STA 294 (a): WAVELETS and STATISTICS - Brani Vidakovic, Spring 1998

Coverage: Theory and practice of wavelets. Applications in statistics. Emphasis on real-life problems and procedures. Examples on practical use of wavelets.

Summary: The course is a mixture of theoretical and practical aspects of wavelets.

Part 1 Introduction to continuous and discrete wavelet transformations and connections with signal processing. Fourier analysis and construction of wavelets. Properties of MRA and scaling functions. Generalizations of wavelets. Biorthogonal Wavelets. Multivariate Wavelets.

Part 2 Statistical Modeling with Wavelets. Wavelet Random Variables and Densities. Wavelet Shrinkage and Applications in Data Denoising, Density Estimation. Applications in Image Processing, Turbulence, etc. The S+Wavelets Software will Support the Course.

Outline Syllabus:

An overview.

History. Mathematical preliminaries.

Continuous and discrete wavelet transformations. Construction and properties of some families of wavelets. Wavelet packets. Multiresolution analysis. Connections with signal precessing.

Software for wavelet analysis. S+, C, Mathematica, Matlab packages.

Wavelet random variables. Wavelet shrinkage, thresholding policies, choice of best basis.

Wavelet data processing. Wavelet regression. De-noising, density and function estimation. Traditional and Bayesian Wavelet shrinkage applications. Wavelet Deconvolutions.

Wavelets and time series. Scalograms. Wavelet transformations of stationary processes. Selfsimilarity. Wavelets in economics.

Multivariate case. Wavelet image processing. Wavelets in medicine.

Related Topics. Local Cosine Transformation. Non-decimated and A trous wavelet transformations. Biorthogonal wavelets. Applications of wavelets in numerical analysis. Wavelets in turbulence.

Notes:

Brani Vidakovic: Statistical Modeling by Wavelets.

Recommended Reading:

Bruce, A. and Gao, H-Y. S+Wavelets Manual.

Walter, G. G. Wavelets and other orthogonal systems with applications, CRC, 1994. Strang and Nguyen. Wavelets and Filterbanks.

Instructor's Handouts

PLAN for STA294a

Classes are Mondays 2:20 pm-5:10 pm in Old Chem 025,

The wavelet part [STA294 a] is on Jan. 19, 26, Feb 9,23, Mar 2, 30, and Apr 13!

The spatial statistics part [STA294 b] is on Jan. 19, Feb 2, 16, Mar 9,23, and Apr 6, 20!

Day	Topics	Notes
Jan. 19	Introduction	1.1 - 1.4
Jan. 26	The Wavelet Transform, Mathematical Background, Continuous Wavelet Transformations, Orthonormal Bases, Definition of an Orthonormal Wavelet. Wavelet Series and Discrete Wavelet Transformations, Fast Algorithms and Connections With Signal Processing.	3.1-3.5; 4.1-4.2,
Feb. 9	Some Genetalizations. Coiflets, Biorthogonal Wavelets, Wavelet Packets, Multivariate Wavelet Transformations. Nondecimated (stationary) Wavelet Transformations.	5.1-5.8
Feb. 23	Software. S-Plus and S+Wavelets. Wavelab and Mathematica Wavelet Packages, Nason's S-Plus Package. Hands-on S+Wavelets Demo.	Chapter 10
Mar. 9	Statistical Modeling by Wavelets. Wavelet Shrinkage. Wavelet Regression. Density Estimation by Wavelets. Bayesian Wavelet Srinkage. Random Processes and Time Series. Deconvolution and Classification Problems. Dimension Reduction Problems.	Chapters 7, 8, 9
Mar. 30	Applications of Wavelets in Various Fields. Data Compression, Image Processing, Ill-posed Problems, Numerical Analysis, Geophysics (Turbulence).	Handouts
Apr. 13	Presentation of Student Projects.

Prerequisites: Consent of instructor. Familiarity with programming in C, FORTRAN, SPlus, or Matlab is desirable.

Please send comments to brani@stat.duke.edu