Frontiers in Physiology (Jun 2012)
Introduction to multifractal wavelet and detrended fluctuation analyses
Abstract
Physiological and behavioural phenomena are often complex, characterized by variations in time series. Variations in time series reflect how these phenomena organize into coherent structures by interactions that span multiple scales in both time and space. The present tutorial is an introduction to multifractal analyses that can identify these scale invariant interactions within time series by its multifractal spectrum. The multifractal spectrum can be estimated directly from scale-dependent measurements or from its q-order statistics. The tutorial emphasizes the most common scale-dependent measurements defined by the wavelet transforms and the detrended fluctuation analyses. The tutorial also emphasizes common features of all multifractal analyses, like the choice of linear regression method, scaling range and elimination of spurious singularities, which are important for a robust estimation of the multifractal spectrum. The tutorial ends with two brief examples where multifractal analyses are employed to time series from multifractal models and the complex phenomena of cognitive performance. References to available software for multifractal analyses are included at the end of the tutorial. The main aim of the tutorial is to give the reader an introduction to multifractal analyses without the extensive technicalities typically provided in mathematical journals.
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