Formulation and comparison of multi-class ROC surfaces

The Receiver Operating Characteristic (ROC) has become a standard tool for the analysis and comparision of classifiers when the costs of misclassification are unknown. There has been relatively little work, however, examining ROC for more than two classes. Here we define the ROC surface for the Qclass problem in terms of a multi-objective optimisation problem in which the goal is to simultaneously minimise the Q(Q − 1) misclassification rates, when the misclassification costs and parameters governing the classifier’s behaviour are unknown. We present an evolutionary algorithm to locate the optimal trade-off surface between misclassifications of different types. The performance of the evolutionary algorithm is illustrated on a synthetic three class problem. In addition the use of the Pareto optimal surface to compare classifiers is discussed, and we present a straightforward multi-class analogue of the Gini coefficient. This is illustrated on synthetic and standard machine learning data.