Scale space view of curve estimation. (English) Zbl 1106.62318

Chaudhuri, Probal; Marron, J. S.

Ann. Stat. 28, No. 2, 408-428 (2000).

Summary: 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 scale 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 effective exploratory data analytic tool called SiZer.

MSC:

62G07	Density estimation
62H35	Image analysis in multivariate analysis
62G05	Nonparametric estimation

Keywords:

Causality; Gaussian kernel; heat diffusion; regression smoothers; mode and anti-mode trees; significance of zero crossings

Software:

SiZer; ggcleveland; KernSmooth

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