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:

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

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

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