Additions/Corrections for Wavelet Methods for Time Series Analysis

Preface

Chapter 1

• 1.0 says "comprehensive mathematical development, which by now is available in many other places," which would be one place to tack on some additional references (in addition to the end of the chapter)
• Q (Dean Billheimer) for 2nd edition: provide some intuition as to what the admissibility condition means.
• Abramovich, F., Sapatinas, T. and Silverman, B. W. (1998) "Wavelet Analysis and Its Statistical Applications," The Statistician, 49, to appear.

Chapter 2

• possible exercise (for Web site): does a_t \FTpair A_k imply A_k \FTpair a_t? The answer is "no", but one of the best students in Stat 530 got tripped up on this trivial point (check that the discussion re the notation is in fact clear).

Chapter 3

Chapter 4

• labels for figures should incorporate \deltt?
• a couple of ideas for the second edition or for Web site
  • get up example of filtering ECG series via ODFT details for a certain range of frequencies versus extraction of peaks via DWT-based scheme in keeping with clustering idea (this would also make a good additional exercise either for the book or for the Web page); we could cross-reference the discussion on clustering in Section 10.8 here.
  • new section discussing how DWT compares to STFT & complex demodulation (this might be best placed in Chapter 5)
• need to add item to index on page 103 for equivalent width (might add item for "effective width -- see equivalent width")

Chapter 5

• labels for figures should incorporate \deltt
• might add illustration of circularizing a time series: $X_0$, \dots, $X_{N-1}$, $X_{N-1}$, \dots, $X_0$.
• might expand discussion on problems of circularity assumption: it doesn't always make sense to treat the first and last observations as being adjacent, so circularity is an assumption that must be combated. In Fourier theory, we combat it by tapering or reflecting time series.
• In Exercise~\exeMODWTbb\ might note that $$ \langle {\cal D}_1, {\cal D}_{\trans,1} \rangle = {\bf W}_1^T {\cal B}_1 {\cal B}_{\trans,1}^T {\bf W}_{\trans,1} $$

Chapter 6

• Cohen, I., Raz, S. and Malah, D. (1997) "Orthonormal and shift-invariant wavelet packet decomposition," Signal Processing, 57, 251-270.
• Pesquet, J.C., Krim, H. and Carfantan, H. (1996) "Time-invariant orthonormal wavelet representations," IEEE Transactions on Signal Processing, 44, 1964-1970 [cited as origin of NWPT]
• Exercise [6.13] is more interesting than suspected: it yields an example of the nonoptimality of a "greedy" algorithm

Chapter 7

• (5/26/00): it might be useful to say more about the posterior density of theta given x. Something along the lines of: the prior expresses what is known (or assumed) about theta prior to observing x; the posterior pdf of theta given x expresses what we learned about theta as a result of observing x. As Section 7.3 now reads, the posterior given x just appears buried in the double integral of Exercise 265, which isn't very instructive. Also, it would be helpful to state that typically d(x) is some sort of decision about the value of theta and to state that a loss function must be nonnegative.
• might be useful to state for FGN, PPL and FD what the large tau form of the ACVS is (or make this an exercise?)
• might consider adding \eta = 2 case to Figure 327 (i.e., plot PDF of log(\chi^2_2) RV); could then reference this in Chapter 10 on the discussion of the non-Gaussianity of the log periodogram
• Possible small C&E for Section 7.6 (or maybe in the section describing white noise): Processes having SDFs that increase with increasing frequency are sometimes called blue noise; those with SDFs that decrease overall as f increases are sometimes called red noise processes.
• add comments from Mark re distinction between short memory and antipersistent (dig up Parzen reference on this material)

Chapter 8

• 2/01: the solution to Exercise [8.11] refers to two unnumbered equation. These should be numbered in the text, and the solution guide should then be updated.
• make proof of asymptotic normality part accessible via Web page
• add reference to scalogram or scalegram at beginning of Chapter 8 (i.e., the name preferred by Jeff Scargle)
• add a reference to Section 9.7 at the end of Section 8.6 (continuation of analysis of atomic clock data)
• Equation (372a) for computing \eta_2 might require a lot of work. A simplification is to assume that the wavelet filters are perfectly bandpass so that we now just sum S_X over 1/2^{j+1} <= f_k <= 1/2^j. It might be good to note this simplification (or make it the subject of an exercise).
• might comment about wavelet variance extensions to evolutionary processes; von Sachs & Schneider (1994): Wavelet Smoothing of Evolutionary Spectra by Non-Linear Thresholding; Tech Report from Kaiserlautern, but I think that it has been published by now; also Wang's work
• auxillary future exercise: Suppose that $S_j(f) = h C_j(f)$, where $C_j(\cdot)$ is known. Show that, by using appropriate Riemann sum approximations to the integrals involved, $\eta_1$ of Equation (357a) reduces to $\eta_2$ of Equation (358a).

Chapter 9

• Table 459 should refer to N'_j rather than M'_j (I think -- compare the discussion in Section 9.8 with the material in 9.6). Also we might want to give the critical values from Equation (451) in Table 459 for the cases with sample sizes 331 and 165). Finally it wasn't clear to Bill how we got the percentage points in the table -- these are independent of the wavelet filter (i.e., can just use white noise).
• possible new reference: "Non-stationary log-periodogram regression," Velasco C, JOURNAL OF ECONOMETRICS, 91: (2) 325-371 AUG 1999
• labels for figures should incorporate \deltt
• need to work in reference to Wang's Annals of Statistics article (24, 1996); this is on function estimation via wavelet shrinkage for long memory processes -- obviously has some relationship to tred estimation question
• Q for more study: how well does the MLE of Section 9.3 work when delta < 0?
• Q for more study: with regard to the MLE of Section 9.3, might use cycle spinning to improve estimate
• Q for more study: with regard to the MLE of Section 9.4, if there is no trend, might use reflection boundary conditions and cycle spinning to improve estimate when $1/2 \le \delta < 1$ (say); if $\delta \ge 1$, difference and use estimator of 9.3 or 9.4; if estimated \delta < 1/2, might switch back to estimator of Section 9.3
• ANOVA
• possible new section for second edition: estimation of stochastic fractal signal (i.e., fBm) in the presence of white noise. There are several articles in the EE literature that address this subject, including Wornell & Oppenheim (1992), Wornell's book, and Hirchoren and D'Attellis, "Estimation of Fractal Signals Using Wavelets and Filter Banks," IEEE Trans. Sig. Proc., vol. 46, June 1998, 1624-30 (I have a copy). This subject is closely related to Wiener filtering, so it would like up with our discussion in Chapter 10.

