Analytical results in calculating the entropy of recurrence microstates

Abstract

Since the development of recurrence plots (RP) and recurrence quantification analysis (RQA), there has been a growing interest in many areas in studying physical systems using recursion techniques. In particular, as part of the RQAs, we observed the development of the concept of recurrence microstates, defined as small blocks obtained from a recurrence graph. It can be shown that some other RQAs can be calculated as a function of recurrence microstates, and the probabilities of occurrences of these microstates can define an information entropy, the so-called entropy of recurrence microstates. It was also observed that recurrence microstates and recurrence entropy can distinguish between correlated and uncorrelated stochastic and deterministic states, due to their symmetry properties. In this paper we propose analytical expressions for calculating the entropy of recurrence microstates, avoiding the need to sample a large set of recurrence microstates. The results can be particularly important in quantifying small amounts of data, where significant sampling of microstates may not be possible. In this paper we propose analytical expressions to compute the entropy of recurrence microstates avoiding the need to sample a large set of recurrence microstates. We show that our results are accurate for cases where the probability distribution function is known. For other situations, the results can be calculated approximately. Another important fact is that the approximate results can be generalized to any size of microstate, making them a powerful tool for calculating the entropy of recurrence. Our approximate methods allow us to know what these properties are and how to exploit this quantifier in the best possible way, with minimal memory usage. Finally, we show that our analytical results are in remarkable agreement with numerical simulations.