Optimal estimation of recurrence structures from time series

¹ Bernstein Center for Computational Neuroscience - Berlin, Germany
² Department of Psychiatry, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
³ Neurobiology Curriculum, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
⁴ Department of Cell Biology and Physiology, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
⁵ Department of Biomedical Engineering, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
⁶ Neuroscience Center, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
⁷ Department of Neurology, University of North Carolina at Chapel Hill - Chapel Hill, NC, USA
⁸ German Meteorological Service - Offenbach am Main, Germany

Received: 2 March 2016
Accepted: 18 May 2016

Abstract

Recurrent temporal dynamics is a phenomenon observed frequently in high-dimensional complex systems and its detection is a challenging task. Recurrence quantification analysis utilizing recurrence plots may extract such dynamics, however it still encounters an unsolved pertinent problem: the optimal selection of distance thresholds for estimating the recurrence structure of dynamical systems. The present work proposes a stochastic Markov model for the recurrent dynamics that allows for the analytical derivation of a criterion for the optimal distance threshold. The goodness of fit is assessed by a utility function which assumes a local maximum for that threshold reflecting the optimal estimate of the system's recurrence structure. We validate our approach by means of the nonlinear Lorenz system and its linearized stochastic surrogates. The final application to neurophysiological time series obtained from anesthetized animals illustrates the method and reveals novel dynamic features of the underlying system. We propose the number of optimal recurrence domains as a statistic for classifying an animals' state of consciousness.