Variable Selection and Basis Learning for Ordinal Classification

We propose a method for variable selection and basis learning for high-dimensional classification with ordinal responses. The proposed method extends sparse multiclass linear discriminant analysis, with the aim of identifying not only the variables relevant to discrimination but also the variables that are order-concordant with the responses. For this purpose, we compute for each variable an ordinal weight, where larger weights are given to variables with ordered group-means, and penalize the variables with smaller weights more severely. A two-step construction for ordinal weights is developed, and we show that the ordinal weights correctly separate ordinal variables from non-ordinal variables with high probability. The resulting sparse ordinal basis learning method is shown to consistently select either the discriminant variables or the ordinal and discriminant variables, depending on the choice of a tunable parameter. Such asymptotic guarantees are given under a high-dimensional asymptotic regime where the dimension grows much faster than the sample size. We also discuss a two-step procedure of post-screening ordinal variables among the selected discriminant variables. Simulated and real data analyses confirm that the proposed basis learning provides sparse and interpretable basis, as it mostly consists of ordinal variables. Supplementary materials for this article are available online.