# Multifractals in Ecology Using R - Day 2

## Characteristic features of fractals

Mandelbrot Originally defined fractals as sets that have fractal dimension strictly greater than its topological dimension.

There is no hard and fast definition but a list of properties.

We refer to

*F*as fractal if:*F*has a fine structure: i.e. detail on small scales.*F*is too irregular to be described by traditional geometrical language*F*has some form of self-similarity, perhaps approximate or statisticalUsually the fractal dimension of

*F*is greater than its topological dimension

## Random walks

Random walk (RW) is a stochastic process in which an object moves in a space by performing random jumps

We can see that the enlarged view of a small part of the trajectory looks similar to the original, is fractal.

The pattern displayed by the one dimensional RW is not self-similar but

*self-affine*because the time and space dimensions do not scale in the same way

## Fractal time series

Fractal properties in time series can be analyzed by means of Hurst’s Rescaled Range Analysis

Let us consider a time series that can be: the number of extinctions of a group of organism or a particular population or the discharge of a river, etc.

\(X_i\) with \(i=1,2,3,…,T\)

The average of \(X_i\) over \(T\) time steps will be \(< X >_T = \left( \sum_{i} X_t \right)/T\)

The departure from the average over a t-year time horizont is given by:

\[X(t,T) = \sum_{i=1}^{t} [X_i - < X >_T ] = \left\{ \sum_{i=1}^{T} X_i \right\} - t < X >_T\]

\(X(t,T)\) is usually calculated dividing the time series in \(M\) segments of size \(T\).

What is the value of \(X(T,T)\) ?

## Rescaled Range Analysis

We need to calculate two more quantities from the previous :

The standard deviation \(S(T) = \left[ (< X_t - <X>_T >)^2 \right]^{1/2}\)

The range \(R(T) = \max_{1 \le t \le T} X(t,T) - \min_{1 \le t \le T} X(t,T)\)

The rescaled range is: \(F(T)=R(T)/S(T)\)

Calculate \(F(T)\) using \(T=5\) and the following series

`3 4 9 2 1 7 8 2 2 9`

When the values of the time series are uncorrelated \(F(T) \propto T^{1/2}\), which is called white noise. The best predictor is the last measured value.

Hurst found a more general scaling relation \(F(T) \propto T^{H}\).

for the natural systems he analyzed \(H > 1/2\)

it can be shown (easily) than the fractal dimension is related:

\[D = 2 - H\]

When the Hurst exponent is greater than 1/2 the system shows persistence on all time scales. An increasing trend in the past implies an increasing trend in the future.

If \(H < 1/2\) an increase in the past implies a decrease in the future, the system shows antipersistence.

## Papers

We will analyze the data from the paper:

- Meltzer MI, Hastings HM (1992) The use of fractals to assess the ecological impact of increased cattle population: case study from the Runde Communal Land, Zimbabwe. Journal of Applied Ecology 29: 635–646.