On the frequencies of patterns of rises and falls

J.M. Luck

Physica A: Statistical Mechanics and its Applications, 2014, vol. 407, issue C, 252-275

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.

Keywords: Data series; Patterns; Rises and falls; Entropy; Multifractals; Permutations (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:407:y:2014:i:c:p:252-275

DOI: 10.1016/j.physa.2014.04.010

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