The ROC manifold for classification systems

We define the ROC manifold and CC manifold as duals in a given sense. Their analysis is required to describe the classification system. We propose a mathematical definition based on vector space methods to describe both. The ROC manifolds for n-class classification systems fully describe each system in terms of its misclassifications and, by conjunction, its correct classifications. Optimal points which minimize misclassifications can be identified even when costs and prior probabilities differ. These manifolds can be used to determine the usefulness of a classification system based on a given performance criterion. Many performance functionals (such as summary statistics) preferred for CC manifolds can also be evaluated using the ROC manifold (under certain constraints). Examples using the ROC manifold and performance functionals to compete classification systems are demonstrated with simulated and applied disease detection data.