the analysis.

Abstract-The recently introduced multiscale entropy (MSE) method accounts for long range correlations over multiple time scales and can therefore reveal the complexity of biological signals. The existing MSE algorithm deals with scalar time series whereas multivariate time series are common in experimental and biological systems. To that cause, in this paper the MSE method is extended to the multivariate case. This allows us to gain a greater insight into the complexity of the underlying signal generating system, producing multifaceted and more robust estimates than standard single channel MSE. Simulations on both synthetic data and brain consciousness analysis support the approach.

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