Multifractals in ecology using R

tags: real-examples math equations

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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 \(r\)

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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.

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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 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:

  \[\mu(B_n) \propto r^\alpha\]

Box counting #

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. https://doi.org/10.1111/j.1461-0248.2009.01345.x.

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