Abstract
AbstractIn this paper, we introduce a sufficient dimension reduction (SDR) algorithm based on distance‐weighted discrimination (DWD). Our methods is shown to be robust on the dimension p of the predictors in our problem, and it also utilizes some new computational results in the DWD literature to propose a computationally faster algorithm than previous classification‐based algorithms in the SDR literature. In addition to the theoretical results of similar methods we prove the consistency of our estimate for fixed p. Finally, we demonstrate the advantages of our algorithm using simulated and real datasets.
Highlights
Sufficient dimension reduction (SDR) is a class of feature extraction techniques introduced in regression settings with high-dimensional predictors
We discuss first distance-weighted discrimination (DWD) as it was presented by Marron et al (2007) and we demonstrate how it can be incorporated into the SDR framework, giving some theoretical results, a sample estimation algorithm and a method for determining the dimension of the central subspace
As the available packages solve the objective function of DWD which does not include Σ in the first term, we demonstrate below that by standardizing the data the objective function of principal distance weighted discrimination (PDWD) becomes equivalent to the objective function of DWD and available packages can be used
Summary
Sufficient dimension reduction (SDR) is a class of feature extraction techniques introduced in regression settings with high-dimensional predictors. Some examples include the work by Wu (2008) and by Yeh, Huang, and Lee (2009) which introduced Kernel SIR, the work by Fukumizu, Bach, and Jordan (2009) which used kernel regression and the work by Li, Artemiou, and Li (2011) who used Support Vector Machine (SVM) algorithms to achieve linear and nonlinear dimension reduction under a unified framework. The interest of DWD lies on the fact that it works much better than SVM as the dimension of the predictors X increases. Results show that DWD works better than SVM for low-dimensional problems and as the dimension increases PSVM gets closer to the performance of principal distance weighted discrimination (PDWD).
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