title: 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.
authors.affiliation: Department of Anesthesia, Critical Care & Pain Medicine, Massachusetts General Hospital, and Harvard Medical School, Boston, Massachusetts, USA, and Massachusetts Institute of Technology, Cambridge, Massachusetts, USA and Research Center E. Piaggio and Department of Information Engineering, University of Pisa, Pisa, Italy.
authors.forename: Gaetano
authors.initials: G
authors.lastname: Valenza
authors.affiliation: Department of Anesthesia, Critical Care & Pain Medicine, Massachusetts General Hospital, and Harvard Medical School, Boston, Massachusetts, USA, and Massachusetts Institute of Technology, Cambridge, Massachusetts, USA and School of Computer Science and Electronic Engineering, University of Essex, Colchester, United Kingdom.
authors.forename: Luca
authors.initials: L
authors.lastname: Citi
authors.affiliation: Research Center E. Piaggio and Department of Information Engineering, University of Pisa, Pisa, Italy.
authors.forename: Enzo Pasquale
authors.initials: EP
authors.lastname: Scilingo
authors.affiliation: Department of Anesthesia, Critical Care & Pain Medicine, Massachusetts General Hospital, and Harvard Medical School, Boston, Massachusetts, USA, and Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.
authors.forename: Riccardo
authors.initials: R
authors.lastname: Barbieri
mesh_terms: D000375:Aging / Q000502:physiology*; D003627:Data Interpretation, Statistical; D019277:Entropy; D005684:Gait / Q000502:physiology*; D006333:Heart Failure / Q000175:diagnosis / Q000503:physiopathology*; D006339:Heart Rate; D006801:Humans; D010363:Pattern Recognition, Automated / Q000379:methods; D015203:Reproducibility of Results; D012680:Sensitivity and Specificity; D012815:Signal Processing, Computer-Assisted