Instructor

• phone numbers: 206-543-1368 (APL office) and 206-543-3866 (Department of Statistics office)
• e-mail address: dbp@uw.edu
• office hours: Mondays and Wednesdays from 11:00AM to 12:15PM in C-310 of Padelford Hall (quite close to the Math Library)

Textbook

In addition to the textbook, here are a list of books that would be good supplements for the material that we will be covering.

1. `A First Course in Wavelets with Fourier Analysis,' by Albert Boggess and Francis Narcowich (Prentice Hall, 2001), a nice blending of wavelets with mathematical Fourier theory;
2. `Practical Time-Frequency Analysis: Gabor and Wavelet Transforms with an Implementation in S,' by Rene Carmona, Wen-Liang Hwang and Bruno Torresani (Academic Press, 1998), a statistically oriented approach;
3. `Ten Lectures on Wavelets,' by Ingrid Daubechies (SIAM, 1992), a thorough look at the mathematical details behind wavelets by a leading researcher;
4. `Time-Frequency/Time-Scale Analysis,' by Patrick Flandrin (Academic Press, 1999), an engineering/mathematical approach that is strong on comparing wavelets with other approaches;
5. `An Introduction to Wavelets and Other Filtering Methods in Finance and Economics,' by Ramazan Gencay, Faruk Selcuk and Brandon Whitcher (Academic Press, 2002), an approach oriented toward applications in finance and economics (this book uses much the same notation as the course textbook, and the third author is a Stat 530 alumnus!);
6. `Wavelets: Algorithms & Applications,' by Stephane Jaffard, Yves Meyer and Robert Ryan (SIAM, 2001), a nice overview of selected aspects of wavelets;
7. `A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way' by Stephane Mallat (Academic Press, 2009), a mathematically oriented approach;
8. `Wavelet Methods in Statistics with R,' by Guy Nason (Springer, 2008), a statistically oriented approach centered around the R package WaveThresh4;
9. `Essential Wavelets for Statistical Applications and Data Analysis,' by Todd Ogden (Springer Verlag, 1996), a statistically oriented approach requiring less of a statistical background than the ones by Carmona et al. or Vidakovic;
10. `Wavelets and Filter Banks,' by Gilbert Strang and Truong Nguyen (Wellesley-Cambridge Press, 1996), a mathematically oriented treatment;
11. `Wavelets and Subband Coding,' by Martin Vetterli and Jelena Kovacevic (Prentice Hall, 1995), a very thorough book emphasizing the engineering approach to the theory of wavelets;
12. `Statistical Modeling by Wavelets,' by Brani Vidakovic (John Wiley & Sons, 1999), a statistically oriented approach; and
13. `Signal Processing with Fractals: A Wavelet-Based Approach,' by Gregory Wornell (Prentice Hall, 1996), an engineering oriented treatment.

There will be seven homework assignments. I will post a problem set on Wednesdays accessible via the Assignments link on this Web site (the first set will be posted on 28 March, and the seventh (and final) set, on 9 May). Each assignment will consist of (1) `self-graded' exercises and (2) exercises that will be collected at the beginning of the Wednesday class one week after the problem set is posted (the first set will be due on 4 April, and the seventh, on 16 May). Late homework will not be accepted. Electronic submissions will not be accepted - you need to turn in a hardcopy of your solutions to me in class (I am only regularly over in Padelford Hall during my office hours, so placing your homework either under my Padelford office door or in my Department of Statistics mailbox won't work). After you have worked on the self-graded exercises, you should check your answers by consulting the solutions given in the appendix to the book starting on page 501 (if you feel that your solution is markedly better than the one given there or if your solution takes an interesting alternative approach, please turn it in to me, and I'll give you extra credit if I agree with your assessment; otherwise, these solutions should not be turned in to me). Some of the self-graded exercises might inspire part of the in-class exam (see below).

