Date Published: May 6, 2019
Publisher: Public Library of Science
Author(s): Ian D. Colley, Roger T. Dean, Lawrence M. Ward.
1/f fluctuations have been described in numerous physical and biological processes. This noise structure describes an inverse relationship between the intensity and frequency of events in a time series (for example reflected in power spectra), and is believed to indicate long-range dependence, whereby events at one time point influence events many observations later. 1/f has been identified in rhythmic behaviors, such as music, and is typically attributed to long-range correlations. However short-range dependence in musical performance is a well-established finding and past research has suggested that 1/f can arise from multiple continuing short-range processes. We tested this possibility using simulations and time-series modeling, complemented by traditional analyses using power spectra and detrended fluctuation analysis (as often adopted more recently). Our results show that 1/f-type fluctuations in musical contexts may be explained by short-range models involving multiple time lags, and the temporal ranges in which rhythmic hierarchies are expressed are apt to create these fluctuations through such short-range autocorrelations. We also analyzed gait, heartbeat, and resting-state EEG data, demonstrating the coexistence of multiple short-range processes and 1/f fluctuation in a variety of phenomena. This suggests that 1/f fluctuation might not indicate long-range correlations, and points to its likely origins in musical rhythm and related structures.
1/f-type correlations have been identified in numerous physical and biological systems, often being described as ‘ubiquitous’ [1, 2]. This phenomenon refers to a pattern of noise over time that exhibits a roughly 1:-1 relationship between power and frequency in a log-log plot of a time series that has been converted to the frequency domain . 1/f can also be thought of as a 1:1 relationship between the amount of fluctuation within a window of observations, and the size of the window as the window is incremented . 1/f is often conceptualized as the center of a continuum of noise color that includes white noise or random fluctuations at one end, and red noise (also called Brownian motion) or deterministic fluctuations at the other end. 1/f, or pink noise, therefore represents a flexible system that fluctuates within a set of constraints . Systems exhibiting 1/f-type noise are stable but adaptable, and thus have been suggested as indicative of a healthily functioning biological system [4, 6–14]. However, their widespread occurrence both in physical and biological systems somewhat undermines this proposed interpretation, suggesting rather that 1/f is simply compatible with normal function.
Before investigating the hi-hat data and running related simulations, we wanted to compare PSD and DFA to see if they would yield consistent results on a dataset known to produce 1/f noise. To do this, we replicated past 1/f simulations  and analyzed the series using both PSD and DFA. The simulation involved three steps. First we generated a white noise series, WN, of length 1,024 and with arbitrary time units. Second, we made three copies of the white noise series, each with a different moving average, MA, filter applied. The MA filters differed in their window size, q. In other words, q is the number of successive events in the WN series that were averaged by the MA filter. Third, we summed all four series with different weights, θ, given to the each series, where θ represents the influence of each timescale/MA window size. As a formula, the process is:
To summarize, first we will revisit our opening questions. 1) ARIMA (short-range) simulations of the hi-hat series (with multiple time lags) did produce moderate-strong 1/f structures. 2) “Long-range” structures can result from a variety of short-range, ARIMA structures, and this is detectable by both PSD and DFA. This was even seen in the unusual case of large AR models with non-significant lags (that is, lags with zero-value coefficients in the model) between significant ones. We have also demonstrated some AR structures that reliably produce 1/f, namely those with positive coefficients at AR1. 3) The coexistence of low order AR and 1/f in a small but diverse set of human processes suggests that the short-range correlations deserve broader consideration as an explanation for 1/f processes, and specifically, those involving multiple lags.
We used various time series analysis functions in R Studio to fit models and simulate data. These included auto.arima in the forecast library, various functions in the tseries library, dfa in the nonlinearTseries library, and arima.sim in the stats package. The R functions are mentioned as the models and simulations are described in the Results.