STA 2270HS -- INTRODUCTION TO WAVELET METHODS IN STATISTICS

Wavelets are a recently developed set of tools with widespread applications in science, engineering, technology and mathematics. They involve new classes of very unusual functions with a surprisingly broad range of cross-disciplinary applications. Due to remarkable `local adaptivity' properties, wavelets have revolutionized the way people think about problems in such fields as signal and image processing, telecommunications, data compression and reconstruction, pattern recognition, and medical imaging, to name only some. What then can wavelet methods offer to statistics? This has been the subject of a considerable amount of work in recent years and the purpose of this course is to explore this question. In doing so, we will have a chance to examine nonparametric statistical estimation in detail, particularly of density functions and regression functions.

To properly understand wavelet methods involves considerable `overhead' because wavelets build upon many classical techniques (such as Fourier analysis, for example). Since it is NOT assumed that students will have had previous exposure to these areas, a significant part of the course will be devoted to carefully developing the needed background. The approach will be largely intuitive, although in a few cases rigorous proofs may be given. Computational issues will be dealt with in detail using both S-Plus and MatLab based wavelets software.

The course content will include:

• Overview of L^p and Hilbert spaces; bases and frames.
  
• Detailed introduction to Fourier series and transforms; applications of the FFT in statistics.
  
• Continuous and discrete wavelet transforms.
  
• Multiresolution analysis and the construction of wavelets.
  
• Nonparametric regression and density function estimation; cross-validation.
  
• Wavelet smoothing; shrinkage and thresholding methods.
  
• Image processing and denoising applications.
  
• Computational aspects; fast wavelet algorithms; introduction to wavelet software in S-Plus and MatLab.

GRADING: Grading will be based primarily on individual projects. Some problems and computing exercises will also be assigned. Students may choose either a theoretical or an applied (e.g. computational) topic (or a combination) depending on their interests. Further details will be available at the first meeting.

Permission of the instructor is required.

Some References:

Abramovich, F., Bailey, T.C. and Sapatinas, T. (2000). Wavelet analysis and its statistical applications. The Statistician -- J. Royal Statist. Soc., Ser D.
Antoniadis, A. (1998). Wavelets in statistics: A review. To appear in J. Italian Statist. Soc.
Antoniadis, A., Gregoire, G. and McKeague, I. (1994). Wavelet methods for curve estimation. J. Amer. Statist. Assoc., 89, 1340-1353.
Burrus, C.S., Gopinath, R.A. and Guo, H. (1998). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice Hall, NJ.
Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. SIAM, Philadelphia, PA.
Debnath, L. (2002). Wavelet transforms and Their Applications. Birkhauser.
Mallat, S.G. (1998). A Wavelet Tour of Signal Processing. Academic Press. San Diego.
Ogden, R.T. (1997). Essential Wavelets for Statistical Applications and Data Analysis. Birkhauser, Boston.
Pinsky, M.A. (2002). Introduction to Fourier Analysis and Wavelets. Brooks/Cole.
Vidakovic, B. (1999). Statistical Modeling by Wavelets. Wiley, New York.
Walnut, D.F. (2001). An Introduction to Wavelet Analysis. Birkhauser.
There are many interesting websites associated with wavelets. One particularly useful one is for the Wavelet Digest -- an electronic bulletin board about wavelet conferences, workshops, courses, software, tech reports, etc.