Analysis of the autocorrelation function for time series with higher-order temporal correlations: An exponential case

TL;DR

This paper develops an analytical model to understand how higher-order temporal correlations, specifically correlated burst sizes, influence the autocorrelation function in time series, using exponential distributions for demonstration.

Contribution

It introduces a copula-based model for higher-order correlations and derives an analytical solution for the autocorrelation function with correlated burst sizes.

Findings

Analytical autocorrelation function derived for correlated burst sizes.

Correlations between burst sizes affect the decay behavior of autocorrelation.

Model demonstrates impact of higher-order correlations on temporal dynamics.

Abstract

Temporal correlations in the time series observed in various systems have been characterized by the autocorrelation function. Such correlations can be explained by heavy-tailed interevent time distributions as well as by correlations between interevent times. The latter is called higher-order temporal correlations, and they have been captured by the notion of bursts; a burst indicates a set of consecutive events that rapidly occur within a short time period and are separated from other bursts by long time intervals. The number of events in the burst is called a burst size. Some empirical analyses have shown that consecutive burst sizes are correlated with each other. To study the impact of such correlations on the autocorrelation function, we devise a model generating a time series with higher-order temporal correlations by employing the copula method. We successfully derive the analytical solution of the autocorrelation function of the model time series for arbitrary distributions of interevent times and burst sizes when consecutive burst sizes are correlated. For the demonstration of our analysis, we adopt exponential distributions of interevent times and burst sizes to understand how the correlations between consecutive burst sizes affect the decaying behavior of the autocorrelation function.