Fast and unbiased estimation of volume under the ordered multi-class ROC hyper-surface with continuous measurements

Abstract

Receiver operating characteristic (ROC) analysis is a popular tool to deal with two-class problems in many science and engineering areas. However, in practice, multi-class problems are frequently encountered, such as in ordinal regression in the area of machine learning. The volume under the multi-class ROC hyper-surface (VUHS) has been proposed to evaluate the performance of multi-class classifiers. Unfortunately, however, the computational loads of current methods are rather heavy, making them impracticable in scenarios where the sample size is large. Moreover, the null distribution, which is mandatory for significance test, is also unknown to the best of our knowledge. To improve such unsatisfactory situations, in this article we develop an efficient algorithm for unbiased estimation of VUHS and the corresponding variance. Exploiting the technique of dynamic programming (DP) as well as Dyck paths, our proposed algorithm outperforms the state-of-the-art algorithm based on graph theoretic for large samples. In addition, we derive the analytical expressions of the mean and variance of VUHS under the null distribution. Theoretical analysis and Monte Carlo simulations verified both the unbiasedness and computational efficiency of our algorithm.