How to compute the minimum diagonal length of recurrence quantifiers to optimize their sensitivity to deterministic and stochastic properties

Abstract

Recurrence Plots (RP) are binary matrices that quantify the recurrent and non-recurrent states of a trajectory. Each element of an RP matrix, denoted by ai,j, consists of an one or a zero, indicating a recurrent or non-recurrent pair of elements (i,j) of the trajectory, respectively. RP represents a graphical mosaic of the recurrent and non-recurrent states of a particular trajectory. The graphical properties of Recurrence Plots can be quantified using tools known as Recurrence Quantification Analysis (RQA), which rely on several structures embedded in an RP, such as diagonal, vertical, or horizontal line lengths. All of these quantifiers depend on free parameters, with the main ones being the threshold used to determine whether two points are recurrent. The recurrence threshold parameter may be computed using the concept of maximum recurrence entropy, which turns it into a self-adjustable parameter. In a similar way, we propose a new method for automatically select the minimum diagonal line length (ℓmin), a critical parameter for recurrence quantifiers such as Determinism. Based on a cost-function, we show how to choose an adequate minimum recurrent diagonal line length to accurately compute diagonal-based recurrences. Our analysis reveals a well-defined limit for the credible applicability of recurrence analysis based on the number of data points. Moreover, we demonstrate a clear dependence of ℓmin on the number of dynamical epochs (pseudo-periods) sampled. We also demonstrate how our method can maximize the sensitivity of determinism with respect to changes in the stationary character of the time series. Finally, we provide two experimental examples to illustrate our approach.