The hypervolume under the ROC hypersurface of "Near-Guessing" and "Near-Perfect" observers in N-class classification tasks

We express the performance of the N-class "guessing" observer in terms of the N²-N conditional probabilities which make up an N-class receiver operating characteristic (ROC) space, in a formulation in which sensitivities are eliminated in constructing the ROC space (equivalent to using false-negative fraction and false-positive fraction in a two-class task). We then show that the "guessing" observer\´s performance in terms of these conditional probabilities is completely described by a degenerate hypersurface with only N-1 degrees of freedom (as opposed to the N²-N-1 required, in general, to achieve a true hypersurface in such a ROC space). It readily follows that the hypervolume under such a degenerate hypersurface must be zero when N>2. We then consider a "near-guessing" task; that is, a task in which the N underlying data probability density functions (pdfs) are nearly identical, controlled by N-1 parameters which may vary continuously to zero (at which point the pdfs become identical). With this approach, we show that the hypervolume under the ROC hypersurface of an observer in an N-class classification task tends continuously to zero as the underlying data pdfs converge continuously to identity (a "guessing" task). The hypervolume under the ROC hypersurface of a "perfect" ideal observer (in a task in which the N data pdfs never overlap) is also found to be zero in the ROC space formulation under consideration. This suggests that hypervolume may not be a useful performance metric in N-class classification tasks for N>2, despite the utility of the area under the ROC curve for two-class tasks.