Statistical Inference for Microstate Distribution in Recurrence Plots

Recurrence is an important property to unveil patterns in dynamical systems. The recurrence plot (RP) is a matrix that summarizes recurrence information from data sequences. Recently, a new approach to the RP have been developed to generalize the standard absence or occurrence of recurrences in the RP. The microstate concept was introduced to statistically characterize samples of any size in the RP. We explore in this paper the relation between microstates of different sizes in the RP. We distinguish two approaches: the top-bottom and the down-up analysis. In the first we have information from larger microstates to make inference about smaller ones, in the second the opposite. We establish analytical results that statistically estimate the microstate probabilities between different sample sizes. The analytical results for the normalized entropy were tested using two time series: a simple white noise case and a series from a logistic equation in the chaotic scenario. For both cases, the top-bottom and the down-up analytical relations were tested with good results. To depict the actual probability distributions, we used the Lorenz oscillator for both, the top-down and the bottom-up strategies, showcasing the limitations and perspectives for future works. We believe our results can open new theoretical paths in the RP analysis.