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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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    References listed on IDEAS

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    1. Bruce Hajek, 1985. "Extremal Splittings of Point Processes," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 543-556, November.
    2. Russell Lyons, 2000. "Singularity of Some Random Continued Fractions," Journal of Theoretical Probability, Springer, vol. 13(2), pages 535-545, April.
    3. Steve Pincus, 2008. "Approximate Entropy as an Irregularity Measure for Financial Data," Econometric Reviews, Taylor & Francis Journals, vol. 27(4-6), pages 329-362.
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