Missing ordinal patterns in correlated noises

Recent research aiming at the distinction between deterministic or stochastic behavior in observational time series has looked into the properties of the ordinal patterns [C. Bandt, B Pompe, Phys. Rev. Lett 88 (2002) 174102] In particular, new insight has been obtained considering the emergence of the so-called forbidden ordinal patterns [J M S Zambrano, M A F Sanjuan, Europhys Lett 79 (2007) 50001]. It was shown that deterministic one-dimensional maps always have forbidden ordinal patterns, in contrast with time series generated by an unconstrained stochastic process in which all the patterns appear with probability one. Techniques based on the comparison of this property in an observational time series and in white Gaussian noise were implemented. However, the comparison with correlated stochastic processes was not considered. In this paper we used the concept of missing ordinal patterns to study their decay rate as a function of the time series length in three stochastic processes with different degrees of correlation. fractional Brownian motion, fractional Gaussian noise and, noises with f(-k) power spectrum. We show that the decay rate of missing ordinal patterns in these processes depend on their correlation structures We finally discuss the implications of the present results for the use of these properties as a tool for distinguishing deterministic from stochastic processes (C) 2010 Elsevier B.V. All rights reserved