The duality of fractals: roughness and self-similarity

Abstract

The fractal dimension (D _HB) is an interesting metrics because it is supposed to quantify by a single value, scale independence and roughness of ecological objects. However, we show here that those two properties may be quantified by a single dimension only in some specific cases. In general, a non-integer D _HB quantifies only the roughness, and self-similarity needs to be evidenced or postulated by other means. Second, we revisit some aspects of the practical estimation of D _HB. We recommend the use of madogram instead of variogram for estimations based on geostatistics. We propose a simplification of its estimation for 2D fields and discuss its possible relationship with self-similarity. We finally underline the problem of scale and resolution. Field data recorded during a scientific acoustic survey on the North Sea herring are used for illustrations. The paper concludes on a synthesis of practical recommendations to ecologists when using fractal dimension.

Appendices

Appendix 1: Summary of the demonstration of the relevance of madogram for estimating D _HB

Let us then consider one particular realisation of a one-dimensional random function Z(x) where x denotes a point in space. For simplicity, the demonstration made by Bruno and Raspa (1989) is made in 1D but it can be generalised to any dimension. Over a small running increment of distance h centred on x, the graph of this realisation can be covered with at most (this is where the principle of economy comes in) $n(h) = \frac{{\left| {Z(x) - Z\left( {x + h} \right)} \right|}}{h} + 1$ squares of side h. As there are 1/h such intervals over a unit distance interval, there is a total of $n(h) \cdot \frac{1}{h} = \left| {Z(x) - Z\left( {x + h} \right)} \right| \cdot {h^{ - 2}} + {h^{ - 1}}$covering squares. The HB measure is then obtained for the limit of h tending towards 0 of the surface area of these d-dimensional squares:

$$ H{B_{\rm{measure}}} = \mathop {{\lim }}\limits_{h \to 0} \left( {g(d) \cdot \left( {\left| {Z\left( {x + h} \right) - Z(x)} \right| \cdot {h^{d - 2}} + {h^{d - 1}}} \right)} \right) $$

Reminding that (1) the fractal dimension of a random function derives from the expected value of the HB measure obtained for each realisation, (2) the expectation can be exchanged with the lim, and finally (3) the first-order variogram also called madogram of a RF is

$$ {\gamma_1}(h) = \frac{1}{2} \cdot {\rm E}\left[ {\left| {{\rm Z}\left( {x + h} \right) - {\rm Z}(x)} \right|} \right], $$

one finally gets that D _HB is obtained by considering the value d for which

$$ \mathop {{{ \lim }}}\limits_{{\rm{h}} \to {0}} { }{\gamma_1}(h) \cdot {h^{d - {\rm N}}} $$

is not degenerated. Considering that the behaviour of the madogram near the origin can be reduced to ${\left| h \right|^\beta }$, we see that d must be such that $\mathop {{{ \lim }}}\limits_{{\rm{h}} \to {0}} { }{\left| h \right|^{\beta + d - N}}$ is not degenerated which is achieved as long as $\beta + d - N = 0$. Finally, we find that the D _BH estimate of Z(x) is

$$ {D_{HB}} = N - { }\beta $$

where β is the slope of the madogram in log-log scale.

Appendix 2: Equivalence is only possible with power laws

Let us show that the function φ is necessarily of the form φ(λ) = λ^ψ.

In one hand, by equivalence, we have:

$$ Z\left( {{\lambda_1}{\lambda_2}x} \right) \equiv \varphi \left( {{\lambda_1}} \right)Z\left( {{\lambda_2}x} \right) \equiv \varphi \left( {{\lambda_1}} \right)\varphi \left( {{\lambda_2}} \right)Z(x) $$

But, in the other hand, we have that:

$$ Z\left( {{\lambda_1}{\lambda_2}x} \right) \equiv \varphi \left( {{\lambda_1}{\lambda_2}} \right)Z(x) $$

The function φ must then be such that

$$ \varphi \left( {{\lambda_1}} \right)\varphi \left( {{\lambda_2}} \right) = \varphi \left( {{\lambda_1}{\lambda_2}} \right) $$

which implies that φ is of the form φ(λ) = λ ^ψ.

Appendix 3: Checking that W(x) is a Levy process

This section considers 1D random functions.

Let Z(x) be a stationary random function with 0 mean and variance σ ²:

$$ {\hbox{Var}}\left\{ {Z(x)} \right\} = {\hbox{Var}}\left\{ {Z\left( {\ln x} \right)} \right\} = {\sigma^2}. $$

And let W(x) be a non-stationary random function defined by

$$ \begin{gathered} W(x) = {x^\psi }Z\left( {\ln \left| x \right|} \right)\quad \quad, x \in \mathbb{R} \hfill \\{\hbox{E}}\left\{ {W(x)} \right\} = 0 \hfill \\{\rm var} \left\{ {W(x)} \right\} = {x^{2\psi }}{\sigma^2} \hfill \\\end{gathered} $$

Levy processes are such that disjoint increments are independent. In particular, for W(x) to be a Lévy process, this implies that:

$$ {\hbox{Cov}}\left\{ {W(x),W\left( {x \vee y} \right) - W\left( {x \wedge y} \right)} \right\} = 0\quad \quad \forall \left( {x,y} \right) \in \left( {\mathbb{R},\mathbb{R}} \right) $$

so that the non-stationary covariance of W(x) should be of the form

$$ \begin{gathered} {C_W}\left( {x,y} \right) = {\hbox{Cov}}\left\{ {W(x),W(y)} \right\} \\= {\hbox{Cov}}\left\{ {W\left( {x \wedge y} \right),W\left( {x \wedge y} \right) + W\left( {x \vee y} \right) - W\left( {x \wedge y} \right)} \right\} \\= {\sigma^2} \cdot {(x \wedge y)^{2\psi }} \\\end{gathered} $$

Given the above definition, we get that

$$ {C_W}\left( {x,y} \right) = {\left( {xy} \right)^\psi }{\hbox{Cov}}\left\{ {Z\left( {\ln x} \right),Z\left( {\ln y} \right)} \right\} = {\left( {xy} \right)^\psi }{C_Z}\left( {\ln \frac{{x \vee y}}{{x \wedge y}}} \right) $$

Taking the following covariance for Z(x)

$$ {C_Z}(h) = {\sigma^2} \cdot {e^{ - \psi \cdot \left| h \right|}} $$

gives

$$ {C_W}\left( {x,y} \right) = {\sigma^2} \cdot {\left( {xy} \right)^\psi }{\left( {\frac{{x \vee y}}{{x \wedge y}}} \right)^{ - \psi }} = {\sigma^2} \cdot {\left( {x \wedge y} \right)^{2\psi }} $$

One gets full characterisation of the random functions through their covariance functions as long as Gaussian random functions are used.