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A Matrix-Based Regularity Measure for Symbolic Sequences

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  • Wine, Trevor

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. The logarithms of the norms of the sequences under test are 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. The random vectors associated with the method are also found to have a well-defined entropy rate, leading to a definition of symbolic sequence homogeneity entropy. Numerical methods for estimating the entropy are also developed. Potential applications are explored, including time series and ergodic systems.

Suggested Citation

  • Wine, Trevor, 2020. "A Matrix-Based Regularity Measure for Symbolic Sequences," OSF Preprints vpg8h_v1, Center for Open Science.
    • Handle: RePEc:osf:osfxxx:vpg8h_v1
    DOI: 10.31219/osf.io/vpg8h_v1
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