ELEC 697: Introduction to Fractal Processes (1998)


Tuesday, Thursday: 2:30 - 3:50 pm, PL 119 (from Duncan Hall cross the Inner Loop)

Selected Homework Problems


My office

Duncan Hall 2025,  x 3020. Please, schedule meetings by email.


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.

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 describe their highly complex dependence structure.

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

Homework will be assigned on an irregular basis. Students who wish to claim credit are expected to do a project which may consist of presenting a research article or programming. The course is fairly distinct of last year's ELEC 697 and does not assume any pre-knowledge on fractals. Only a basic course in probability (ELEC 533) is required.

The structure of the course is as follows.