The goal of quantification learning is to induce models capable of accurately predicting the class distribution for new bags of unseen examples. These models only return the prevalence of each class in the bag because prediction of individual examples is irrelevant in these tasks. A prototypical application of ordinal quantification is to predict the proportion of opinions that fall into each category from one to five stars. Ordinal quantification has hardly been studied in the literature, and in fact, only one approach has been proposed so far. This article presents a comprehensive study of ordinal quantification, analyzing the applicability of the most important algorithms devised for multiclass quantification and proposing three new methods that are based on matching distributions using Earth mover's distance (EMD). Empirical experiments compare 14 algorithms on synthetic and benchmark data. To statistically analyze the obtained results, we further introduce an EMD-based scoring function. The main conclusion is that methods using a criterion somehow related to EMD, including two of our proposals, obtain significantly better results.

Quantification (or prevalence estimation) algorithms aim at predicting the class distribution of unseen sets (or bags) of examples. These methods are useful for two main tasks: 1) quantification applications, for instance when we need to track the proportions of several groups of interest over time, and 2) domain adaptation problems, in which we usually need to adapt a previously trained classifier to a different --albeit related-- target distribution according to the estimated prevalences. This paper analyzes several binary quantification algorithms showing that not only do they share a common framework but are, in fact, equivalent. Inspired by this study, we propose a new method that extends one of the approaches analyzed. After an empirical evaluation of all these methods using synthetic and benchmark datasets, the paper concludes recommending three of them due to their precision, efficiency, and diversity.