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

Area of Science:

• Complex Systems
• Statistical Physics
• Time Series Analysis

Background:

• Many complex systems generate time series with diverse autocorrelation behaviors, often power-law.
• Standard algorithms generate Gaussian time series, but real-world data is frequently non-Gaussian.
• Transforming Gaussian time series to non-Gaussian distributions can alter autocorrelation functions.

Purpose of the Study:

• To analytically and numerically investigate how transforming Gaussian time series affects autocorrelation functions.
• To understand the impact of various destination distributions (bounded, unbounded, symmetric, asymmetric, heavy-tailed) on correlation.
• To develop a generalized method for creating non-Gaussian time series with controlled power-law correlations.

Main Methods:

• Analytical derivation of how Pearson's correlation changes under distribution transformation.
• Numerical simulations of Gaussian time series transformed into various non-Gaussian distributions.
• Extension of findings to time series and development of a generalized algorithm.
• Application to generating synthetic stock return time series.

Main Results:

• The study quantifies how marginal distribution transformations impact the autocorrelation function of time series.
• Specific behaviors of bounded, unbounded, symmetric, asymmetric, and heavy-tailed distributions on correlation are identified.
• A method is proposed to generalize existing algorithms for generating Gaussian power-law correlated time series.
• The generalized algorithm successfully creates synthetic time series mimicking real stock return data distributions and autocorrelation.

Conclusions:

• The transformation of Gaussian time series to non-Gaussian distributions can be controlled to preserve or modify autocorrelation properties.
• The developed method allows for the generation of synthetic non-Gaussian time series with arbitrary marginal distributions and controlled power-law correlations.
• This approach has practical applications, such as modeling financial market dynamics accurately.