On Nonparametric Ordinal Classification with Monotonicity Constraints

We consider the problem of ordinal classification with monotonicity constraints. It differs from usual classification by handling background knowledge about ordered classes, ordered domains of attributes, and about a monotonic relationship between an evaluation of an object on the attributes and its class assignment. In other words, the class label (output variable) should not decrease when attribute values (input variables) increase. Although this problem is of great practical importance, it has received relatively low attention in machine learning. Among existing approaches to learning with monotonicity constraints, the most general is the nonparametric approach, where no other assumption is made apart from the monotonicity constraints assumption. The main contribution of this paper is the analysis of the nonparametric approach from statistical point of view. To this end, we first provide a statistical framework for classification with monotonicity constraints. Then, we focus on learning in the nonparametric setting, and we consider two approaches: the "plug-in" method (classification by estimating first the class conditional distribution) and the direct method (classification by minimization of the empirical risk). We show that these two methods are very closely related. We also perform a thorough theoretical analysis of their statistical and computational properties, confirmed in a computational experiment.