Inhomogeneous point-process entropy: An instantaneous measure of complexity in discrete systems

Abstract

Measures of entropy have been widely used to characterize complexity, particularly in physiological dynamical systems modeled in discrete time. Current approaches associate these measures to finite single values within an observation window, thus not being able to characterize the system evolution at each moment in time. Here, we propose a new definition of approximate and sample entropy based on the inhomogeneous point-process theory. The discrete time series is modeled through probability density functions, which characterize and predict the time until the next event occurs as a function of the past history. Laguerre expansions of the Wiener-Volterra autoregressive terms account for the long-term nonlinear information. As the proposed measures of entropy are instantaneously defined through probability functions, the novel indices are able to provide instantaneous tracking of the system complexity. The new measures are tested on synthetic data, as well as on real data gathered from heartbeat dynamics of healthy subjects and patients with cardiac heart failure and gait recordings from short walks of young and elderly subjects. Results show that instantaneous complexity is able to effectively track the system dynamics and is not affected by statistical noise properties.

Assessment of spontaneous cardiovascular oscillations in Parkinson's disease

“…Of note, this instantaneous assessment of complexity opens the possibility of analyzing the proposed measures also in terms of variability of their evolution along time -an idea which we refer to as our complexity variability framework [38]. Moreover, we have shown that the dynamical estimates and are not substantially affected by the type of noise underlying the complex system, thus ensuring truly instantaneous tracking of the complexity of the underlying dynamical system [38].…”

“…Within this framework, Laguerre expansions of the Wiener-Volterra linear and nonlinear autoregressive terms account for long-term nonlinear information [36][37][38]. As major advantages, instantaneous measures can be estimated without applying any interpolation techniques to the RR interval series, and are associated to effective goodness-of-fit measures.…”

“…In addition, while the physiology underlying complexity-related measures of heartbeat dynamics is still unknown, it is well known that their quantification can provide meaningful information on psychophysiological as well as pathological states [8,10,14,24,39]. In this context, we have previously shown that measures of time-varying complexity provide enhanced discriminatory power when compared to standard complexity measures [36,38], and that the stability and complexity of autonomic dynamics is altered in PD patients when compared to healthy controls as well as within different PD subgroups [4,6].…”

“…In this study, the estimated instantaneous entropy measures refer to the inhomogeneous point-process approximate and sample entropy, and , respectively [38].…”

“…To this end, we employ our instantaneous inhomogeneous point-process framework [36][37][38], which augments standard ANS-related HRV metrics defined in the time and frequency domains [5] with measures of instantaneous complexity (dominant Lyapunov exponent and entropy) [36,38] and higher-order statistics (bispectra) [37].…”