ROC representation for the discriminability of multi-classification markers

•A proper assessment of performance probabilities is shown to be the optimal ROC manifold.•The necessary and sufficient conditions are addressed for the existence of the HUM.•The scientific groundwork is extended to a more general multi-class ROC analysis.

In this paper, the receiver operating characteristic (ROC) representation and its accuracy measures are well-defined and meaningful assessments for the discriminability of multi-classification markers are shown. Given a set of classifiers CC, a parameterized system can be used to characterize the corresponding optimal ROC manifold. A connection with the decision set further leads to a better understanding of some geometric features of optimal ROC manifolds and preserves the simplicity in computing the hypervolume under the ROC manifold (HUM). In addition, it motivates us to address the necessary and sufficient conditions for the existence of the HUM. To sum up, this work provides working scientists with an extension of the two-class ROC analysis to the multi-classification ROC analysis in a theoretically sound manner.