FractalsBasic Concept

Entities are composed of features that are reproduced at different scales.

And whole entities can be described as a sum of smaller features that have the same characteristics.

Typically based on lines and examines line segments that have the same pattern.

Examples in nature: Trees and Rivers

A stem has the same structure as a branch, as a branch has the same structure as a tree.

Why use Fractals• Used as a descriptor of spatial pattern• They can be used at different scales

– Whole landscape– Overall for each cover type– Individual patches

• Used to create models of landscape patterns

Fractal Characteristics• They embody the idea of self-similarity, the

manner in which variations at one scale are repeated at another scale;

• Their dimensions are not an integer, but rather a fraction.

• Euclidean dimensions are common:- Point = 0; Line = 1; Area = 2; Volume =

3• Fractals may range between 1 and 2, indicating

that complex lines take up space.

Example

Self-similarity• The pattern stays the same regardless of scale (or level

of magnification).• When the parts of an object are similar to entire object it

is called self-similarity.• Self-similarity can be “perfect,” but this is rare in natural.• Approximate Self-similarity exhibits small variations at

different scales.• Brownian (statistical) self-similarity exhibits random

variations. The patterns are statistically similar but the lines are random instead of fixed.

• Using Brownian self-similarity in simulations create more realistic representations of coastlines and landscapes

Examples

Fractal DimensionMandelbrot defined the level ofvariation present at all scales as the fractal dimension (d) as:

d = log N / log r

where N = # of steps used to measure a pattern and r is scale ratio.

In the first case (a and b), N = 4 and r = 3 and d = 1.2619

What is the second case (c and d)? Answer: d = 1.4649

Fractal Dimension of PatchesMandelbrot defined the relationship between perimeter (P) and area (A) for a 2D object as

A = (kP)d

If P is the length of one side of an object, k = 1 and d can be computed as

d = ln A / ln P Krummel et al. 1987 found that complexity

increased with larger patches

Areas also can show self-similarity

Four area representations at

different scales. Note that

area/perimeter relationships

appear similar.

Complexity and other landscape measurements

Other landscape metrics are

Related to shape complexity.

Relationship between

dominance, contagion and

shape complexity for 3

subregions in the S.E. USA.

2D Box Counting Method• Common method for calculating the fractal dimension of lines

and easy to implement in a GIS environment.• It is implemented by covering the entity with a grid and

counting the number of cells intersecting with the entity (N) as a function of grid size (r) .

• Since d = -log N / log r, the box method computes the fractal dimension (d) by plotting the log N vs. log r for different grid sizes and then find the slope of a linear regression. The -slope is the fractal dimension.

Application

• The Box Method can lead to computing a wrong fractal dimension for non-perfect self-similarity.

• Brown (1989) suggested:– Choose equal number of intervals along both axes (i.e. square grids)– Make the range of the ordinate of the fractal function equal to the range

of the profile (i.e. include the whole).

• Grid method: border or line image. Two box ‘lengths’ are

• shown in (a) and (b). Boxes including the image are shaded.

Example

Modeling• Any fractal has some infinitely repeating pattern. When creating

such fractal, you would suspect that the easiest way is to repeat a certain series of steps which create that pattern. Instead of the word "repeat" we use a mathematical synonym "iterate" and the process is called iteration. In fact, any fractal can be made by the iteration of a certain rule. For example, the rule for creating the Koch Snowflake is:

• To create a true fractal, we have to iterate an infinite amount of

times. However, when doing it on a computer, we are limited by speed and resolution, so we only iterate a certain number of times. Increasing the number of iterations makes the fractal more accurate.

•

Modeling

ModelingThere are three basic types of iteration:1. Generator Iteration — Creates fractals by

repeatedly substituting certain geometric shapes with other shapes.

2. IFS Iteration — Creates fractals by repeatedly applying geometric transformations (such as rotation and reflection) to points.

3. Formula Iteration — Includes several ways of creating fractals by repeating a certain mathematical formula or several formulas.

Modeling• One way of creating fractals is by generator iteration. To do that,

we start with a figure called the base. Then, every part of it is substituted with another figure, called the motif or generator. In the new figure, we again substitute every part with the motif. If we iterate these substitutions an infinite number of times, we end up with a fractal. For example, let’s try making a fractal with the following base and motif:

• We start with a triangle and substitute every side with the motif:

Modeling• We again substitute each of the 12 segments with the motif

and continue the process to form a fractal called the Koch Snowflake:

Fractal Landscapes• Based on Brownian Fractal – i.e. random elements• Midpoint Displacement Method is used to create surfaces of real numbers.• Brownian Fractal motion is controlled by two parameters:

– Variance displacement of points (usually set as 1.0), Normal distribution is common with a mean = 0 and variance = 1)

– H, that controls the correlation between iterations– The fractal dimension of the surface is D = 3.0 – H– The value at a new point is equal to Ym = (Y1 + Y2) / 2 + f(C(d))

• The difference between two points is proportional to the square of the distance, d, and the correlation, C(d), where– C(d) = 22H-1 – 1– H = 0.5, no correlation– H < 0.5 negative correlation, rough surface– H > 0.5 positive correlation, smooth surface

Fractal Landscape

• A fractal landscape is recursively self-similar, that appears similar at all scales of magnification.

