Fractals in time  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 selfsimilarity, perhaps approximate or statistical

Usually 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 selfsimilar but selfaffine 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 tyear 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.