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Gaussian Regularized Sliced Inverse Regression

Caroline Bernard-Michel 1 Laurent Gardes 1 Stephane Girard 1
1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology, LJK - Laboratoire Jean Kuntzmann, Inria Grenoble - Rhône-Alpes
Abstract : Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. In this paper, a Gaussian prior distribution is introduced on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data is provided.
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Submitted on : Tuesday, April 23, 2013 - 10:44:25 AM
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Caroline Bernard-Michel, Laurent Gardes, Stephane Girard. Gaussian Regularized Sliced Inverse Regression. Statistics and Computing, Springer Verlag (Germany), 2009, 19 (1), pp.85-98. ⟨10.1007/s11222-008-9073-z⟩. ⟨inria-00180458v3⟩



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