Model-Free Variable Selection With Matrix-Valued Predictors

ABSTRACT

We introduce a novel framework for model-free variable selection with matrix-valued predictors. To test the importance of rows, columns, and submatrices of the predictor matrix in terms of predicting the response, three types of hypotheses are formulated under a unified framework. The asymptotic properties of the test statistics under the null hypothesis are established and a permutation testing algorithm is also introduced to approximate the distribution of the test statistics. A maximum ratio criterion (MRC) is proposed to facilitate the model-free variable selection. Unlike the traditional stepwise regression procedures that require calculating p-values at each step, the MRC is a noniterative procedure that does not require p-value calculation and is guaranteed to achieve variable selection consistency under mild conditions. Performance of the proposed method is evaluated in extensive simulations and demonstrated through the analysis of an electroencephalography data. Supplementary materials for this article are available online.

KEYWORDS:

• Matrix normal distribution
• Matrix-valued predictors
• Maximum ratio criterion
• Model-free variable selection
• Permutation test

Supplementary Materials

Supplementary materials are available online, including a pdf file that includes additional simulation studies, details of the matrix normal distribution, and proofs. R code for implementing the proposed procedures is provided.

Acknowledgments

The authors thank the referees, associate editor, and editor for providing insightful comments that greatly improved the article.