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Not all (possibly) "random" sequences are created equal.

S Pincus, R E Kalman

    Proceedings of the National Academy of Sciences of the United States of America
    |April 15, 1997
    PubMed
    Summary
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    This study introduces approximate entropy (ApEn) to assess the randomness of finite sequences, a gap in traditional probability theory. ApEn quantifies sequence irregularity, offering a novel approach to understanding randomness in real-world data.

    Area of Science:

    • Information Theory
    • Probability Theory
    • Statistical Analysis

    Background:

    • Axiomatic probability theory does not adequately address the assessment of randomness for finite sequences.
    • Assessing randomness in single, finite sequences is a ubiquitous but unaddressed problem.
    • Classical measures of randomness have limitations, especially for finite sequences.

    Purpose of the Study:

    • To introduce and validate approximate entropy (ApEn) as a computable measure for assessing the randomness of single, finite sequences.
    • To demonstrate the utility of ApEn and associated deficit functions (def(m)) in refining concepts of probabilistic independence and normality.
    • To explore the application of ApEn to mathematical constants and sequences derived from irrational numbers.

    Main Methods:

    Related Experiment Videos

  • Application of approximate entropy (ApEn), a measure of sequential irregularity, to single sequences of finite and infinite length.
  • Identification of maximally irregular sequences for finite, finite-state sequences.
  • Development and application of deficit (def(m)) functions to quantify deviations from maximal irregularity.
  • Analysis of sequences including digits of mathematical constants (e, pi, radical2, radical3) and sequences from fractional parts of irrational multiples.

    • Main Results:

    • Approximate entropy (ApEn) provides a novel, multidimensional approach to assessing sequence randomness, applicable to finite sequences.
    • Deficit functions (def(m)) refine notions of probabilistic independence and normality, demonstrating utility with mathematical constants and irrational sequences.
    • Analytic results confirm the role and validity of axiomatic probability properties for specified sequences in the physical world.

    Conclusions:

    • Approximate entropy (ApEn) offers a practical and effective method for evaluating the randomness of finite sequences, addressing a key limitation in probability theory.
    • The developed deficit functions provide a quantitative measure of deviation from maximal irregularity, enhancing the analysis of sequence properties.
    • The study bridges theoretical probability with practical sequence analysis, offering insights applicable to real-world data and mathematical sequences.