Generalized (m,k)-Zipf law for fractional Brownian motion-like time series with or without effect of an additional linear trend

TL;DR

This study explores how the Zipf law applies to fractional Brownian motion signals, analyzing the influence of trends and persistence on the Zipf exponent, and proposing a generalized law for such time series.

Contribution

It introduces a novel application of the Zipf method to FBM signals, revealing how the Zipf exponent varies with signal persistence, trend effects, and signal length, and proposes a generalized (m,k)-Zipf law.

Findings

Zipf exponent varies as a power law with word length m.

Persistent signals are more affected by linear trends than antipersistent ones.

The law ζ' = |2H - 1| holds only near H=0.5.

Abstract

We have translated fractional Brownian motion (FBM) signals into a text based on two ''letters'', as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the $\zeta '$ exponent relating the word frequency and its rank on a log-log plot. We have studied the variation of the Zipf exponent(s) giving the relationship between the frequency of occurrence of words of length $m<8$ made of such two letters: $\zeta '$ is varying as a power law in terms of $m$. We have also searched how the $\zeta '$ exponent of the Zipf law is influenced by a linear trend and the resulting effect of its slope. We can distinguish finite size effects, and results depending whether the starting FBM is persistent or not, i.e. depending on the FBM Hurst exponent $H$. It seems then numerically proven that the Zipf exponent of a persistent signal is more influenced by the trend than that of an antipersistent signal. It appears that the conjectured law $\zeta ' = |2H-1|$ only holds near $H=0.5$. We have also introduced considerations based on the notion of a {\it time dependent Zipf law} along the signal.