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
Instructor
Dr. Rudolf Riedi
Duncan Hall 2025, 713 / 348 3020,
Office Hours: By appointment
Time/Location
Tuesday/Thursday 2:30-3:50, AL B209
Requirement
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
Introduction
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.
Syllabus
-
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)
To get a rough idea on the material covered in this course point your browser
to an earlier year's
course schedule.
Grade
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
-
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. Dr.
Rudolf Riedi