Students in the past who have gotten the most out of this course have worked on the homework assignments by themselves. I thus discourage group efforts, but feel free to discuss homework problems with your classmates as long as the solutions you turn in reflect your own work and not a group effort (copying is certainly not permitted either from each other or from solutions to certain problems that can be tracked down from the Internet or elsewhere). While grading of your homework will be based primarily on technical correctness, lack of clarity and sloppiness can affect your grade. If I have to spend extra time trying to figure out exactly what your solution is or if it is not clear what steps you took in getting to your answer, I will deduct points - take this as fair warning to be as clear as possible! Also I frown upon excessive reliance on Mathematica (or software of a similar ilk). If a homework problem says `assuming X to be true, show that Y follows,' I do not consider `I took X, put it into Mathematica, and Y popped out' to be an acceptable answer.

There will be a 50 minute exam on Friday, 25 May, which will count for 20% of your course grade. The exam will be closed book, and use of any sort of electronic device during the exam is not allowed; however, you will be allowed to bring in a single sheet of standard 8.5 by 11 inch paper on which you can write anything you so desire (you may use both sides of the paper). Some of the problems on the exams might be inspired by the self-graded exercises in the homework assignments.

The term project is an essential and important part of this course. It could be a data analysis, a simulation study, methodological or theoretical research, or a report on a research article of interest to you. Topics for the project must be approved by me no later than Friday, 11 May. You are expected to provide a concise written report (approximately 5 pages to a maximum of 10 pages in length; double-spaced single-column pages using a font size that won't strain my eyes - a 9 point font is recommended). The report will be due by 2PM on Friday, 8 June (the last day of the quarter). I will need both a paper copy (to be handed to me in my office (C-310, Padelford) or placed in my mailbox in the Department of Statistics) and a PDF file (to be e-mailed to me (dbp@uw.edu)). You should plan on giving a short presentation about your term project (5 to 10 minutes in length, depending on class size - the exact length will be announced in April). The presentations will be given during either one of the final two class periods (Wednesday, 30 May, and Friday, 1 June, 12:30PM to 1:20PM) or the time slot allocated for a final exam for the class (Thursday, 7 June, starting at 8:30AM). For more information, see the Term Project link on this Web site.

Course Outline

• introduction to wavelets via the continuous wavelet transform
• review of fundamental properties of the discrete Fourier transform and of filtering theory; orthonormal transforms of time series; the orthonormal discrete Fourier transform; the discrete Fourier power (variance) spectrum
• the orthonormal discrete wavelet transform (DWT); Haar and other Daubechies class wavelets; the wavelet empirical power spectrum; multiresolution analysis (wavelet details, smooths and roughs); wavelet filters; quadrature mirror filters; scaling filters; pyramid algorithm for computing DWT; the Daubechies `extremal phase' and `least asymmetric' classes of wavelet filters; phase properties of Daubechies wavelet filters; coiflets; examples; practical considerations
• the maximal-overlap discrete wavelet transform (MODWT); shift invariant properties of MODWT; pyramid algorithm for computing MODWT; variance analysis using MODWT; multiresolution analysis using MODWT; examples
• discrete wavelet packet transform; wavelet packet tables; best basis algorithm; maximal overlap discrete wavelet packet transform; matching pursuit
• review of basic theory for random variables and stochastic processes, including stationary processes, long memory processes, fractionally differenced processes and processes with stationary backward differences
• the wavelet variance and its application to stochastic processes with stationary backward differences; estimation of the wavelet variance; confidence intervals for the wavelet variance; examples
• smoothing and denoising time series using wavelet shrinkage; thresholding schemes for wavelet shrinkage; spectral density function estimation via wavelet shrinkage
• analysis and synthesis of long-memory processes using wavelets
• translation and dilation of signals; scaling functions and approximation spaces; theory of multiresolution analysis; wavelet functions and detail spaces; two-scale relationship for scaling functions; relationship between multiresolution theory and filtering theory; construction of wavelet functions; vanishing moments and regularity