Abstract
Sufficient dimension reduction (SDR) is a successful tool for reducing data dimensionality without stringent model assumptions. In practice, data often display heteroscedasticity which is of scientific importance in general but frequently overlooked since a primal goal of most existing statistical methods is to identify conditional mean relationship among variables. In this article, we propose a new SDR method called principal quantile regression (PQR) that efficiently tackles heteroscedasticity. PQR can naturally be extended to a nonlinear version via kernel trick. Asymptotic properties are established and an efficient solution path-based algorithm is provided. Numerical examples based on both simulated and real data demonstrate the PQR’s advantageous performance over existing SDR methods. PQR still performs very competitively even for the case without heteroscedasticity.
Highlights
Sufficient dimension reduction (SDR) as long as the notion of the reduced space is clearly defined at the sample level, which is the case for the kernel principal quantile regression (PQR)
We investigate the performance of the proposed cross validation Bayesian information criterion (CVBIC) procedure to estimate the structural dimension from the linear PQR candidate matrix
In this paper we proposed a new SDR method, PQR by exploiting quantile regression (QR) that is useful in the presence of heteroscedasticity
Summary
Dimension reduction is often of primary interest in high dimensional data analysis. The principal component analysis (PCA) is widely used in this regard, but it suffers when identifying relationship between response and covariates. Variable selection can be regarded as another type of dimension reduction. It often relies on specific model assumptions which can possibly be violated in practice. Sufficient dimension reduction (SDR) has recently received much attention thanks to its promising performance in reducing data dimensionality under relatively mild model assumptions. SDR achieves dimension reduction of pdimensional predictor X by finding a matrix B = (b1, · · · , bd) ∈ Rp×d which satisfies. The SDR model (1.1) is often called the linear SDR since the dimension reduction is achieved through finding a linear mapping, B X. The linear SDR can naturally be extended in a nonlinear fashion by assuming. The principal support vector machine [PSVM, 18] is the first attempt to tackle both linear and nonlinear SDR in a unified framework. The PSVM can be readily extended to nonlinear SDR via kernel trick, just like the SVM
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