CNR IAMI
Istituto per le Applicazioni
della Matematica e dell'Informatica

via A. M. Ampère 56 - 20131 Milano (Italy)

COURSE on
WAVELETS: BASICS AND STATISTICAL APPLICATIONS

December, 13-15, 1999

Brani Vidakovic, ISDS, Duke University

Schedule

                    Monday,    December, 13th, 14.30-16.30

                    Tuesday,   December, 14th, 10.30-12.30

                    Tuesday,   December, 14th, 14.30-16.30

                    Wednesday, December, 15th, 10.30-12.30

This mini-course on wavelets and statistics will have 2 parts consisting of four 1-hour lectures each. The first part introduces multiscale methods to a novice. Familiarity with the Fourier transformation and Hilbert spaces is desirable, but not crucial.

The second part discusses various applications of wavelets in statistics. It covers standard statistical applications, such as regression, density estimation and analysis of time series, and some non-traditional methods as well. Exposure to statistics at an introductory or intermediate level is desirable.

Extensive references are provided as well as pointers to web sources. Plans for the lectures are provided next.

Recommended Text: Vidakovic, B. Statistical Modeling By Wavelets, Wiley 1999.


Recommended Additional Reading: Daubechies, I. 10 Lectures on Wavelets, Siam.
                                Nguyen, Strang, Wavelets
                                Mallat, Wavelet Tour, AP 1998.

PART 1

Lecture 1. Wavelets: Why?

This introductory lecture is a ``commercial'' for usage of wavelet based tools. It discusses, at an informal level, decorrelating, self-similarity, and ``disbalancing'' properties of wavelets.
Interesting applications are presented and ramifications of wavelet methods are discussed.

Lecture 2. Wavelets: How?

By utilizing the Haar basis the construction and basic properties of wavelets are illustrated. Several techniques of constructing wavelets are explored as well. Standard families of wavelets (Shannon, Meyer, Franklin, Daubechies) are briefly overviewed.

Lecture 3. Discrete Wavelet Transformations and Variations

In this lecture we discuss Mallat's algorithm and its connection with continuous wavelet transformations and wavelet series. Wavelet packets and stationary wavelet transforms are discussed. Links between regularity of functions and rates of decay of their wavelet coefficients are discussed. Moment conditions (Strang-Fix condition, interpolating wavelets, coiflets) are overviewed, as well.

Lecture 4. Beyond Orthogonal Wavelet Transformations

In this lecture we discuss bivariate wavelets, wavelet-like bases, matching pursuit methods and more general classes of time/scale representations (Cohen Class Distributions).

PART 2.

Lecture 5. Regression via Wavelet Shrinkage.

Standard wavelet regression problem is presented. Shrinkage paradigm, Minimax, Exact risk, Bayesian and Bayesian decision-theoretic shrinkage and their optimality properties are discussed.

Lecture 6. Standard Statistical Wavelet Based Methods.

In this lecture an application of wavelets in density estimation and time series is covered. Wavelet methods of dealing with self-similar data and use of wavelets for generating random objects (random functions and densities) is discussed.

Lecture 7. Non-Standard Statistical Wavelet Based Methods.

Lecture 8. A Case Study: Functional Data Analysis via Wavelets

Utilizing multiresolution methods in functional data analysis (FDA) is beneficial in several respects. The fact that wavelets decorrelate the data makes the functional inference in the wavelet domain easy to implement and interpret.
We discuss two FDA applications: Dimension reduction problem and wavelet-based ANOVA.
The application in the second problem is illustrated by an ongoing research project at Cancer Center at Duke University.