The Annals of Statistics

Contour regression: A general approach to dimension reduction

Bing Li, Hongyuan Zha, and Francesca Chiaromonte

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We propose a novel approach to sufficient dimension reduction in regression, based on estimating contour directions of small variation in the response. These directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response, leading to simple and general contour regression (SCR and GCR) methodology. In comparison with existing sufficient dimension reduction techniques, this contour-based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions, while maintaining $\sqrt{n}$-consistency and computational ease. Moreover, it proves robust to departures from ellipticity. We establish population properties for both SCR and GCR, and asymptotic properties for SCR. Simulations to compare performance with that of standard techniques such as ordinary least squares, sliced inverse regression, principal Hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses. We demonstrate the use of contour-based methods on a data set concerning soil evaporation.

Article information

Ann. Statist., Volume 33, Number 4 (2005), 1580-1616.

First available in Project Euclid: 5 August 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G09: Resampling methods 62H05: Characterization and structure theory

Central subspace empirical directions PCA nonparametric regression data visualization


Li, Bing; Zha, Hongyuan; Chiaromonte, Francesca. Contour regression: A general approach to dimension reduction. Ann. Statist. 33 (2005), no. 4, 1580--1616. doi:10.1214/009053605000000192.

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