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On the frequencies of patterns of rises and falls

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  • Luck, J.M.

Abstract

We investigate the probability of observing a given pattern of n rises and falls in a random stationary data series. The data are modeled as a sequence of n+1 independent and identically distributed random numbers. This probabilistic approach has a combinatorial equivalent, where the data are modeled by a random permutation on n+1 objects. The probability of observing a long pattern of rises and falls decays exponentially with its length n in general. The associated decay rate α is interpreted as the embedding entropy of the pattern. This rate is evaluated exactly for all periodic patterns. In the most general case, it is expressed in terms of a determinant of generalized hyperbolic or trigonometric functions. Alternating patterns have the smallest rate αmin=ln(π/2)=0.451582…, while other examples lead to arbitrarily large rates. The probabilities of observing uniformly chosen random patterns are demonstrated to obey multifractal statistics. The typical value α0=0.806361… of the rate plays the role of a Lyapunov exponent. A wide range of examples of patterns, either deterministic or random, is also investigated.

Suggested Citation

  • Luck, J.M., 2014. "On the frequencies of patterns of rises and falls," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 252-275.
  • Handle: RePEc:eee:phsmap:v:407:y:2014:i:c:p:252-275
    DOI: 10.1016/j.physa.2014.04.010
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    References listed on IDEAS

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    1. Le Doussal, Pierre & Monthus, Cécile, 2003. "Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 140-198.
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