Recurrence Time Distribution, Renyi Entropy, and Pattern Discovery

Entropy and recurrence times are two of the most important complexity measures for both random fields and nonlinear dynamical systems. We report a fundamental relation between recurrence time distribution and Renyi entropy of arbitrary integer order for both ergodic random fields and ergodic nonlinear dynamical systems, thus provide an elegant and comprehensive characterization for these two important systems. The fundamental relation is obtained by parameterizing the dynamics in the state space by a discrete symbol sequence and collectively consider recurrences to all non-empty sub-regions of the state space. Event detection using recurrence time statistics is also considered, including speech endpoint detection, epileptic seizure detection/prediction from continuous EEG measurements, and gene identification from genomic DNA sequences.