Explicit construction of joint multipoint statistics in complex systems

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.

Superstatistical wind fields from point-wise atmospheric turbulence measurements

“…In the following, we first derive a joint multipoint statistics for such a superstatistical random field (see also [48]) and prove that its velocity increment statistics (a two-point quantity) is consistent with the Kolmogorov-Oboukhov (K62) model of turbulence. Hence, in a first iteration, we approximate the atmospheric wind field as a homogeneous and isotropic random field.…”

“…fluctuate with respect to the parameter ξ, which follows a lognormal distribution (we also refer the reader to [48])…”

“…Nonetheless, a statistical model for the velocity increments v(r) is not sufficient for generating a complete random field u(x), as it solely relies on two-point statistical quantities [54]. A generalization to a multipoint statistics that consist of a superposition of multivariate Gaussian statistics is however possible [48] and yields the joint n-point PDF…”

“…where S 2ξ,r (r) = 2C ξ,rr (0) − 2C ξ,rr (r). Latter relation holds only for even n and structure functions of odd order are zero as the model does not account for skewness [48].…”

“…where ln + denotes the positive branch of the logarithm and where we introduced the same model parameters as for the original K62 model in Eq. ( 2) (see also [48] for further information). Moreover, we introduce Kummer's confluent hypergeometric function…”