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5 Multifractal characteristics of singular signals

Paweł Oświęcimka and Ludovico Minati


Multifractal formalism is one of the most potent tools for characterizing the singular behavior of signals encountered across various scientific and engineering situations. This conceptually advanced methodology is made accessible to experimentalists via the development of algorithms capable of determining a time-serie’s local scaling properties through a set of scaling exponents and an associated singularity spectrum. By determining a signal’s multifractal properties, the temporal organization of the underlying various amplitude fluctuations can be quantitatively described within a unique scheme of the correlation structure. In this work, we demonstrate that the degree of complexity of a signal and its hierarchical organization’s character are reflected in the shape of the singularity spectrum. A stark example of this is offered by financial time series, wherein well-developed multifractal spectra quantify the hierarchical structure of the data and the heterogeneity of singular behavior. On the other hand, for signals without a cascade-like organization of the singularities, the singularity spectrum’s interpretation must be undertaken with extreme care. In such cases, artifactual singularities can be produced by processes that are not firmly interrelated. Thus, the fluctuation structure does not truly reveal the hierarchy of the organization. We show examples of such time series produced by nonlinear dynamical systems, particularly the Saito chaos generator, within which twofold dynamics, one related to periodic component and the other related to hysteresis, are not hierarchically nested. Consequently, a local scaling analysis based on the wavelet transform must be adequately applied to identify the structure of isolated singularities.

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