### 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)∼R-ζ and P(ω)∼ω-ψ. 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.

Original language | English |
---|---|

Pages (from-to) | 446-452 |

Number of pages | 7 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1995 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*52*(1), 446-452. https://doi.org/10.1103/PhysRevE.52.446

**Correlations in binary sequences and a generalized Zipf analysis.** / Czirók, A.; Mantegna, Rosario N.; Havlin, Shlomo; Stanley, H. Eugene.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 52, no. 1, pp. 446-452. https://doi.org/10.1103/PhysRevE.52.446

}

TY - JOUR

T1 - Correlations in binary sequences and a generalized Zipf analysis

AU - Czirók, A.

AU - Mantegna, Rosario N.

AU - Havlin, Shlomo

AU - Stanley, H. Eugene

PY - 1995

Y1 - 1995

N2 - 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)∼R-ζ and P(ω)∼ω-ψ. 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.

AB - 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)∼R-ζ and P(ω)∼ω-ψ. 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.

UR - http://www.scopus.com/inward/record.url?scp=0001012115&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001012115&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.52.446

DO - 10.1103/PhysRevE.52.446

M3 - Article

AN - SCOPUS:0001012115

VL - 52

SP - 446

EP - 452

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 1

ER -