Optimal Estimation of Recurrence Structures from Time Series

The present work proposes a solution of a pertinent problem for recurrence plots and recurrence quantification analysis of time series: the optimal selection of distance thresholds for estimating the recurrence structure of complex dynamic systems. We approximate this recurrence structure through Markov chains obtained from recurrence grammars. The goodness of fit is assessed with a utility function derived from a stochastic Markov transition matrix. It assumes a local maximum for that distance threshold which reflects 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 the segmentation of 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.