Contour regression: A general approach to dimension reduction

By Bing Li, Hongyuan Zha, Francesca Chiaromonte. 2005

335 citations recorded by the source corpus.

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 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.

Subfields: Analytical Chemistry, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4e54229df4e9b4415d8fe96fd9c491acca0f210e

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In Analytical Chemistry

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Citations in the corpus

Combining eigenvalues and variation of eigenvectors for order determination (2016, background): “Intuitively, this assumption means that the relation between M ∗ and M̂ is similar to the relation between M̂ and M ; so resampling is like taking a selfie! This condition is proved for many statistics; see Bickel & Freedman (1981), Parr (1985), Liu, Singh & Lo (1989) and Gill (1989).”
On a Projective Resampling Method for Dimension Reduction With Multivariate Responses (2008, background): “…(Hristache, Juditsky, Polzehl, and Spokoiny 2001), iterative Hessian transformations (Cook and Li 2002, 2004), outer product gradient estimator (Xia et al. 2002), canonical correlation estimator (Fung et al. 2002), contour regression (Li et al. 2005), and directional regression (Li and Wang 2007).”
Nonlinear sufficient dimension reduction for functional data (2017, background): “On a different front, linear SDR was generalized to the nonlinear case by Li, Artemiou and Li (2011) and Lee, Li and Chiaromonte (2013), which seek a set of nonlinear functions f1(X), . . . , fd(X) such that Y ⊥ X|f1(X), . . . , fd(X).”
Kernel dimension reduction in regression (2009, methodology): “poration (Evap). The data were collected daily during 46 days; thus, the number of data points is 46. This data set was studied in the context of contour regression methods for dimension reduction in [22]. We standardize each variable so that the sample variance is equal to 5.0, and use the Gaussian”
Sufficient Dimension Reduction via Distance Covariance (2016, methodology): “To precisely present the bootstrap idea, we introduce a measure of distance based on Li, Zha, and Chiaromonte (2005): m(S1,S2) =‖ PS1 − PS2 ‖, where S1 and S2 are two d-dimensional subspaces of Rp and PS1 , PS2 be the orthogonal projections onto S1 and S2, and ‖ · ‖ is the maximum singular value of…”
ENVELOPE MODELS FOR PARSIMONIOUS AND EFFICIENT MULTIVARIATE LINEAR REGRESSION (2010, background): “Many of these, ranging from various versions of principal components to a multivariate implementation of sliced inverse regression (Li et al. (2003)) are algorithmic in nature, making it difficult to determine postapplication standard errors and other inference-related quantities.”
Sufficient dimension reduction based on an ensemble of minimum average variance estimators (2011, methodology): “Alternative transformations such as those proposed by Manly (1976), John and Draper (1980) and Bickel and Doksum (1981), for instance, that do not require Y to be positive may be used to form different family F.”
On Directional Regression for Dimension Reduction (2007, methodology): “By lemma 2.1 of Li et al. (2005) and Cook (1998, prop.”
Sequential sufficient dimension reduction for large p, small n problems (2015, background): “However, there is a distinction between them (Li et al., 2005).”
Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds (2011, background): “[37] in the context of dimensionality reduction in regression.”
Sliced Regression for Dimension Reduction (2008, methodology): “For an arbitrary estimate B̄, the estimation accuracy was evaluated by (B̄,B0) = |B̄(B̄ B̄)−1B̄ − B0(B 0 B0)−1B 0 | (Li et al. 2005).”
Gradient-Based Kernel Dimension Reduction for Regression (2014, methodology): “Many methods have been proposed in this vein of inverse regression such as SAVE (Cook and Weisberg, 1991), directional regression (Li and Wang, 2007) and contour regression (Li et al., 2005), which use statistics such as mean and variance in each slice or contour of Y .”