Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

摘要

The need of understanding and modelling the space–time variability ofnatural processes in hydrological sciences produced a large body ofliterature over the last thirty years. In this context, a multifractalframework provides parsimonious models which can be applied to a wide-scalerange of hydrological processes, and are based on the empirical detection ofsome patterns in observational data, i.e. a scale invariant mechanismrepeating scale after scale. Hence, multifractal analyses heavily rely onavailable data series and their statistical processing. In such analyses,high order moments are often estimated and used in model identification andfitting as if they were reliable. This paper warns practitioners againstthe blind use in geophysical time series analyses of classical statistics, which isbased upon independent samples typically following distributions ofexponential type. Indeed, the study of natural processes reveals scalingbehaviours in state (departure from exponential distribution tails) and intime (departure from independence), thus implying dramatic increase of biasand uncertainty in statistical estimation. Surprisingly, all thesedifferences are commonly unaccounted for in most multifractal analyses ofhydrological processes, which may result in inappropriate modelling, wronginferences and false claims about the properties of the processes studied.Using theoretical reasoning and Monte Carlo simulations, we find that thereliability of multifractal methods that use high order moments(>3) is questionable. In particular, we suggest that, because ofestimation problems, the use of moments of order higher than two should beavoided, either in justifying or fitting models. Nonetheless, in mostproblems the first two moments provide enough information for the mostimportant characteristics of the distribution.