Monofractal

To study the fractal properties of a set we must first define what a fractal is. This is done by considering the Haussdorff dimension. Although the definition is rather technical, we shall only consider it in a theoretical context and introduce the box- counting dimension afterwards, which gives a much more intuitive sense of dimen- sion. We should briefly here discuss an intuitive idea of what dimension is. For simple Euclidean objects the idea of dimension seems obvious and we can appeal to the linear algebra definition for a dimension as the size of the basis set i.e. how many co-ordinates you would need to describe a position in a set. For example a circle is

a one-dimensional object as only one co-ordinate, namely the angle θ is needed to describe the position.

To make the ideas of dimension more explicit consider the prototypical example of fractal: the Cantor set. First, consider the unit interval [0,1]. The construction is iterative; in the first step the middle third is removed leaving a disjoint union of two intervals [0,1₃]∪[2₃,1]. At this stage we have two identical copies of the original set, the procedure continues by removing the middle third of each of the intervals, leaving [0,1/9]∪[2/9,1/3]∪[2/3,7/9]∪[8/9,1]. This iteration continues to infinity leaving the resulting Cantor set. Now we may ask the question how many co-ordinates are needed in order to describe the set, this can be done by considering how it was constructed. Consider a point in the Cantor set, at each stage of the construction there is a choice over whether the point is in the left third or the right third of the set being divided, which can be represented as a co-ordinate that takes the value 0 or 1. Hence a point that has the co-ordinates (0,1,1. . .) will be in the interval [8/27,1/3]. The problem then comes as to describe a point in the Cantor set it appears you need an infinite number of co-ordinates. On the other hand, we know that the Cantor set is a subset of the unit interval which has dimension one. For a definition of dimension to hold we would require that if the setsE ⊂F ⊂G then the dimension of each satisfies dim(E) ≤dim(F) ≤dim(G). We would also desire a definition of dimension to coincide with the dimension of an Euclidean space. In the next section we will explore two such dimensions: The Hausdorff dimension and the Box-Counting dimension.

The idea behind the Haussdorff Dimension is to consider various coverings for a set and consider the sum of the size of these coverings raised to a powers. The sum is minimised over all coverings of sizeδ. As the size of the coverings reduces as the sum increases, they reach a supremum asδ → 0. This provides the Haussdorff measure. More formally, letF be a set with a δ-covering {Ui} i.e. |Ui|< δ for all i andF ⊂ ∪∞

i=0Ui, then the s-dimensional Haussdorff measure is defined as

Hs(F) = lim δ→0inf ( _∞ X i=0 |Ui|s : {Ui} is aδ-cover of F ) . (4.1)

The Haussdorff dimension is defined from the s-dimensional Haussdorff measure as

As an example of a calculation of the Haussdorff dimension consider the unit interval [0,1]. A minimal δ-covering for this set would ben=d1/δe. Hence,

n X i=0 |Ui|s= d1/δe X i=0 |Ui|s = d1/δe X i=0 δs=δs−1, (4.3)

where we assume that 1/δ has an integer value. As δ→0 we can see that

Hs(F) =

(

∞ :s <1

0 :s >1 . (4.4)

The corresponding Haussdorff dimension is calculated directly from definition as dimH[0,1] = 1. In general, the Haussdorff dimension need not take an integer value, as for the Cantor set the Haussdorff dimension is log(3)/log(2) [Falconer, 2013].

Although the Haussdorff dimension is appealing from a theoretical point of view, it is in general intractable to calculate for real data. The box-counting or Minkowski-Boulingand dimension is defined on a set F embedded in a Euclidean spaceRn. For our purposes we should only concern ourselves with sets embedded in R2 , since we are considering vegetation occupancy patterns. The box-counting di-

mension is calculated via the box-counting algorithm. Boxes of length are placed in a regular grid fashion over the space. The number of boxes of length that intersect the setF is denotedN(). The box-counting dimension is then defined as

dimBC(F) := lim

→0

logN()

log(1/). (4.5)

For instance for the Cantor set, if we take a box size of length (1/3)n, then the num- ber of boxes that are filled are 2n. It is then straightforward to calculate dimBC(F)

as

dimBC(Cantor set) = lim

n→∞ log(2n) log(3n₎ = lim n→∞ nlog(2) nlog(3) = log(2) log(3), (4.6)

which is the same value as the Haussdorff dimension. The Haussdorff dimension often gives an equivalent value as the box-counting dimension, however they are not

the same and in general the box-counting dimension is greater than or equal to the Haussdorff dimension.

There are various definitions of a fractal set. It is often defined as a set that roughly repeats itself on finer scales [Gisiger, 2001; Hastings and Sugihara, 1993]. Mandelbrot [1983] gave the definition of a fractal as a set whose Haussdorff Dimen- sion strictly exceeds its Topological Dimension. For our purposes, we define a set to be fractal if the number of non-empty boxes of length scales as a power-law over some range of. This does mean that certain trivial sets such as the empty set or the spatial Poisson Process would be classed as fractals under this definition. We shouldn’t necessarily be concerned with the definition of fractal and non-fractal too much, as we are more focused on being able to robustly measure scaling properties such as the box-counting dimension. A set that we class as non-fractal is then one where the developed fractal analyses such as the box-counting dimension are not applicable.

4.2.1 Measuring monofractality from data

In the previous section, we have concerned ourselves with what theoretical measure- ments we wish to apply to data in order to detect the underlying scaling properties. This lead to establishing the box-counting dimension as a measure that can be ef- ficiently calculated and easily applied to several different types of data including time-series and occupancy. The standard method of calculating the box-counting dimension is to measure the number of boxes of lengththat are occupied and plot it against 1/ on a log-log plot. The relationship between the two for a fractal is linear on the plot and the gradient is taken to be the box-counting dimension. For real data, there is an issue that the same type of scaling might not be present over all ranges being considered. As an example, over smaller spatial scales seagrass is buffeted by small currents and wave action and is also undergoing turnover in the form of clonal and seed spreading. Over larger ranges, patches form and these coag- ulate into meadows. On even larger scales, the vegetation is affected by large-scale currents and geographic features such as coastline and coastal shelf. It is therefore not expected that there should be a similar scaling throughout the entirety of the spatial scales. In order to deal with this problem, a number of ideas can be pro- posed. The first is to use splines to fit to the log-log plot in order to detect different forms of scaling. There are a few issues with this: firstly, there may be a smooth transition between one form of scaling and another making it difficult to find the optimal scale to place the spline; secondly, it may not be obvious how many forms

of scaling and thus how many splines should be used to fit to the data.

The second method is to find an appropriate range of scales over which the power law relationship does hold for a single dimension. It is this method that shall be employed to analyse the Isles of Scilly data. The method we use is outlined in Seuront [2009]; for all ranges of log(δ) where the number of δ points is 5, fit a line by linear regression and calculate the coefficient of determination r2 and the sum of squared residuals (SSR). Then, find the range that minimises theSSR and see if this range corresponds with the range that maximises the r2_{. If it does, then}

use the gradient calculated by linear regression from this range to determine the box-counting dimension. If the ranges do not match up, there is no best-fit for any range and we say that the data is non-fractal as it has no range over which the scaling is constant.