The area under the ROC curve is a common measure that is often used to rank the relative performance of different binary classifiers. However, as has been also previously noted, it can be a measure that ill-captures the benefits of different classifiers when either the true class values or misclassification costs are highly unbalanced between the two classes. We introduce a third dimension to capture these costs, and lift the ROC curve to a ROC surface in a natural way. We study both this surface and introduce the VOROS, the volume over this ROC surface, as a 3D generalization of the 2D area under the ROC curve. For problems where there are only bounds on the expected costs or class imbalances, we restrict consideration to the volume of the appropriate subregion of the ROC surface. We show how the VOROS can better capture the costs of different classifiers on both a classical and a modern example dataset.

翻译：ROC曲线下面积是衡量不同二分类器相对性能的常用指标。然而，正如先前研究指出的，当真实类别值或误分类成本在两个类别间高度不均衡时，该指标可能无法准确反映不同分类器的效益。我们引入第三维度来捕捉这些成本，并以自然方式将ROC曲线提升为ROC曲面。我们研究了该曲面，并引入VOROS（ROC曲面上方体积）作为二维ROC曲线下面积的三维泛化形式。对于仅有预期成本或类别不平衡界限的问题，我们将关注范围限定在ROC曲面适当子区域的体积。通过经典与现代示例数据集，我们展示了VOROS如何更准确地捕捉不同分类器的成本差异。