| Abstract: | ![]()
This article deals with a criterion for selection of variables for the multiple group discriminant analysis in high-dimensional data. The variable selection models considered for discriminant analysis in Fujikoshi (1985 Fujikoshi , Y. ( 1985 ). Selection of variables in discriminant analysis and canonical correlation analysis . In: Krishnaiah , P. R. , ed. Multivariate Analysis . Vol. VI. Amsterdam : North-Holland , pp. 219 – 236 . [Google Scholar], 2002 Fujikoshi , Y. ( 2002 ). Selection of variables for discriminant analysis in a high-dimensional case . Sankhya Ser. A 64 : 256 – 257 . [Google Scholar]) are the ones based on additional information due to Rao (1948 Rao , C. R. ( 1948 ). Tests of significance in multivariate analysis . Biometrika 35 : 58 – 79 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], 1970 Rao , C. R. ( 1970 ). Inference on discriminant function coefficients . In: Bose , R. C. , ed. Essays in Probability and Statistics . Chapel Hill , NC : University of North Carolina Press , pp. 537 – 602 . [Google Scholar]). Our criterion is based on Akaike information criterion (AIC) for this model. The AIC has been successfully used in the literature in model selection when the dimension p is smaller than the sample size N. However, the case when p > N has not been considered in the literature, because MLE can not be estimated corresponding to singularity of the within-group covariance matrix. A popular method used to address the singularity problem in high-dimensional classification is the regularized method, which replaces the within-group sample covariance matrix with a ridge-type covariance estimate to stabilize the estimate. In this article, we propose AIC-type criterion by replacing MLE of the within-group covariance matrix with ridge-type estimator. This idea follows Srivastava and Kubokawa (2008 Srivastava , M. S. , Kubokawa , T. ( 2008 ). Akaike information criterion for selecting components of the mean vector in high dimensional data with fewer observations . J. Japan Statist. Soc. 38 : 259 – 283 . [Google Scholar]). Simulations revealed that our proposed criterion performs well. |
