# Multifractals in Ecology Using R - 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 box-counting 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:

$\mu(B_n) \propto r^\alpha$

## The generalized dimensions

• The fractal dimension $$D$$ already defined is actually one of an infinite spectrum of so-called correlation dimension of order $$q$$ or also called Renyi entropies.

$D_q = \lim_{r \to 0} \frac{1}{q-1}\frac{log \left[ \sum_{i=1}^{N(r)}p_i^q \right]}{\log r}$

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 so-called moment or partition function of order $$q$$.

• Varying q allows to measure the non-homogeneity 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 so-called 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

1. Kellner JR, Asner GP (2009) Convergent structural responses of tropical forests to diverse disturbance regimes. Ecology Letters 12: 887–897. doi:10.1111/j.1461-0248.2009.01345.x.