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Multifractal Analysis

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Encyclopedia of Nonlinear Science

MULTIFRACTAL ANALYSIS

A fractal is a geometrical shape that shows self-similarity, meaning that parts appear similar to the set as a whole (See Fractals). A multifractal is a fractal with a probability measure attributed to its geometrical support set. A typical multifractal can have different fractal dimensions in different parts of its support set, depending on the multifractal measure chosen. Typical examples of multifractals are attractors of nonlinear mappings, where the relevant probability measure is given by the invariant measure of the map. Often, quite a complicated structure is observed, and the invariant density may diverge at infinitely many points in the phase space with different exponents, that is, an entire spectrum of singularities is generated.

To statistically analyze this complex behavior, it is useful to evaluate the so-called Rényi dimensions D_q (Rényi, 1970). These are generalizations of the usual box dimensions (or capacity) that contain information not only on the topological structure of the fractal but on the probability measure as well. The simplest definition of the Rényi dimensions is as follows. Cover the fractal with small d-dimensional cubes (“boxes”) of volume ε^d, where ε is the side length of the box and d is an integer dimension large enough to embed the fractal. For each such box i we consider the probability p_i that is associated with it:

(1)

Here is the probability density considered. The Rényi dimensions are defined for any real parameter q as

(2)

where the sum is over all i with p_i≠0. There are also more sophisticated definitions of the Rényi dimensions based on boxes of variable size (analogous to the definition of the Hausdorff dimension), see Beck & Schlögl (1993) for more details.

Generally, for a complicated multifractal (with lots of singularities of the probability density) the Rényi dimensions D_q depend on q, whereas for “simple” multifractals all D_q are the same and equal to the Hausdorff dimension. By changing the parameter q, one “scans” the structure of the multifractal. Large q values give more weight to large probabilities p_i, small q favor small probabilities p_i. Useful special cases of the Rényi dimensions are the box dimension (or capacity) DO (usually equal to the Hausdorff dimension, up to pathological cases), the information dimension D₁ (more precisely given by the limit lim_q→1 D_q), and the correlation dimension D₂. The correlation dimension can be easily extracted from experimental time series using the Grassberger-Procaccia algorithm (Grassberger & Procaccia, 1983). It is also of relevance

Figure 1. Construction of a classical Cantor set with a multiplicative non-uniform measure.

for the estimation of typical period lengths that are generated due to computer roundoff errors in chaotic dynamical systems (Beck, 1989). Also important are the limit dimensions D±∞ obtained for q→±∞. They describe the scaling behavior of the invariant density in the region of the phase space where the measure is most concentrated (D_∞) and least concentrated (D_−∞).

From the Rényi dimensions, one can proceed to f(α), the spectrum of local scaling exponents α of the measure, by a Legendre transformation. The basic idea is very similar to thermodynamics. In fact, many of the ideas of the multifractal formalism are related to early work by Sinai, Ruelle, and Bowen on the so-called “thermodynamic formalism” of dynamical systems (Ruelle, 1978). For multifractals, one can regard the function τ(q)=(q−1)D_q as a kind of free energy, and q as a kind of inverse temperature (Tél, 1988; Beck & Schlögl, 1993). One then defines the variable α (the “internal energy”) by ∂τ/∂q=:α and proceeds to the f(α) spectrum (the “entropy”) by Legendre transformation:

f(α)=qα−τ(q).

(3)

The advantage of the f(α) spectrum is that it has a kind of “physical meaning.” Roughly speaking, it is the fractal dimension of the subset of points for which the probability density scales with a local exponent α, that is, with Hence, this is a kind of statistical mechanics of local Hölder indices.

Let us consider a simple example of a multifractal, given by the classical Cantor set with a multiplicative (non-uniform) measure (Figure 1). The classical Cantor set is constructed by cutting the middle third out of the unit interval, then cutting out the middle third out of the remaining two intervals, and so on (See Fractals). In this way, at the kth construction step there are 2^k intervals of length We now attribute a product measure to each of the intervals, according to the rule sketched in Figure 1, with but In the limit k→∞ we obtain a multifractal.

Figure 2. (a) Rényi dimensions and (b) f(α) spectrum of the Feigenbaum attractor.

Let us calculate the corresponding Rényi dimensions. We cover the multifractal with small intervals of size The number of boxes with probability is

(4)

Hence,

(5)

and for the Rényi dimensions, one obtains from definition (2)

(6)

In particular, the box dimension D₀ is given by the value D₀=log 2/log 3, independent of and The other dimensions depend on the probabilities for example, for the information dimension one obtains by considering the limit q→1. The Legendre transform of (6) yields the f(α) spectrum.

Another interesting example is the attractor of the logistic map at the critical point of period doubling accumulation. Figure 2a shows the corresponding Rényi dimensions and Figure 2b the corresponding f(α) spectrum. Generally, the value of α where the function f(α) has its maximum is equal to the Hausdorff dimension of the attractor. The value of f(α) where the function has slope 1 is equal to the information dimension.

In practice, one often wants to evaluate the Rényi dimensions (or the f(α) spectrum) for a given time series (Kantz & Schreiber, 1997) of experimental data without explicitly knowing the underlying dynamics or the invariant measure. Here, various interesting methods are known (Grassberger-Procaccia algorithm (Grassberger & Procaccia, 1983), wavelet analysis (Arneodo et al., 1995), etc.). These algorithms can be implemented without explicitly knowing the underlying dynamics. The wavelet transform of an experimental signal s(x) is defined as

(7)

(* indicates complex conjugate), where the analyzing wavelet Ψ is some localized function, often chosen to be the Nth derivative of a Gaussian function. For small a, the wavelet transform extracts local Hölder exponents from the signal s, and qth moments of W_Ψ can then be used to define suitable partition functions which yield the Rényi dimensions.

CHRISTIAN BECK

Multifractal Analysis

MULTIFRACTAL ANALYSIS

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