Chapter 4 Spatial analysis
4.3 Fractional Brownian motion
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 β d p β = 0.1 β = 0.25 β = 0.4
Figure 4.1: Realisations of fBm for a range of power spectrum scaling β = 1 + 2H as defined in Eq. 4.8. The calculated boundary dimension (db) for a number of realisations is given in the top figure, whilst the bottom figure gives example realisations at differentβ levels. db scales linearly with β over a small range due to finite size effects.
In studies of spatial analysis, it is useful to be able to construct a neutral landscape model where certain statistical properties of the landscape are at specific
values, whilst other properties are allowed to vary randomly. Fractional Brownian Motion is one of the canonical examples of a neutral landscape model [Keitt, 2000]. The idea is to construct a two-dimensional random fieldX, where level sets of the random field form boundaries that have the desired fractal properties. Fractional Brownian motion (fBm) is a unique probability distribution with independent incre- ments, stationarity and finite variance and is controlled by the Hurst exponentH. For (x, y)∈R2, we define a random fieldX(x, y) such that the following conditions
are satisfied:
1. X(0,0) = 0 with probability 1. 2. X(x, y) is a continuous function.
3. The height increments X(x+h, y+k)−X(x, y) have a normal distribution with zero mean and variance (h2+k2)H for (x, y),(h, k)∈R2 .
The model then has one parameter H and this can be interpreted as the persis- tence of the process. IfH = 12, the increments become uncorrelated. With H > 12, correlations are positive and realisations of the surface are smoother or more per- sistent. Conversely for H < 12 points are anti-correlated and realisations become rougher or anti-persistent. Fractal Brownian patches are then defined as a level set of this process, i.e. X−1(c) = {(x, y) : X(x, y) = c}, which defines the boundaries of the patches. Note that, in this context when talking about persistence or anti- persistence, it is about the spatial process only and has nothing to say about the underlying dynamics that caused it. fBm can be easily generalised to n-dimensions and thus can produce spatio-temporal models of vegetation patches, however each dimension may have its own Hurst exponent and hence spatial persistence may be independent of dynamic persistence for this general statistical model. However, fBm does have some desirable properties. The graph of fBm (referring to the set
{(x, y, z) : x, y ∈R, z ∈X(x, y)} as opposed to the trail of fBm which only refers to the set {z : z ∈ X(x, y)}) can be shown to have both Haussdorff and box- counting dimension 3−H. This has lead to the assumption that the box-counting dimensions and the Hurst exponent are the same, even though this only applies to fBm and other derivatives of this model. In general the Hurst exponent may be different or even independent of the box-conting dimension [Gneiting and Schlather, 2004]. The level-set of a fBmX−1(c) has box-counting dimension one less than the box-counting dimension of the corresponding fBm field. This then provides a rela- tionship between the Hurst exponent for the underlying fBm and the box-counting dimension as
Note that this only refers to the collection of boundaries of the level set that has this fractal property. A single patch boundary may have a different dimension due to finite-size effects. It also only refers to the boundary as opposed to the patch itself, which will necessarily have a box-counting dimension 2.
A realisation of fBm can be simulated by considering the spectral properties of the field. It can be shown via the WienerKhinchin theorem that a fBm generated with Hurst exponentH has the power spectrum
S(f)∼1/f1+2H (4.8)
This formula gives insight into the scaling properties of fBm, for example when H = 0, the process has 1/f noise. Also note that white noise corresponds to a constant power spectrum or 1/f0, which means that the corresponding Hurst exponent would be H =−1/2. Although this is not defined for the model it does give insight into how the model changes smoothly from Brownian motionH = 1/2 to Gaussian noise H = −1/2, where there is a transition from the path being continuous to discontinuous at H= 0.
An approximate realsiation of fBm can be constructed via a spectral method [Hastings and Sugihara, 1993; Peitgen et al., 1988]. For an N ×N grid an i.i.d. Gaussian white noise process is simulated for each point. A discrete two-dimensional Fourier transform can then be taken on this grid. The Fourier transformed Gaussian noise can then be multiplied by 1/(fx2+fy2)β/2, wherefx,fy are the wavelengths of thex and y component respectively. The corresponding power spectrum then has power law scaling ofβ. The inverse Fourier can be taken and the absolute value used as an approximation to the fBm process. Examples of level set of this can be seen in Fig. 4.1. Notice that due to finite size effects, there is a range of corresponding Hurst values for each value of β, this can be used to characterise the error in the estimation of a Hurst exponent using the box-counting dimension of the boundary. Since the simulated realisations of fBm can have a level set at any value, we may vary the constant continuously until the desired density is reached (for Fig. 4.1 all outputs are held constant with density at a half). We may, therefore, use this to simulate landscapes with the desired density and Hurst exponent, but with other properties allowed to vary randomly.