- Email: [email protected]
Discussion: Time-threshold maps: Using information from wavelet reconstruction with all threshold values simultaneously
Journal of the Korean Statistical Society 41 (2012) 169–170
Contents lists available at SciVerse ScienceDirect
Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Discussion
Discussion: Time-threshold maps: Using information from wavelet reconstruction with all threshold values simultaneously Thomas C.M. Lee University of California at Davis, United States
article
info
Article history: Received 30 January 2012 Available online 19 February 2012
In the paper under discussion, Prof. Fryzlewicz proposes a novel graphical technique that helps with identifying structures hidden in data. This new technique, termed the time-threshold map (TTM) technique, utilizes and combines all the information extracted from various wavelet reconstructions computed at different scales. This paper is well written, and studies an interesting and popular topic. Prof. Fryzlewicz is to be congratulated for his achievement. After reading Prof. Fryzlewicz’s paper, the first thing that came to my mind was the graphical technique SiZer, proposed by Chaudhuri and Marron (1999). I do not think that SiZer and TTM are competitors. Rather, I see them complementing each other: SiZer is very good at picking up increasing/decreasing trends in the data, while TTM is excellent in revealing breakpoints and testing for homogeneities. TTM has an advantage, though: the computation is quicker, as the construction of a SiZer map requires the execution of many hypothesis tests. TTM has another attractive property: it can be easily modified to handle different situations. For example, wavelet techniques for nonparametric regression are sensitive to outliers, which would in turn affect the performance of TTM. This can be simply fixed by the use of robust wavelet methods when estimating the signal at different scales (e.g., Oh, Nychka, & Lee, 2007; Sardy, Tseng, & Bruce, 2001). As another example, for the purposes of examining various quantiles of the signal, one could simply use wavelet quantile smoothing techniques (e.g., Oh, Lee, & Nychka, 2011) when constructing the TTM map. Of course, additional analytical work may be required to provide supporting theories (e.g., like Theorem 4.1 in the paper) for these TTM variants, but, at least in principle, it shows the great flexibility of the TTM methodology proposed by Prof. Fryzlewicz. My last comment concerns the artificial signal X˜ t defined by Eq. (7) of the paper. It is a clever construction: the integration operation ‘‘accumulates’’ the trends (i.e., low frequency components) in the original signal Xt with the result that these trends can be detected more easily, while at the same time it stabilizes the noise (i.e., high frequency components). Perhaps one could iterate this idea further: for example, apply the integration operation to X˜ t to obtain yet another signal, which should make the detection task even easier. Again, Prof. Fryzlewicz is to be congratulated for such a thought provoking paper.
References Chaudhuri, P., & Marron, J. S. (1999). SiZer for exploration of structures in curves. Journal of the American Statistical Association, 94, 807–823. Oh, H.-S., Nychka, D., & Lee, T. C. M. (2007). The role of pseudo data for robust smoothing with application to wavelet regression. Biometrika, 94, 893–904.
E-mail address: [email protected]. 1226-3192/$ – see front matter © 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2012.01.009
170
T.C.M. Lee / Journal of the Korean Statistical Society 41 (2012) 169–170
Oh, H.-S., Lee, T. C. M., & Nychka, D. (2011). Fast nonparametric quantile regression with arbitrary smoothing methods. Journal of Computational and Graphical Statistics, 20, 510–526. Sardy, S., Tseng, P., & Bruce, A. (2001). Robust wavelet denoising. IEEE Transactions on Signal Processing, 49, 1146–1152.
Contents lists available at SciVerse ScienceDirect
Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Discussion
Discussion: Time-threshold maps: Using information from wavelet reconstruction with all threshold values simultaneously Thomas C.M. Lee University of California at Davis, United States
article
info
Article history: Received 30 January 2012 Available online 19 February 2012
In the paper under discussion, Prof. Fryzlewicz proposes a novel graphical technique that helps with identifying structures hidden in data. This new technique, termed the time-threshold map (TTM) technique, utilizes and combines all the information extracted from various wavelet reconstructions computed at different scales. This paper is well written, and studies an interesting and popular topic. Prof. Fryzlewicz is to be congratulated for his achievement. After reading Prof. Fryzlewicz’s paper, the first thing that came to my mind was the graphical technique SiZer, proposed by Chaudhuri and Marron (1999). I do not think that SiZer and TTM are competitors. Rather, I see them complementing each other: SiZer is very good at picking up increasing/decreasing trends in the data, while TTM is excellent in revealing breakpoints and testing for homogeneities. TTM has an advantage, though: the computation is quicker, as the construction of a SiZer map requires the execution of many hypothesis tests. TTM has another attractive property: it can be easily modified to handle different situations. For example, wavelet techniques for nonparametric regression are sensitive to outliers, which would in turn affect the performance of TTM. This can be simply fixed by the use of robust wavelet methods when estimating the signal at different scales (e.g., Oh, Nychka, & Lee, 2007; Sardy, Tseng, & Bruce, 2001). As another example, for the purposes of examining various quantiles of the signal, one could simply use wavelet quantile smoothing techniques (e.g., Oh, Lee, & Nychka, 2011) when constructing the TTM map. Of course, additional analytical work may be required to provide supporting theories (e.g., like Theorem 4.1 in the paper) for these TTM variants, but, at least in principle, it shows the great flexibility of the TTM methodology proposed by Prof. Fryzlewicz. My last comment concerns the artificial signal X˜ t defined by Eq. (7) of the paper. It is a clever construction: the integration operation ‘‘accumulates’’ the trends (i.e., low frequency components) in the original signal Xt with the result that these trends can be detected more easily, while at the same time it stabilizes the noise (i.e., high frequency components). Perhaps one could iterate this idea further: for example, apply the integration operation to X˜ t to obtain yet another signal, which should make the detection task even easier. Again, Prof. Fryzlewicz is to be congratulated for such a thought provoking paper.
References Chaudhuri, P., & Marron, J. S. (1999). SiZer for exploration of structures in curves. Journal of the American Statistical Association, 94, 807–823. Oh, H.-S., Nychka, D., & Lee, T. C. M. (2007). The role of pseudo data for robust smoothing with application to wavelet regression. Biometrika, 94, 893–904.
E-mail address: [email protected]. 1226-3192/$ – see front matter © 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2012.01.009
170
T.C.M. Lee / Journal of the Korean Statistical Society 41 (2012) 169–170
Oh, H.-S., Lee, T. C. M., & Nychka, D. (2011). Fast nonparametric quantile regression with arbitrary smoothing methods. Journal of Computational and Graphical Statistics, 20, 510–526. Sardy, S., Tseng, P., & Bruce, A. (2001). Robust wavelet denoising. IEEE Transactions on Signal Processing, 49, 1146–1152.