Fractal Landscape

• A way to create this fractal landscape is to employ random midpoint displacement algorithm.

x

y

x

y

x

y

Fractal Landscape

Another method is to divide and purturb.

Fractal Landscapes

Neutral Landscape Models

• Neutral landscape models generate raster maps in which complex habitat structures are generated with analytical algorithms. Thus, they are neutral to the biological and physical processes that shape real landscape patterns.

• What is the value of neutral models?– Statistical: How do structural properties of landscapes

deviate from theoretical spatial distributions?– Modeling: How are ecological processes affected by

landscape pattern?– Neutral Models DO NOT represent actual

landscapes!!!!!

Three Model Types

• Operationally, useful to think of three general types of landscape models – Neutral Models– Landscape change models

• Land cover classes, ecosystem types, or habitats• Influenced by natural or anthropogenic processes• Includes landscape process models

– Individual-based models

Fractal LandscapesThe numbers can be classified into types. You can define the proportion of the landscape used by type.

Fractal Landscapes

H = 0.2 H = 0.8

Other methods

• Landscape models may be generated by random, hierarchical, or fractal algorithms.

Simple Random Maps

Percolation Theory

• As an increasingly large proportion of the landscape is occupied, the occupied cells coalesce into larger patches.•

• Once p= 0.5928 (0.41 for the 8-neighbor rule), the largest cluster will span the map edge-to-edge.

• Important since all landscape metrics covary with p.

Neutral Landscape Models General Insights

• Threshold effects occur as nonlinear relationships between patterns or processes and p.

• Neutral landscape models are very important for calibrating and understanding different measures of landscape pattern -what is the expected range?

• Concepts from Neutral Models can be applied to Landscape Change Models -What happens if I turn on/off process X?

• Specific results of neutral models do not necessarily apply to any actual landscapes, but the insights of the models do apply.

Landscape Change Models

• Landscape change models simulate pattern change or state change in a landscape.

• Most landscape models are different ways of conceptualizing the interactions between three general areas: abiotic template, biotic interactions, disturbances.

• Depending on needs, a model may need to include processes operating within any of these three areas.

• All landscape change models include some processes.• Questions and scales determine which processes to include.

Markov Model (Time Series)Transition in time based on past events.

A 1st-order Markov model assumes that you can predict the state of a system at time t + 1 by knowing the state of the system in time t. The Markov model is based on a transition matrix P, that summarizes the probability that a unit (e.g. a raster cell) in state type i will change to state type j during a single time step.

 

A Markov Model is projected with the equation:

xt + 1 = xt P

where the state vector is multiplied by the transition matrix. The next projection for time t + 2 is continued:

xt + 2 = xt + 1 P = xt PP = xt P2

which provides the general form of the equation at t = t + k as:

xt + k = xt Pk

 where xt represents the initial conditions. Consequently, the

model can project into the future simply by iterating through the matrix operation.

Markov models can be solved by iteration to project the state of a system. Given the state of the system as a vector (a single row matrix):

 

xt = [x1 x2 x3 ….]

 

where xi is the proportion of the cells (assume you are

dealing with a discrete raster) in type i at time t.

[1] [2] [3]P12

P21

P23

P31

P11 P12 P13 0.9 0.1 0.0

P21 P22 P23 = 0.1 0.8 0.1

P31 P32 P33 0.1 0.0 0.9

1 = Disturbed

2 = Early seral stage

3 = Mature

Problems with Simple Model

• Historical Influences: If the transition probabilities depend on more than the immediately prior state, then the system retains a “memory” of antecedent conditions. If so, the dynamics are not first-order.

• History (time lags):The transition probabilities become conditional based on the age of the site (transitions occur only after the site has been in a certain state for some time).Time lags are particularly important for disturbance events.

Problems with Simple Model

• History (antecedent events):The transition probabilities become conditional based on whether a particular antecedent event occurred.

• Spatial dependencies -Covariates: The transition probabilities become conditional probabilities based on some ancillary information about the covariates.

Problems with Simple Model

• Spatial dependencies -Neighborhood Effects: The transition probabilities become conditional probabilities based on the state of neighboring cells surrounding the focal cell.– Sleuth Growth Types

• Nonstationarity: The transition matrix varies over time (i.e., the probabilities are not constant) –which implies that the rules governing landscape change are changing over time.– Mesquite expansion in the 1970-80s - it did not continue growing– Calculate new transition matrices for each time period of interest, or

calculate transitions as functions of time.

Problems with Simple Model

• Disturbance: Disturbances are a special case in modeling, because they are an integration of all the special cases affecting transition probabilities.– Disturbances (e.g., fires) may be physically

constrained (spatial covariates), may spread contagiously (neighborhood effects), may be lagged in time (time lags), and may change through time (nonstationarity), and may be stochastic.

Application

• Cellular Automata – cellular models – But with a multivariate, multi-temporal conditional

probabilities.– Should be based on “real world” observations. – Stochastic element – uniform random generator