arXiv subjects

Dean M. Young

Publications and source records attributed to Dean M. Young.

3 recordsLinked to original sources

arXiv2025

Subspace Ordering for Maximum Response Preservation in Sufficient Dimension Reduction

Sufficient dimension reduction (SDR) methods aim to identify a dimension reduction subspace (DRS) that preserves all the information about the conditional distribution of a response given its predictor. Traditional SDR methods determine the DRS by solving a method-specific generalized eigenvalue problem and selecting the eigenvectors corresponding to the largest eigenvalues. In this article, we argue against the long-standing convention of using eigenvalues as the measure of subspace importance and propose alternative ordering criteria that directly assess the predictive relevance of each subspace. For a binary response, we introduce a subspace ordering criterion based on the absolute value of the independent Student's T-statistic. Theoretically, our criterion identifies subspaces that achieve the local minimum Bayes' error rate and yields consistent ordering of directions under mild regularity conditions. Additionally, we employ an F-statistic to provide a framework that unifies categorical and continuous responses under a single subspace criterion. We evaluate our proposed criteria within multiple SDR methods through extensive simulation studies and applications to real data. Our empirical results demonstrate the efficacy of reordering subspaces using our proposed criteria, which generally improves classification accuracy and subspace estimation compared to ordering by eigenvalues.

stat.ME↗

arXiv2025

Precision Matrix Regularization in Sufficient Dimension Reduction for Improved Quadratic Discriminant Classification

Sufficient dimension reduction (SDR) methods, which often rely on class precision matrices, are widely used in supervised statistical classification problems. However, when class-specific sample sizes are small relative to the original feature-space dimension, precision matrix estimation becomes unstable and, as a result, increases the variability of the linear dimension reduction (LDR) matrix. Ultimately, this fact causes suboptimal supervised classification. To address this problem, we develop a multiclass and distribution-free SDR method, stabilized SDR (SSDR), that employs user-specified precision matrix shrinkage estimators to stabilize the LDR projection matrix and supervised classifier. We establish this technique with the theoretical guarantee of preserving all classification information under the quadratic discriminant analysis (QDA) decision rule. We evaluate multiple precision matrix shrinkage estimators within our proposed SSDR framework through Monte Carlo simulations and applications to real datasets. Our empirical results demonstrate the efficacy of the SSDR method, which generally improves classification accuracy and frequently outperforms several well-established competing SDR methods.

stat.ME↗

arXiv2017

High-Dimensional Regularized Discriminant Analysis

Regularized discriminant analysis (RDA), proposed by Friedman (1989), is a widely popular classifier that lacks interpretability and is impractical for high-dimensional data sets. Here, we present an interpretable and computationally efficient classifier called high-dimensional RDA (HDRDA), designed for the small-sample, high-dimensional setting. For HDRDA, we show that each training observation, regardless of class, contributes to the class covariance matrix, resulting in an interpretable estimator that borrows from the pooled sample covariance matrix. Moreover, we show that HDRDA is equivalent to a classifier in a reduced-feature space with dimension approximately equal to the training sample size. As a result, the matrix operations employed by HDRDA are computationally linear in the number of features, making the classifier well-suited for high-dimensional classification in practice. We demonstrate that HDRDA is often superior to several sparse and regularized classifiers in terms of classification accuracy with three artificial and six real high-dimensional data sets. Also, timing comparisons between our HDRDA implementation in the sparsediscrim R package and the standard RDA formulation in the klaR R package demonstrate that as the number of features increases, the computational runtime of HDRDA is drastically smaller than that of RDA.

stat.ML↗