Generating synthetic data for analysing multifractal Brownian signals

Abstract: A signal that is a realisation from an experiment or from a process characterised by a set of differential equations may exhibit intermittent behaviour. That is, sudden and dramatic oscillations that are dynamically important but can be difficult to represent with simple stochastic models, and result in long tails to the observed probability distribution function. Such phenomena are typically attributes of a nonlinear process and nonlinearity may be tested for in several ways. The Gottwald and Melbourne 0-1 test is one means if the signal is the output of a deterministic process. More broadly, the surrogate data hypothesis testing framework (Schreiber and Schmitz, 1996) may be adopted. This is an empirical approach for investigating signs of intermittency, chaos, or nonlinearity depending on the means used to characterise the signal and its surrogates. However, it cannot be used to test for additional properties of a complex, nonlinear signal, conditioned on the observed intermittency. In this talk I will present a method in the spirit of the surrogate data framework that solves this problem because the surrogate data preserve the intermittent nature of the signal (in terms of its pointwise Holder regularity) as well as the values of the signal, but destroy the dependence between these two characteristics.  Examples are given in terms of pointwise and local Holder regularity of time-series and the coupling between elevation and pointwise Holder regularity in landscape surfaces.