A Matrix-Based Regularity Measure for Symbolic Sequences

Abstract

A set of statistics is developed for defining and determining the regularity of symbolic sequences. This is achieved by testing a given sequence against a template set with fixed asymptotic symbol proportions $p_i$, $\sum_i p_i = 1$. The process centers on casting the sequence into matrix product form, and defining a parametrized probability distribution via the entrywise norms. The parameter allows varying the weighting between strict adherence to the template sequences, and a generalized Bernoulli randomness. Numerical methods for estimating the entropy of the resulting probability distributions are also developed. The logarithms of the norms of the sequences under test are further shown to satisfy a central limit theorem. This allows the assignment of z-scores, and rigorous comparison of the regularity between sequences of different types. Potential applications are explored, including time series and ergodic systems.