Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

Abstract

The observable outputs of many complex dynamical systems consist in time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorre- lation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time se- ries are used to study and characterize such systems. Typically, those algorithms produce Gaussian time series. However, real, experimentally observed time series are often non-Gaussian, and may follow distributions with a diversity of behaviors concerning the support, the symmetry or the tail properties. Given a correlated Gaussian time series, it is always possible to transform it into a time series with a different distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson’s cor- relation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties, from decays faster than exponential to heavy tail cases including power-laws, and we find how these properties affect the correlation of the final variables. We extend these results to Gaussian time series which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution.