The Annals of Statistics

Scale space view of curve estimation

Probal Chaudhuri and J. S. Marron

Full-text: Open access


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.

Article information

Ann. Statist., Volume 28, Number 2 (2000), 408-428.

First available in Project Euclid: 15 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation

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


Chaudhuri, Probal; Marron, J. S. Scale space view of curve estimation. Ann. Statist. 28 (2000), no. 2, 408--428. doi:10.1214/aos/1016218224.

Export citation


  • Adler, R. J. (1990). An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA.
  • Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656-1670.
  • Chaudhuri, P. and Marron, J. S. (1998). Scale Space view of curve estimation. North Carolina Institute of Statistics, Mimeo Series 2357.
  • Chaudhuri, P. and Marron, J. S. (1999). SiZer for exploration of structures in curves. J. Amer. Statist. Assoc. To appear.
  • Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74 829-836.
  • Cleveland, W. S. (1993). Visualizing Data. Hobart Press, Summit NJ.
  • Cleveland, W. S. and Devlin, S. J. (1988). Locally weighted regression: an approach to regression analysis by local fitting, J. Amer. Statist. Assoc. 84 596-610.
  • Cleveland, W. L. and Loader, C. (1996). Smoothing by local regression: principles and methods. In Statistical Theory and Computational Aspects of Smoothing (W. H¨ardle and M. G. Schimek, eds.) 10-49. Physica, Heidleberg.
  • Donoho, D. L. (1988). One sided inference about functionals of a density. Ann. Statist. 16 1390- 1420.
  • Eubank, R. L. (1988). Spline Smoothing and Nonparametric Regression. Dekker, New York.
  • Fan, J. (1992). Design adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998-1004.
  • Fan, J. (1993). Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21 196-216.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications. Chapman and Hall, London.
  • Fisher, N. I., Mammen, E. and Marron J. S. (1994). Testing for multimodality. Comput. Statist. Data Anal. 18 499-512. Godtliebsen, F., Marron, J. S. and Chaudhuri, P. (1999a) Significance in scale space. Unpublished manuscript. Godtliebsen, F., Marron, J. S. and Chaudhuri, P. (1999b) Significance in scale space for density estimation. Unpublished manuscript.
  • Good, I. J. and Gaskins, R. A. (1980). Density estimation and bump hunting by the penalized maximum likelihood method exemplified by scattering and meteorite data (with discussion). J. Amer. Statist. Assoc. 75 42-73.
  • Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models. Chapman and Hall, London.
  • H¨ardle, W. (1990). Applied Nonparametric Regression. Cambridge Univ. Press.
  • Hartigan, J. A. and Hartigan, P. M. (1985). The DIP test of multimodality. Ann. Statist. 13 70-84.
  • Hartigan, J. A. and Mohanty, S. (1992). The RUNT test for multimodality. J. Classification 9 63-70.
  • Hastie, T. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.
  • Hirschman, I. I. and Widder, D. V. (1955). The Convolution Transform. Princeton Univ. Press.
  • Karlin, S. (1968). Total Positivity. Stanford Univ. Press. Kim, C. S. and Marron, J. S. (1999) SiZer for jump detection. Unpublished manuscript.
  • Koenderink, J. J. (1984). The structure of images. Biological Cybernatics 50 363-370.
  • Lindeberg, T. (1994). Scale Space Theory in Computer Vision. Kluwer, Boston.
  • Mammen, E., Marron, J. S. and Fisher, N. I. (1992). Some asymptotics for multimodality tests based on kernel density estimates. Probab. Theory Related Fields 91 115-132.
  • Marchette, D. J. and Wegman, E. J. (1997). The filtered mode tree. J. Comput. Graph. Statist. 6 143-159. Marron, J. S. and Chaudhuri, P. (1998a). Significance of features via SiZer. In Statistical Modelling. Proceedings of 13th International Workshop on Statistical Modelling (B. Marx and H. Friedl, eds.) 65-75. Marron, J. S. and Chaudhuri, P. (1998b) When is a feature really there? The SiZer approach. In Automatic Target Recognition VII (F. A. Sadjadi, ed.) 306-312. SPIE Press, Bellingham, WA.
  • Marron, J. S. and Chung, S. S. (1997). Presentation of smoothers: the family approach. Unpublished manuscript.
  • Minnotte, M. C. (1997). Nonparametric testing of the existence of modes. Ann. Statist. 25 1646- 1660.
  • Minnotte, M. C. and Scott, D. W. (1993). The mode tree: a tool for visualization of nonparametric density features. J. Comput. Graph. Statist. 2 51-68.
  • M ¨uller, H. G. (1988). Nonparametric Regression Analysis of Longitudinal Data. Lecture Notes in Statist. Springer, Berlin.
  • M ¨uller, H. G. and Sawitzki, G. (1991). Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 738-746.
  • Muzy, J. F., Bacry, E. and Arneodo, A. (1994). The multifractal formalism revisited with wavelets. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4 245-302.
  • Rosenblatt, M. (1991). Stochastic Curve Estimation. IMS, Hayward, CA. Ruppert, D., Sheather, S. J. and Wand, M. P. (1995) An effective bandwidth selector for local least squares regression, J. Amer. Statist. Assoc. 90 1257-1270.
  • Schoenberg, I. J. (1950). On P´olya frequency functions, II: variation diminishing integral operators of the convolution type. Acta Sci. Math. (Szeged) 12B 97-106.
  • Silverman, B. W. (1981). Using kernel density estimates to investigate multimodality. J. Roy. Statist. Soc. Ser. B 43 97-99.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer, New York.
  • Stone, C. J. (1977). Consistent nonparametric regression. Ann. Statist. 5 595-620.
  • Tukey, J. W. (1970). Exploratory Data Analysis. Addison-Wesley, Reading, Mass.
  • Wahba, G. (1991). Spline Models for Observational Statistics. SIAM, Philadelphia.
  • Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.
  • Weickert J. (1997). Anisotropic Diffussion in Image Processing. Teubner, Stuttgart.
  • Witkin, A. P. (1983). Scale space filtering. In Proceedings of the 8th International Joint Conference on Artificial Intelligence 1019-1022. Morgan Kaufman, San Francisco.
  • Wong, Y. F. (1993). Clustering data by melting. Neural Computation 5 89-104.