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1 BJTU - Beijing Jiaotong University

Abstract : Modem advances in computing power have greatly widened scientists- scope in gathering and investigating information from many variables. We describe sliced inverse regression SIR, for reducing the dimension of the input variable x without going through any parametric or nonparametric model-fitting process. This method explores the simplicity of the inverse view of regression. Instead of regressing the univariate output variable y against the multivariate x, we regress x against y. Forward regression and inverse regression are connected by a theorem that motivates this method. The theoretical properties of SIR are investigated under a model of the form, y = f β ′x, e where the $\beta$-s are unknown vectors. This model looks like a nonlinear regression, except for the crucial difference that the functional form off is completely unknown. For effectively reducing the dimension, one only needs to estimate the effective dimension reduction e.d.r. space generated by the $\beta$-s. If the distribution of $x$ has been standardized to have the zero mean and the identity covariance, the inverse regression curve falls into the e.d.r. space. Hence a principal component analysis on the covariance matrix for the estimated inverse regression curve can be conducted to locate its main orientation, yielding our estimates for e.d.r. directions. Furthermore, a simple step function can be used to estimate the inverse regression curve.

Autor: Li Kevin -



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