On Directional Regression for Dimension Reduction

We introduce directional regression (DR) as a method for dimension reduction. Like contour regression, DR is derived from empirical directions, but achieves higher accuracy and requires substantially less computation. DR naturally synthesizes the dimension reduction estimators based on conditional m...

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Bibliographic Details
Published in:	Journal of the American Statistical Association 2007-09, Vol.102 (479), p.997-1008
Main Authors:	Li, Bing, Wang, Shaoli
Format:	Article
Language:	eng
Subjects:	Applications Asymptotic methods Computational methods Contour regression Datasets Dimensionality reduction Distribution theory Efficiency Estimation Estimation methods Estimators Exact sciences and technology Exhaustive estimation General topics Linear inference, regression Mathematical vectors Mathematics Matrices P values Probability and statistics Random variables Regression analysis Sample size Sciences and techniques of general use Sliced average variance estimation Sliced inverse regression Statistical methods Statistical variance Statistics Theory and Methods Variance
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Summary:	We introduce directional regression (DR) as a method for dimension reduction. Like contour regression, DR is derived from empirical directions, but achieves higher accuracy and requires substantially less computation. DR naturally synthesizes the dimension reduction estimators based on conditional moments, such as sliced inverse regression and sliced average variance estimation, and in doing so combines the advantages of these methods. Under mild conditions, it provides exhaustive and -consistent estimate of the dimension reduction space. We develop the asymptotic distribution of the DR estimator, and from that a sequential test procedure to determine the dimension of the central space. We compare the performance of DR with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models. Finally, we apply DR to analyze a data set concerning the identification of hand-written digits.
ISSN:	0162-1459 1537-274X