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ROC analysis has become a standard tool in the design and evaluation of 2-class classification problems. It allows for an analysis that incorporates all possible priors, costs, and operating points, which is important in many real problems, where conditions are often nonideal. Extending this to the multiclass case is attractive, conferring the benefits of ROC analysis to a multitude of new problems. Even though ROC analysis does extend theoretically to the multiclass case, the exponential computational complexity as a function of the number of classes is restrictive. In this paper we show that the multiclass ROC can often be simplified considerably because some ROC dimensions are independent of each other. We present an algorithm that analyses interactions between various ROC dimensions, identifying independent classes, and groups of interacting classes, allowing the ROC to be decomposed. The resultant decomposed ROC hypersurface can be interrogated in a similar fashion to the ideal case, allowing for approaches such as cost-sensitive and Neyman-Pearson optimisation, as well as the volume under the ROC. An extensive bouquet of examples and experiments demonstrates the potential of this methodology.
T.C.W. Landgrebe and R.P.W. Duin are in the Information and Communication Theory Group, Delft University of Technology Mekelweg 4, 2628 CD Delft, The Netherlands

R. P. Duin and T. C. Landgrebe, "Efficient Multiclass ROC Approximation by Decomposition via Confusion Matrix Perturbation Analysis," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 30, no. , pp. 810-822, 2007.
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