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

TL;DR

This paper studies how transforming Gaussian variables affects their correlations, enabling the generation of non-Gaussian time series with desired long-range power-law autocorrelation properties for applications like financial data modeling.

Contribution

It provides a theoretical framework for understanding correlation transformations and introduces a generalized algorithm to generate non-Gaussian correlated time series with arbitrary distributions.

Findings

Correlation properties depend on the target distribution's support and tail behavior.

The proposed method successfully generates synthetic time series with specified power-law autocorrelations.

Application demonstrated by mimicking stock return time series.

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 autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, algorithms able to generate correlated noises reproducing the properties of real time series produce \textsl{Gaussian} outputs, while 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. Here, we study how the correlation of two Gaussian variables changes when they 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. As an application of our results, we propose how to generalize standard algorithms producing Gaussian power-law correlated time series in order to create synthetic time series with arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of a real time series: the absolute returns of stock prices.