A Brief Overview of Multifractal Time Series

Part 4: The singularity spectra of multifractal signals

Our analysis becomes more complex if instead of a single type of singularity, the signal of interest has multiple types of singularities. As an example, consider the signal in Fig. 5 which is also a Devil's staircase (i.e., Fig. 4) because of its many singularities. But in contrast to the signal of Fig. 4, the types of singularities vary considerably. The reason for this variation is made clear by the top panel in Fig. 5. The type of fluctuations in local increments vary considerably even for the fourth iteration.

Multiplicative binomial cascade
Figure 5: A multifractal Devil's staircase. Top: Four iteration steps in the building of a multiplicative binomial cascade. The set is generated by partitioning the mass of the segment into two parts of equal length but un-equal densities. For the case shown, the left half of the segment receives 1/4th of the mass while the right half receives 3/4th of the mass. Bottom: One can generate a Devil's staircase type of signal by integrating the set generated according to the previous rule. Such a signal is shown in this panel. Note the presence of numerous cusp-like features in the signal. These cusps indicate the times where singularities occur. Because of the local variations in the mass distribution of the binomial cascade of the Top panel, the singularities in this case are of several different types.

To quantify the variation in the local singularities of the signal of Fig. 5, we calculate the value of h at every singularity. Figure 6 shows the signal again and also, by a color coding, the value of h. Clearly hi can take many different values. Moreover, by focusing on a single color, i.e., a single value of h, one can uncover the fractal structure of the corresponding set of singularities.

Singularity spectrum of the multiplicative binomial cascade
Figure 6: Singularity decomposition of the multiplicative binomial process of Fig. 5. (a) Devil staircase after 9 iterations. (b) Position and value of the different singularities for the signal in (a). (c) Color coding of (b). The dark blue background indicates absence of singularities. The color spectrum goes from dark blue to green to yellow and to reddish brown. Blue indicates small values of h while reddish brown indicates large values of h. Note that no singularities appear at the edges because we do not enforce periodic boundary conditions on the signal and hence cannot perform calculations close to the edges. (d) Decomposition of the singularities into different sets corresponding to different values of h. The top panel displays singularities with values of h approximately two standard deviations smaller than the mean h = 0.6. The middle panel displays singularities with the average h = 1.1. Finally, the bottom panel displays singularities with values of h approximately two standard deviations larger than the mean h = 1.6. (Note: The color panels in (d) have bars of a single color, unfortunately color and resolution conflicts may give rise to bars of different colors.)
Previous: Part 3: The fractal dimension of the singular behavior