New Statistical Approach to the Analysis of Hierarchical Data

Abstract:

Many variables possess a hierarchical structure reflected in how their increments vary in space and/or time. Quite commonly the increments (a) fluctuate in a highly irregular manner; (b) possess symmetric, non-Gaussian frequency distributions characterized by heavy tails that often decay with separation distance or lag; (c) exhibit nonlinear power-law scaling of sample structure functions in a midrange of lags, with breakdown in such scaling at small and large lags; (d) show extended power-law scaling (ESS) at all lags; and (e) display nonlinear scaling of power-law exponent with order of sample structure function. Some interpret this to imply that the variables are multifractal, which explains neither breakdowns in power-law scaling nor ESS. We offer an alternative interpretation consistent with all above phenomena. It views data as samples from stationary, anisotropic sub-Gaussian random fields subordinated to truncated fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). The fields are scaled Gaussian mixtures with random variances. Truncation of fBm and fGn entails filtering out components below data measurement or resolution scale and above domain scale. Our novel interpretation of the data allows us to obtain maximum likelihood estimates of all parameters characterizing the underlying truncated sub-Gaussian fields. These parameters in turn make it possible to downscale or upscale all statistical moments to situations entailing smaller or larger measurement or resolution and sampling scales, respectively. They also allow one to perform conditional or unconditional Monte Carlo simulations of random field realizations corresponding to these scales. Aspects of our approach are illustrated on field and laboratory measured porous and fractured rock permeabilities, as well as soil texture characteristics and neural network estimates of unsaturated hydraulic parameters in a deep vadose zone near Phoenix, Arizona. We also use our approach to investigate the scaling of statistics characterizing vertical increments in neutron porosity data, and extreme values of these increments, from six deep boreholes in three oil and/or gas producing depositional environments.