Extensions: Random fractals' Selfaffine fractals' Multifractals'

1
Extensions Random fractals. Self-affine
fractals. Multifractals.

• Edith Perrier, IRD France,
• Visiting scientist at UCT,
• Applied Maths Dept., room 417.1, Ext. 3205,
• E-mail edith_at_maths.uct.ac.za
• Web site where to download previous lectures
  lect.ppt (or lect.zip if links included)
  http//www.mth.uct.ac.za/Affiliations/BioMaths/Ind
  ex.html

2
Random fractals examples

DLog(n)/log (1/r)
DLog4/log3


Random spatial repartition of n replicates
Statistical fractal the mean value of the
number of replicates is n
3
Ex soil structures fractal models and random
extensions
Calculations on regular fractal. Extension to
statistical fractals by simulation
links between structural properties and
hydraulic properties (Perrier et al.,
1996-2001) Model 1 Density data
water retention curve Model 2
Particle size distribution
water retention curve
4
Self-affine extensions for porous structures
To model soil anistropy and preferential water
paths
5
Self-Affine Fractals

This fractal F is self-affine instead of
self-similar because the pieces are scaled by
different amounts in the x- and y-directions. The
coloring of the pieces on the right emphasizes
this. Each piece is scaled by 1/3 in the
x-direction and by 1/2 in the y-direction. One
can show (Falconer, 1990) that the Hausdorff
dimension of F equals With in the above
example p2,q3, pq6 subrectangles at each
iteration and Ni is the number of rectangles
selected in each row i between 1 and p at each
iteration
6
Randomness and selfaffinity

• Example 1 (topographical maps)
• Other methods include brownian motion (see lect
  6) or Fourier transforms.
• Example 2 (a self affine time series, Gouyet
  1992)
• Time series (see lect. 5)

7
Multifractals

• From Michael Frame, Benoit Mandelbrot, and Nial
  Neger , http//classes.yale.edu/fractals/
• 7. Multifractals
• 7A. Gold Distribution
• 7B. Unequal Probabilities
• 7C. Histograms
• 7D. Histogram Tops
• 7E. Second-Highest Probability
• 7F. Another Example
• 7G. Coarse Dimensions
• 7H. Local Dimensions
• 7I. Multifractals from IFS
• 7J. f(a) curves and probabilities
• 7K. f(a) from financial data

8
Lecture series

• Lecture 1. March. 3rd. Introduction to fractal
  geometry . Measures and power laws.
• Lecture 2. March 5th. Definitions of non-integer
  dimensions and mathematical formalisms
• Lecture 3. March 10th. Iteration of functions and
  fractal patterns
• Lecture 4. March 12th. Extensions Self-similar
  and self-affine sets . Multifractals.
• Lecture 5. March 17th. Fractals / Geostatistics
  / Time series analysis
• Lecture 6. March 19th. Dynamical processes
  Fractal and random walks
• Lecture 7. March 24th. Dynamical processes
  Fractal and Percolation
• Lecture 8. March 26th. Dynamical processes
  Fractal and Chaos