Scale space view of curve estimation

Probal Chaudhuri, J. S. Marron
2000 Annals of Statistics  
Scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation. The family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface. The smoothing parameter here plays the same role as that played by the scale of resolution in a visual system. In this paper, we study in detail various features of that surface from a statistical viewpoint. Weak convergence of the empirical
more » ... e space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions. Our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data. In particular, we have used the scale space approach for the development of an e®ective exploratory data analytic tool called SiZer. SiZer is a graphical device for evaluating statistical signi¯cance of features (e.g. peaks and valleys) visible in a curve estimate by assessing the signi¯cance of zero crossings of the derivatives of that curve estimate at di®erent levels of smoothing.
doi:10.1214/aos/1016218224 fatcat:fzb65rbs75b3xk77bfchcp2bfi