A Theoretical Approach to Ordinal Classification: Feature Space-Based Definition and Classifier-Independent Detection of Ordinal Class Structures

Abstract

Ordinal classification (OC) is a sub-discipline of multi-class classification (i.e., including at least three classes), in which the classes constitute an ordinal structure. Applications of ordinal classification can be found, for instance, in the medical field, e.g., with the class labels order, early stage-intermediate stage-final stage, corresponding to the task of classifying different stages of a certain disease. While the field of OC was continuously enhanced, e.g., by designing and adapting appropriate classification models as well as performance metrics, there is still a lack of a common mathematical definition for OC tasks. More precisely, in general, a classification task is defined as an OC task, solely based on the corresponding class label names. However, an ordinal class structure that is identified based on the class labels is not necessarily reflected in the corresponding feature space. In contrast, naturally any kind of multi-class classification task can consist of a set of arbitrary class labels that form an ordinal structure which can be observed in the current feature space. Based on this simple observation, in this work, we present our generalised approach towards an intuitive working definition for OC tasks, which is based on the corresponding feature space and allows a classifier-independent detection of ordinal class structures. To this end, we introduce and discuss novel, OC-specific theoretical concepts. Moreover, we validate our proposed working definition in combination with a set of traditionally ordinal and traditionally non-ordinal data sets, and provide the results of the corresponding detection algorithm. Additionally, we motivate our theoretical concepts, based on an illustrative evaluation of one of the oldest and most popular machine learning data sets, i.e., on the traditionally non-ordinal Fisher’s Iris data set.

Keywords

• ordinal classification
• detection of ordinal class structures
• Fisher’s discriminant ratio