Multifractal theory  Day 3
Multifractals

Many natural systems cannot be characterized by a single number such as the fractal dimension. Instead an infinite spectrum of dimensions must be introduced.
Multifractal definition

Consider a given object $\Omega$, its multifractal nature is practically determined by covering the system with a set of boxes ${B_i(r)}$ with $(i=1,…, N(r))$ of side lenght $r$

These boxes are nonoverlaping and such that
$$\Omega = \bigcup_{i=1}^{N(r)} B_i(r)$$
This is the boxcounting method but now a measure $\mu(B_n)$ for each box is computed. This measure corresponds to the total population or biomass contained in $B_n$, in general will scale as:
$$\mu(B_n) \propto r^\alpha$$
Box counting
The generalized dimensions

The fractal dimension $D$ already defined is actually one of an infinite spectrum of socalled correlation dimension of order $q$ or also called Renyi entropies.
$$D_q = \lim_{r \to 0} \frac{1}{q1}\frac{log \left[ \sum_{i=1}^{N(r)}p_i^q \right]}{\log r}$$
where $p_i=\mu(B_i)$ and a normalization is assumed:
$$\sum_{i=1}^{N(r)}p_i=1$$

For $q=0$ we have the familiar definition of fractal dimension. To see this we replace $q=0$
$$D_0 = \lim_{r \to 0}\frac{N(r)}{\log r}$$
Generalized dimensions 1

It can be shown that the inequality $D_q’ \leq D_q$ holds for $q’ \geq q$

The sum
$$M_q(r) = \sum_{i=1}^{N(r)}[\mu(B_i(r))]^q = \sum_{i=1}^{N(r)}p_i^q$$
is the socalled moment or partition function of order $q$.

Varying q allows to measure the nonhomogeneity of the pattern. The moments with larger $q$ will be dominated by the densest boxes. For $q<0$ will come from small $p_i$'s.

Alternatively we can think that for $q>0$, $D_q$ reflects the scaling of the large fluctuations and strong singularities. In contrast, for $q<0$, $D_q$ reflects the scaling of the small fluctuations and weak singularities.
Exercise

Calculate the partition function for the center and lower images of the figure:
Two important dimensions

Two particular cases are $q=1$ and $q=2$. The dimension for $q=1$ is the Shannon entropy or also called by ecologist the Shannon’s index of diversity.
$$D_1 = \lim_{r \to 0}\sum_{i=1}^{N(r)} p_i \log p_i$$
and the second is the socalled correlation dimension:
$$D_2 = \lim_{r \to 0} \frac{\log \left[ \sum_{i=1}^{N(r)} p_i^2 \right]}{\log r} $$
the numerator is the log of the Simpson index.
Application

Salinity stress in the cladoceran Daphniopsis Australis. Behavioral experiments were conducted on individual males, and their successive displacements analyzed using the generalized dimension function $D_q$ and the mass exponent function $\tau_q$
both functions indicate that the successive displacements of male D. australis have weaker multifractal properties. This is consistent with and generalizes previous results showing a decrease in the complexity of behavioral sequences under stressful conditions for a range of organisms.

A shift between multifractal and fractal properties or a change in multifractal properties, in animal behavior is then suggested as a potential diagnostic tool to assess animal stress levels and health.
Mass exponent and Hurst exponent

The same information contained in the generalized dimensions can be expressed using mass exponents:
$$M_q(r) \propto r^{\tau_q}$$
This is the scaling of the partition function. For monofractals $\tau_q$ is linear and related to the Hurst exponent:
$$\tau_q = q H  1$$
For multifractals we have
$$\tau_q = (q 1) D_q$$
Note that for $q=0$, $D_q = \tau_q$ and for $q=1$, $\tau_q=0$
Paper
 Kellner JR, Asner GP (2009) Convergent structural responses of tropical forests to diverse disturbance regimes. Ecology Letters 12: 887–897. doi:10.1111/j.14610248.2009.01345.x.