Title: Extensions: Random fractals' Selfaffine fractals' Multifractals'
1Extensions 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
2Random 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
3Ex 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
4Self-affine extensions for porous structures
To model soil anistropy and preferential water
paths
5Self-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
6Randomness 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)
7Multifractals
- 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
8Lecture 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