Chapter 10

• correct threhold (page 483)
• There is still a question regarding differentiability viz a viz SURE. The text starting "Using integration by parts" is based on a result in Hardle (p72, 2), namely, $$ \int f(u) a'(u) du = - \int a(u) f'(u) du, $$ for which we probably should add a reference; however, it seems that Hardle p72 assumes the fn a(.) to be of bounded support, but in the SURE application it is the exp(.) fn, which surely does not have bounded support??
• I am not getting good agreement with S+Wavelets in the SURE-bases estimate of the NMR. These calculations should be carefully checked while CUP is editing the book.
• If time permits, add brief reference to confidence limits for waveshrink ala Bruce & Gao's Biometriak article.
• If time permits, add brief reference to Clyde's work on multiple shrinkage.
• compute expected value of the periodogram and MT estimator for Moulin's MRC spectrum (more for general interest than for inclusion in the book)
• supplementary exercise for Section 10.7: apply mt+waveshrink to beam-1 time series
• application of mt+ws to speech data for getting cepstrum (Jim Pitton has data)

Chapter 11

• several figures should have a \vfill before \endinsert
• add reference for analytic derivation for D(6) scaling filter (ATW has it)
• get Exercise [11.2] sorted out (proof that wavelet functions provide a basis for W_0 etc)
• possible new exercise from ATW
• possible new exercise: create figure showing the four possible L=8 Daubechies wavelet filters (impulse response sequences and phase plots, after shift giving least deviation from linear phase); verify that stated choice satisfies the LA criterion

Appendix

• replace solutions for Exercises [116] and [201b] with improved ones by J. Cooke (might add his name to Preface)
• do NOT end a solution with a displayed equation!!!

References

• "North Holland" should be "North--Holland"
• Graybill, 1983, is referenced in the Appendix and nowhere else.

General Comments

• 2/5/01: go through and change all "\,\bmod\," to just \bmod (there are several in Chapter 8 at least).
• need to carefully distinguish between the width and length of a filter
  • length is either N (circular filters) or infinite (i.e., impulse response function is defined for all t, even if all but a finite number are zero)
  • width L says only L contiguous values are (possibly) nonzero
  • equivalent width
• use "--" rather than "-" between hypenated names (e.g., Prentice--Hall, Box--Jenkins and Foufoula--Georgiou); checked this on 7/7/99 in Chapter 1 and references; need to check eventually in remaining chapters (search or grep for "-").
• Somewhere it would be good to summarize the basic properties of the DWT somewhat along the lines of the hidden Markov model article. One aspect that needs to be emphasized is the decorrelation of stationary processes. The point to make is that the DWT will be an approximate decorrelator if the process has little structure over the nominal passband. Interesting thought: with the reversal of frequencies caused by downsampling the result of high pass filtering, it would seem that the wavelet coefficients for fdp are in fact slightly anticorrelated; i.e., the sdf is slightly bluish.
• time series that might make good examples
  • Kobe earthquake data
  • Melanie's brut data (spelling?)
  • whale data from John Hopkins APL
  • snapping shrimp data
  • Julian Douglass's data

Things to check

• na{\" \i}ve instead of naive (use grep to detect)
• SDF instead of sdf (use grep to detect); ACVS instead of acvs etc
• level (rather than order)
• DWT coefficients = wavelet coefficients + scaling coefficients; on occasion, can say "DWT wavelet coefficients" if they are being contrasted with "MODWT wavelet coefficients"
• sequence of things to check when getting ready to go:
  • spell check (preferably Excalibar and emacs)
  • check TeX log
  • NEWRIGHT check; orphan check
  • special figure placement; heading check
  • MCL double-check of equation numbers
• check for \NEWRIGHT{\ChapterNumber.\DWTwfilter\ \ The Wavelet Filter}%
• check that \Websitepage is used properly as per examples in all-the-global-eqndefs (have done this just for Chapter 10 so far); grep on xxv to check for places where \Websitepage should be substituted
• data are plural (checked in all chapters and appendix on 9/7/99 -- might double-check Chapter 9 after it is complete)
• tick (not tic)
• hypenation conventions used in SAPA (check via grep)
  • low-pass, high-pass, band-pass, all-pass, pass-band (checked all chapters for these 1/9/98)
  • bandwidth
  • sidelobe
  • nonstationary
  • non-Gaussian
• t=0, \pm 1, \pm 2, \ldots might be confusing in that the sequence could be interpreted as X_0, X_1, X_{-1}, X_2, X_{-2}, ...; it might be better to use t = \ldots, -1, 0, 1, \ldots or possibly t \in \Zvecspace; in any case, clarify choice in section on notation
• use "full DWT" (as opposed to "complete DWT"); try to avoid "standard DWT", but, if used, this should refer to the partial DWT rather than the full DWT
• ANOVA (introduced this in all chapters except 9 as of 9/9/99 -- Chapter 9 was not in decent shape on that date)