Explicit construction of joint multipoint statistics in complex systems

TL;DR

This paper develops an explicit formula for the joint multipoint probability density function of a multifractal field, enabling the systematic modeling of synthetic multifractal phenomena in complex systems with multiscale interactions.

Contribution

It introduces a novel explicit construction of joint multipoint statistics for multifractal fields using a scale mixture of fractional Ornstein-Uhlenbeck processes.

Findings

Derived an explicit formula for joint multipoint PDFs of multifractal fields

Modelled synthetic multifractal phenomena with a new statistical approach

Potential applications in wind, financial, and geophysical data modeling

Abstract

Complex systems often involve random fluctuations for which self-similar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent $H$, provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov-Oboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional Ornstein-Uhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or non-Gaussian features in geophysical or geospatial settings.