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August 1996 Extremal properties of principal curves in the plane
Tom Duchamp, Werner Stuetzle
Ann. Statist. 24(4): 1511-1520 (August 1996). DOI: 10.1214/aos/1032298280


Principal curves were introduced to formalize the notion of "a curve passing through the middle of a dataset." Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it. This idea can be made precise by defining principal curves for probability densities. In this paper we study principal curves in the plane. Like linear principal components, principal curves are critical points of the expected squared distance from the data. However, the largest and smallest principal components are extrema of the distance, whereas all principal curves are saddle points. This explains why cross-validation does not appear to be a viable method for choosing the complexity of principal curve estimates.


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Tom Duchamp. Werner Stuetzle. "Extremal properties of principal curves in the plane." Ann. Statist. 24 (4) 1511 - 1520, August 1996.


Published: August 1996
First available in Project Euclid: 17 September 2002

zbMATH: 0867.62025
MathSciNet: MR1416645
Digital Object Identifier: 10.1214/aos/1032298280

Primary: 62G07
Secondary: 62H25 , 62H30 , 62J02

Keywords: calculus of variations , curve fitting , least squares , Nonlinear regression , Principal curves

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 4 • August 1996
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