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

(i.e. a basic understanding of conditional expectation, random processes, and auto-correlation).

**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

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.*

- Introduction: Self-similarity, iteration, dimensions
- Brownian motion: a fractal
- fBm: Self-similarity and LRD
- Data traffic modelling (via self-similar processes)
- Cascades: paradigm of multifractal behavior
- Large deviations: the multifractal formalism
- Multifractal properties and LRD of cascades
- Data traffic modelling (via multifractal processes)
- Image processing (via multiplicative de-compositions)

**Topics covered in class**

- Space filling curves, Self-similarity
- Hausdorff- and Box-dimensions
- Hoelder continuity, dim of graphs and images
- IFS, Wiener process as random walk
- Wiener process by midpoint displacement, dim of path
- Self-similar processes
- Finite variance, Gaussian case
- Self-similarity and Long Range Dependence (LRD)
- Stochastic `white noise' integral
- Infinite variance, stable case
- Traffic modeling: Aggregation and Self-similarity
- ON-OFF model: Fractional Brownian limit
- ON-OFF model: Stable Levy limit
- Estimators of LRD: time domain
- Estimators of LRD: spectral domain
- Wavelets: estimation and modeling of LRD
- Cascades: positive LRD processes
- Large Deviation Principles (LDP) and LLN
- LDP and multifractal scaling (beyond LRD)
- Multifractal formalism

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.