A Brief Overview of Multifractal Time Series

Part 5: What one learns from the singularity spectra of multifractal signals

The singularity spectrum D(h) quantifies the degree of nonlinearity in the processes generating the output f(t) in a very compact way (see Fig. 7). For a linear fractal process the output of a system will have the same fractal properties (i.e., the same type of singularities) regardless of initial conditions or of driving forces. In contrast, nonlinear fractal processes will generate outputs with different fractal properties that depend on the input conditions or the history of the system. That is, the output of the system over extended periods of time will display different types of singularities.

Singularity spectra
Figure 7: Singularity spectra of the two signals considered in Figs. 4 and 5. Note the broad range of values of h with non-zero fractal dimensions for the multiplicative binomial process and contrast it to the "pulse"-like spectrum for the Cantor signal.
A classical example from physics is the Navier-Stokes equation for fluid dynamics (10). In the turbulent regime, this nonlinear equation generates a multifractal output with a characteristic singularity spectrum D(h) similar, for some types of turbulence, to D(h) for the binomial multiplicative process.
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