A Brief Overview of Multifractal Time Series

Part 3: The fractal dimension of the singular behavior

The next problem is to quantify the "frequency" in the signal of a particular value h of the singularity exponents hi. Let us first assume that our signal is monofractal. Different possibilities can be considered. For example, the set of times with singular behavior {ti} may be a finite fraction of the time series and homogeneously distributed over the signal. But {ti} may also be an asymptotically infinitesimal fraction of the entire signal and have a very heterogeneous structure. That is, the set {ti} may be a fractal itself. In either case, it is useful to quantify the properties of the sets of singularities in the signal by calculating their fractal dimensions (8).

Consider the signal in Fig. 4. This type of signal is usually called a Devil's staircase because it takes constant values except at a subset of points where it changes discontinuously (2, 4). At those points, the function f(t) has singularities. Moreover, all singularities are of the same type --i.e., the signal is monofractal.

Figure 4: A monofractal Devil's staircase. Top: Four iteration steps in the building of a Cantor set. The set is generated by removing from the middle of a segment a region with half the length of the segment. This rule generates a "dust'' of points with equal mass and a fractal distribution in time. The distribution is fractal because there are holes of all sizes between the dust. The fractal dimension of this dust is D = 1/2 (see text for details). Bottom: One can generate a Devil's staircase type of signal by integrating the fractal dust generated according to the previous rule. Such a signal is shown in this panel. Note the discontinuities in the signal (2 k for the kth iteration). These discontinuities are the times where singularities occur. The singularities in this case are all of the same type. Hence one has a single value of h in the signal. The corresponding fractal dimension is also D = 1/2.

Because the signal of Fig. 4 is deterministic, we can easily identify the position of the singularities. Their positions are shown in the top panel of Fig. 4. One can see that the singularity points arise from the iteration of a Cantor set rule. The signal in the bottom panel of Fig. 4 arises from integrating the "dust'' generated by the Cantor rule (8).

One can easily calculate the fractal dimension of the Cantor set of singularities by using box counting methods. The fractal dimension is, as usual, given by the relation

(4)
where n(r) is the number of boxes of radius r needed to cover the fractal dust. For the deterministic fractal shown, after k iterations, one has r = (1/4) k, and n(r) = 2 k, yielding
(5)
In the multifractal formalism, one says that the signal of Fig. 4 has a single type of singularity hi = 1/2 and that the support of that singularity has fractal dimension D(h=1/2) = 1/2. The curve D(h) is called the singularity spectrum of the time series, which for this case is zero everywhere except at a single point h = 1/2.

The signals in Figs. 1a,c are also monofractal. They are usually called fractional Brownian motion. For the signal in Fig. 1a we have h = -0.8 while for the signal in Fig. 1c we have h = 0.2. But in contrast with the devil's staircase of Fig. 4, for which singularities appear only for a very small and heterogeneous set of times, singularities appear uniformly throughout the signals in Figs. 1a,c. Hence, the fractal dimension of the set of singularities is one, the dimension of a line.

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