On the frequencies of patterns of rises and falls

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 α min = 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 α 0 = 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.