A Source Model with Probability Distribution over Word Set and Recurrence Time Theorem

Masayuki Goto, Toshiyasu Matsushima, Shigeichi Hirasawa

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    Nishiara and Morita defined an i.i.d. word-valued source which is defined as a pair of an i.i.d. source with a countable alphabet and a function which transforms each symbol into a word over finite alphabet. They showed the asymptotic equipartition property (AEP) of the i.i.d. word-valued source and discussed the relation with source coding algorithm based on a string parsing approach. However, their model is restricted in the i.i.d. case and any universal code for a class of word-valued sources isn't discussed. In this paper, we generalize the i.i.d. word-valued source to the ergodic word-valued source which is defined by an ergodic source with a countable alphabet and a function from each symbol to a word. We show existence of entropy rate of the ergodic word-valued source and its formula. Moreover, we show the recurrence time theorem for the ergodic word-valued source with a finite alphabet. This result clarifies that Ziv-Lempel code (ZL77 code) is universal for the ergodic word-valued source.

    Original languageEnglish
    Pages (from-to)2517-2525
    Number of pages9
    JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
    VolumeE86-A
    Issue number10
    Publication statusPublished - 2003 Oct

    Fingerprint

    Recurrence
    Probability distributions
    Probability Distribution
    Theorem
    Entropy
    Model
    Countable
    Equipartition
    Source Coding
    Parsing
    Strings
    Transform
    Generalise

    Keywords

    • Recurrence time
    • Word sequences
    • Word set
    • Word-valued source
    • Ziv-Lempel code

    ASJC Scopus subject areas

    • Electrical and Electronic Engineering
    • Hardware and Architecture
    • Information Systems

    Cite this

    @article{1bf1af604bcc4ef29099130ddd116d28,
    title = "A Source Model with Probability Distribution over Word Set and Recurrence Time Theorem",
    abstract = "Nishiara and Morita defined an i.i.d. word-valued source which is defined as a pair of an i.i.d. source with a countable alphabet and a function which transforms each symbol into a word over finite alphabet. They showed the asymptotic equipartition property (AEP) of the i.i.d. word-valued source and discussed the relation with source coding algorithm based on a string parsing approach. However, their model is restricted in the i.i.d. case and any universal code for a class of word-valued sources isn't discussed. In this paper, we generalize the i.i.d. word-valued source to the ergodic word-valued source which is defined by an ergodic source with a countable alphabet and a function from each symbol to a word. We show existence of entropy rate of the ergodic word-valued source and its formula. Moreover, we show the recurrence time theorem for the ergodic word-valued source with a finite alphabet. This result clarifies that Ziv-Lempel code (ZL77 code) is universal for the ergodic word-valued source.",
    keywords = "Recurrence time, Word sequences, Word set, Word-valued source, Ziv-Lempel code",
    author = "Masayuki Goto and Toshiyasu Matsushima and Shigeichi Hirasawa",
    year = "2003",
    month = "10",
    language = "English",
    volume = "E86-A",
    pages = "2517--2525",
    journal = "IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences",
    issn = "0916-8508",
    publisher = "Maruzen Co., Ltd/Maruzen Kabushikikaisha",
    number = "10",

    }

    TY - JOUR

    T1 - A Source Model with Probability Distribution over Word Set and Recurrence Time Theorem

    AU - Goto, Masayuki

    AU - Matsushima, Toshiyasu

    AU - Hirasawa, Shigeichi

    PY - 2003/10

    Y1 - 2003/10

    N2 - Nishiara and Morita defined an i.i.d. word-valued source which is defined as a pair of an i.i.d. source with a countable alphabet and a function which transforms each symbol into a word over finite alphabet. They showed the asymptotic equipartition property (AEP) of the i.i.d. word-valued source and discussed the relation with source coding algorithm based on a string parsing approach. However, their model is restricted in the i.i.d. case and any universal code for a class of word-valued sources isn't discussed. In this paper, we generalize the i.i.d. word-valued source to the ergodic word-valued source which is defined by an ergodic source with a countable alphabet and a function from each symbol to a word. We show existence of entropy rate of the ergodic word-valued source and its formula. Moreover, we show the recurrence time theorem for the ergodic word-valued source with a finite alphabet. This result clarifies that Ziv-Lempel code (ZL77 code) is universal for the ergodic word-valued source.

    AB - Nishiara and Morita defined an i.i.d. word-valued source which is defined as a pair of an i.i.d. source with a countable alphabet and a function which transforms each symbol into a word over finite alphabet. They showed the asymptotic equipartition property (AEP) of the i.i.d. word-valued source and discussed the relation with source coding algorithm based on a string parsing approach. However, their model is restricted in the i.i.d. case and any universal code for a class of word-valued sources isn't discussed. In this paper, we generalize the i.i.d. word-valued source to the ergodic word-valued source which is defined by an ergodic source with a countable alphabet and a function from each symbol to a word. We show existence of entropy rate of the ergodic word-valued source and its formula. Moreover, we show the recurrence time theorem for the ergodic word-valued source with a finite alphabet. This result clarifies that Ziv-Lempel code (ZL77 code) is universal for the ergodic word-valued source.

    KW - Recurrence time

    KW - Word sequences

    KW - Word set

    KW - Word-valued source

    KW - Ziv-Lempel code

    UR - http://www.scopus.com/inward/record.url?scp=0242496295&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0242496295&partnerID=8YFLogxK

    M3 - Article

    VL - E86-A

    SP - 2517

    EP - 2525

    JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

    JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

    SN - 0916-8508

    IS - 10

    ER -