Efficient Computation of Multiscale Entropy over Short Biomedical Time Series Based on Linear State-Space Models

Abstract

The most common approach to assess the dynamical complexity of a time series across multiple temporal scales makes use of the multiscale entropy (MSE) and refined MSE (RMSE) measures. In spite of their popularity, MSE and RMSE lack an analytical framework allowing their calculation for known dynamic processes and cannot be reliably computed over short time series. To overcome these limitations, we propose a method to assess RMSE for autoregressive (AR) stochastic processes. The method makes use of linear state-space (SS) models to provide the multiscale parametric representation of an AR process observed at different time scales and exploits the SS parameters to quantify analytically the complexity of the process. The resulting linear MSE (LMSE) measure is first tested in simulations, both theoretically to relate the multiscale complexity of AR processes to their dynamical properties and over short process realizations to assess its computational reliability in comparison with RMSE. Then, it is applied to the time series of heart period, arterial pressure, and respiration measured for healthy subjects monitored in resting conditions and during physiological stress. This application to short-term cardiovascular variability documents that LMSE can describe better than RMSE the activity of physiological mechanisms producing biological oscillations at different temporal scales.

A Study of Brain Neuronal and Functional Complexities Estimated Using Multiscale Entropy in Healthy Young Adults

“…See [46] for a review of these refined methods. More recently, a theoretical approach was presented to analytically calculate a new MSE measure using state-space models [47][48][49]. In addition, the importance of considering signal normalization and spectral content was shown using simulated and empirical data [50].…”

“…On each box, the central mark shows the median, bottom and top edges show the 25th and 75th percentiles, respectively, and the whiskers extend to the most extreme data points that are not considered as outliers, which are defined as 1.5 times the interquartile range away from the top or bottom of the box. Nodes belonging to the RSNs are shown in parentheses-auditory (1-3), basal ganglia(4-7), dorsal default mode network(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), primary visual(18)(19), language (20-26), left executive control network(27)(28)(29)(30)(31)(32), sensorimotor(33)(34)(35)(36)(37)(38), posterior salience(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50), precuneus(51)(52)(53)(54), higher visual (55-56), right executive control network (57-62), anterior salience (63-69), ventral default mode network (70-79), visuospatial (80-90).…”

Improved Permutation Entropy for Measuring Complexity of Time Series under Noisy Condition

“…To remedy this, Zunino et al proposed to calculate PE as a function of the time delay, offering a way to unveil the presence of structures on multiple temporal scales [20]. Moreover, Costa et al proposed a coarse-graining technique [19,21,22]. Based on Costa's work, Aziz introduced the multiscale permutation entropy (MPE) [23] by combining the coarsegraining procedure with PE.…”

Entropy-Based Pattern Learning Based on Singular Spectrum Analysis Components for Assessment of Physiological Signals

“…Entropy not only can be used to measure the additional information needed to determine the state of a system, but also can be used to quantify the irregular, random and chaotic behavior of physiological signals [12]. ere have been reported many application cases of entropy-based pattern learning by the use of a variety of entropy measures (such as approximate entropy (ApEn), sample entropy (SampEn), permutation entropy, spectral entropy, short-term Rényi entropy and Shannon entropy, and so on) [13][14][15][16][17][18][19][20][21]. For instance, Raghu et al proposed a novel minimum variance modi ed fuzzy entropy to identify epileptic seizures in real time from electroencephalogram (EEG) signals, which achieved the classi cation accuracy of 100% [22].…”