Multifractals in ecology using R
tags: realexamples math equations
Description/Summary
Disclaimer This post is from the link posted by GitHub user lsaravia in this comment. All credit for this post goes to the original author. 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 length \(r\)
Content
 Disclaimer
 This post is from the link posted by GitHub user lsaravia in this comment. All credit for this post goes to the original author.
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 length \(r\)

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. https://doi.org/10.1111/j.14610248.2009.01345.x.
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