ELEC 697: Introduction to Fractal
Tuesday, Thursday: 2:30 - 3:50 pm, PL 119 (from Duncan Hall cross the Inner
Selected Homework Problems
Brownian motion. Statistical scaling, iterative construction (midpoint
Dimensions: geometrical scaling. Self-similar paths.
Hoelder continuity and dimension I. Intro + Ex: path of Brownian motion
Hoelder continuity and dimension II. Rigorous arguments.
Box dimension, Diophantine examples
Self-similar processes, simple properties
The unique Gaussian self-similar process: fBm (definition through correlation
Martingales, stochastic integrals I
Stochastic integrals II, Ito's formula
Second definition of fBm (as Ito integral of Brownian motion)
Second order self-similarity
Synthesis of fBm I: Midpoint, FFT, Cholesky
Synthesis of fBm II: Crouse, FARIMA, Weierstrass-Mandelbrot. Stable laws.
Wavelet decomposition of fBm: decorrelation and synthesis.
Wavelet-based estimation of LRD for self-similar.
Large deviation principles (LDP), ex: coin toss, appl: buffer-overflow
of fGn-based traffic model
Local singularity exponents, via increments and...
...via wavelets. Two-microlocalization.
Binomial cascade: singularities (via increments) and dyadic expansions,
Binomial cascade: multifractal structure (via increments and wavelets).
Multifractal formalism as a LDP
Modeling data traffic via cascades, statistical advantages over fGn
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.
Introduction: Self-similarity, iteration, dimensions
Brownian motion: a fractal
fBm: Self-similarity and LRD
Data traffic modeling (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)