Predicting ordinal classes via classification trees

CART (Classification and Regression Trees) is a non-parametric tree-structured recursive partitioning method, introduced by Breiman et al. (1984), to predict a response variable $Y$ on the basis of $p$ predictors: $X_1,\ldots,X_p$ observed on a learning sample of $N$ units. In this paper the case where the response $Y$ is an ordered categorical variable with $k$ levels $y_1\prec \ldots\prec y_k$ is considered. The aim of the classification tree is thus to predict the level of $Y$ on the basis of the vector $\mathbf{X}$ of the $p$ explanatory variables. Instead of accommodating existing algorithm to the ordinal classification task, some papers have faced directly the problem of identifying a suitable rule to grow ordinal classification trees. We will consider these papers into details and propose some improvements based on a known decomposition result regarding Gini's mean difference.