Approximating the multiclass ROC by pairwise analysis

The use of Receiver Operator Characteristic (ROC) analysis for the sake of model selection and threshold optimisation has become a standard practice for the design of two-class pattern recognition systems. Advantages include decision boundary adaptation to imbalanced misallocation costs, the ability to fix some classification errors, and performance evaluation in imprecise, ill-defined conditions where costs, or prior probabilities may vary. Extending this to the multiclass case has recently become a topic of interest. The primary challenge involved is the computational complexity, that increases to the power of the number of classes, rendering many problems intractable. In this paper the multiclass ROC is formalised, and the computational complexities exposed. A pairwise approach is proposed that approximates the multi-dimensional operating characteristic by discounting some interactions, resulting in an algorithm that is tractable, and extensible to large numbers of classes. Two additional multiclass optimisation techniques are also proposed that provide a benchmark for the pairwise algorithm. Experiments compare the various approaches in a variety of practical situations, demonstrating the efficacy of the pairwise approach.