Codifference as a practical tool to measure interdependence

Abstract

Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as from real experimental data. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.

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• First passage time moments of asymmetric Lévy flights
• First passage properties of asymmetric Lévy flights
• Probabilistic properties of detrended fluctuation analysis for Gaussian processes
• Tempered relaxation equation and related generalized stable processes
• Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates
• Cross-codifference for bidimensional VAR(1) models with infinite variance
• Nonlinear friction in underdamped anharmonic stochastic oscillators
• Discrete $q$-exponential limit order cancellation time distribution
• Scaling characteristics of fractional diffusion processes in the presence of power-law distributed random noise
• On the distribution of the product of two continuous random variables with an application to electricity market transactions. Finite and infinite-variance case