Contents

Course Syllabus
Handouts
HW
Student Projects
Reveille's Wavelets

WAVELET-BASED SCALING ASSESSMENT AND APPLICATIONS

MAKING SENSE OUT OF NOISE

Wavelets, Fractals, Hurst Exponent, Scaling, and Applications

Philosophy

The course will cover advanced wavelet-based algorithms designed primarily for summarizing large and noisy data sets for a subsequent statistical inference and processing. The theoretical component of the course will provide unified multiresolution-based framework for efficient modeling, synthesizing, analyzing, and processing several broad classes of fractal signals. Applications are numerous. Whenever observed data are noisy, the properties of noise may provide important and unutilized modality in subsequent data modeling and analysis.

The objective of the course is to familiarize students with wavelets and use of wavelet domains for assessing self-similarity, degree of fractality, and scaling in data. Methods will be motivated by real-life applications. The resulting low-dimensional summaries could serve as inputs to machine learning tools. The course will primarily have an applied and computational focus, although the theoretical framework will be covered as well.

Texts

• Instructor's Handouts
• Brani Vidakovic, Statistical Modeling by Wavelets. Wiley 1999.
• Gregory Wornell, Signal Processing with Fractals. A Wavelet Based Approach, Prentice Hall 1995.
• Patrick Flandrin, Time-Frequency/Time-Scale Analysis. Academic Press 1999.

• Agenda

• Colored noise. Self-similar time series. Short and long memory. Brownian motion. Fractional and multifractional Brownian motions. Fractals.
• Fourier transform. Statistical signal processing. Spectral analysis. Introduction to wavelets.
• Software implementation of wavelets. Matlab (Octave) and Python.
• Multiscale domains, Continuous and discrete wavelet transform, Nondecimated wavelets, Wavelet packets, Wavelets in n-D.
• Estimation of Hurst exponent. Various methods.
• Multifractals via a wavelet lens, Bin's triangle, Testing for monofractality. Scalograms and energygrams.
• Wavelet image processing. Assessment of self-similarity in images and applications.
• Applications of wavelet-based scaling indices in Geosciences (turbulence), Genomics (selfsimilarity in DNA), Microeonomics (stock market and exchange rates), and Health (breast cancer diagnostics)
• Brani Vidakovic