True and false forbidden patterns in deterministic and random dynamics

Abstract

In this letter we discuss some properties of order patterns both in deterministic and random orbit generation. As it turns out, the orbits of one-dimensional maps have always forbidden patterns, i.e., order patterns that cannot occur, in contrast with random time series, in which any order pattern appears with probability one. However, finite random sequences may exhibit "false" forbidden patterns with non-vanishing probability. In this case, forbidden patterns decay with the sequence length, thus unveiling the random nature of the sequence. Last but not least, true forbidden patterns are robust against noise and disintegrate with a rate that depends on the noise level. These properties can be embodied in a simple method to distinguish deterministic, finite time series with very high levels of observational noise, from random ones. We present numerical evidence for white noise.

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“…Statistical methods in order to obtain a model of the mean process have been the classical approach, leading to autoregressive, integrated and moving average models [3]. Characterization of non-linear time series, mainly chaotic, has also attracted the interest of the scientific community [8,9], where phase space reconstruction, spectral analysis or wavelets methods have been revealed to be good indicators of the underlying dynamics of a real time series.Recently the study of the order patterns has been proposed as a technique of evaluating the determinism of a given time series [10,11,12]. Consider a discrete information source emitting a series of observable values [x 1 , x 2 , .…”

“…Nevertheless, when the series corresponds to a chaotic variable there are some patterns that cannot be encountered in the data due to the underlying deterministic structure: they are the so-called forbidden patterns. It has been demonstrated that most chaotic systems exhibit forbidden patterns, and that in many cases (ergodic finitealphabet information sources) the measure of the number of this patterns is related to other classic metric entropy rates (e.g., the Lyapunov exponent) [12].…”

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