Correlations in binary sequences and a generalized Zipf analysis

Andras Czirók, Rosario N. Mantegna, Shlomo Havlin, and H. Eugene Stanley
Phys. Rev. E 52, 446 – Published 1 July 1995
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Abstract

We investigate correlated binary sequences using an n-tuple Zipf analysis, where we define ‘‘words’’ as strings of length n, and calculate the normalized frequency of occurrence ω(R) of ‘‘words’’ as a function of the word rank R. We analyze sequences with short-range Markovian correlations, as well as those with long-range correlations generated by three different methods: inverse Fourier transformation, Lévy walks, and the expansion-modification system. We study the relation between the exponent α characterizing long-range correlations and the exponent ζ characterizing power-law behavior in the Zipf plot. We also introduce a function P(ω), the frequency density, which is related to the inverse Zipf function R(ω), and find a simple relationship between ζ and ψ, where ω(R)∼Rζ and P(ω)∼ωψ. Further, for Markovian sequences, we derive an approximate form for P(ω). Finally, we study the effect of a coarse-graining ‘‘renormalization’’ on sequences with Markovian and with long-range correlations.

  • Received 12 January 1995

DOI:https://doi.org/10.1103/PhysRevE.52.446

©1995 American Physical Society

Authors & Affiliations

Andras Czirók, Rosario N. Mantegna, Shlomo Havlin, and H. Eugene Stanley

  • Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215
  • Department of Atomic Physics, Eötvös University, Puskin u. 5-7, Budapest, 1088 Hungary
  • Dipartimento di Energetica ed Applicazioni di Fisica, Università, di Palermo, Palermo, I-90128, Italy
  • Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

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Vol. 52, Iss. 1 — July 1995

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