Introduction to Fractal Processes

 ELEC 697, Spring 2001

 Rice University

This course develops the basic concepts of fractal processes
---Long range dependence, scaling, and multifractal properties---
and their relevance in applications such as network traffic modelling and image processing


Dr. Rudolf Riedi
Duncan Hall 2025, 713 / 348 3020,
Office Hours: By appointment


Tuesday/Thursday 2:30-3:50, AL B209


ELEC 533 or equivalent course on the basics of probability theory
(i.e. a basic understanding of conditional expectation, random processes, and auto-correlation).

Text books

On reserve at Fondren Library (ask for books for elec533):

Falconer Fractal geometry (1990)

Papoulis Probability, Random Variables, and Stochastic Processes
Samorodnitsky and Taqqu Stable non-Gaussian random processes
Wong and Hajek Stochastic Processes in Engineering Systems
Falconer Techniques in fractal geometry (1997)

See also:
Michael Barnsley Fractals everywhere


In 1975, Mandelbrot coined the term `fractal', which up to now has been used more informally to describe a basic concept, rather than being defined in a mathematical rigorous way. Roughly speaking, a fractal entity is characterized by the inherent, ubiquitous occurrence of irregularities which governs its shape and complexity. It has become generally accepted that the theory of fractals is certainly more suitable for a comprehensive description of the physical world than many other theories which usually handle mainly completely regular phenomena.

The best known fractal process is Brownian motion which can be constructed through a simple iteration, a property shared by many fractal objects which can be studied analytically. Brownian motion strikes through its erratic - fractal - appearance which finds a natural description through self-similarity and fractal dimensions.

With the siblings of Brownian motion, the fractional Brownian motions (fBm) highly irregular behavior becomes linked to the concept of Long Range Dependence (LRD). LRD stands simply for the presence of strong auto-correlations even over large time lags. In the class we will show how LRD relates to the concept of self-similarity, i.e. the ``looking alike'' on all scales. With the example of Internet data traffic it will be developed how self-similarity can be used in modelling, numerically as well as analytically, and in particular it's implications to asymptotic queuing performance.

Multiplicative measures, in short cascades, were introduced by Mandelbrot in 1974 as models for intermittency in turbulence. Cascades can be viewed as generalizing the self-similarity of fBm and providing a class of processes with greater flexibility. Cascades possess a multi-fractal structure which goes beyond LRD and which is also present in network traffic, with relevance to queuing performance not only in the asymptotic regime. This makes cascades an ideal modelling tool.

In this course we balance an introduction to the mathematical background of fractals and multifractals with applications of practical importance, e.g. in Internet traffic modelling (in particular queuing and path-inference) and in image processing. Thereby, we will keep things as simple as possible, making the course accessible to a wide audience.

The course does not assume any pre-knowledge on fractals. Only a basic course in probability (ELEC 533) is required.


To get a rough idea on the material covered in this course point your browser to an earlier year's course schedule.


The grade will be computed from Homework, a Quiz and a Project, which consists in presenting a paper from the instructor's list. 

Topics covered in class

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities should also contact Disability Support Services in the Ley Student Center.
May 23, 2001.  Dr. Rudolf Riedi