Equivalences among Some Information Measures for Individual Sequences and Their Applications for Fixed-Length Coding Problems

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  • UYEMATSU Tomohiko
    Department of Information and Communications Engineering, Tokyo Institute of Technology
  • MATSUTA Tetsunao
    Graduate School of Science and Engineering, Saitama University

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<p>This paper proposes three new information measures for individual sequences and clarifies their properties. Our new information measures are called as the non-overlapping max-entropy, the overlapping smooth max-entropy, and the non-overlapping smooth max-entropy, respectively. These measures are related to the fixed-length coding of individual sequences. We investigate these measures, and show the following three properties: (1) The non-overlapping max-entropy coincides with the topological entropy. (2) The overlapping smooth max-entropy and the non-overlapping smooth max-entropy coincide with the Ziv-entropy. (3) When an individual sequence is drawn from an ergodic source, the overlapping smooth max-entropy and the non-overlapping smooth max-entropy coincide with the entropy rate of the source. Further, we apply these information measures to the fixed-length coding of individual sequences, and propose some new universal coding schemes which are asymptotically optimum.</p>

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