Comments

This splendid paper provides an irresistible opportunity to comment on several issues connected with the analysis of functional data or the use of functional models. The most important of these is, of course, to congratulate the authors warmly on beginning a new chapter on the analysis of data characterized by sharply localized events separated by long, smooth gaps. Wavelets are the obvious choice among bases for this problem, and who is better positioned than they are to put these fascinating little wigglers to work on this, the principal components analysis, problem? Getting along without wavelets is like an orchestra playing music without a percussion section. In the design stages of our work on functional data (Ramsay and Silverman 2002, 2005), we reflected on giving wavelets more than passing mention as basis systems, but they were still a comparatively exotic topic during the early ’90s and we had already committed ourselves to optimizing readability and to a general focus on the estimation of smooth variation. The article by Johnstone and Lu effectively highlights the contrast between smooth and local variation in functional data, and it was gratifying to see that smooth principal components analysis did such a fine job of showing itself to be the wrong tool. But are smoothness and sparsity really such different concepts? Functions can be ‘‘small’’ in two quite orthogonal ways—most obviously in terms of their amplitude variation, but perhaps less clearly to those less familiar with higher analysis, in terms of the dimension of a function space required to represent the signal adequately. The latter concept, function complexity, is often expressed in terms of the rate of decay of an infinite series of smooth basis functions, but wavelets put the matter up front where nobody can miss the point. The high-dimensional modeling context, referred to often as p >> n, is receiving a lot of attention these days, but in the end our strategy does not seem to change much. We seek a model structure that is usefully identified by the available data. What has changed is the range of options for model structures, with the possibility of defining simplicity in new ways, such as the number and complexity of a set of nonlinear differential equations with solutions that exhibit catastrophic or chaotic variation, as deviations from the quantile function for the uniform distribution, or as the number lines of computer code required to display data structure. Combining wavelets with principal components analysis is the ideal strategy for the exploration of sparse variation, and both concepts are deeply rooted in the traditions of applied mathematics and statistics. Where do we go from here? The focus of the article by Johnstone and Lu is rather exclusively on variation defined by a single principal component, and maybe this problem has rather more to do with data smoothing than it has to do with classic principal components analysis. For example, would it be better to treat the scores on this single component as multiplicative random effects, so that their variances have parametric status rather than being just handy indices for defining hard thresholds? Substantial contributions from even a few additional principal components makes the problem much tougher by deemphasizing eigenvectors or eigenfunctions, and calling for coordinate-free methods for assessing subspace identification. I hope to see a flurry of new papers extending this work in this direction, as well as in the direction of functional linear modeling and dynamic systems identification. The bipolar spikes in the right panels of the displays of ECG variation are, in my experience, the signature of phase variation, and seem most likely to be the result of slight misalignments of the R wave, perhaps related to the use of dyadic sampling point sequences. It seems intuitive that sparse signals of this nature will routinely show horizontal as well as vertical displacements, and that registration methodology will wind up being as important for the analysis of sparsity as it has been for smoothness.