Why Does Rebalancing Class-Unbalanced Data Improve AUC for Linear Discriminant Analysis?

Many established classifiers fail to identify the minority class when it is much smaller than the majority class. To tackle this problem, researchers often first rebalance the class sizes in the training dataset, through oversampling the minority class or undersampling the majority class, and then use the rebalanced data to train the classifiers. This leads to interesting empirical patterns. In particular, using the rebalanced training data can often improve the area under the receiver operating characteristic curve (AUC) for the original, unbalanced test data. The AUC is a widely-used quantitative measure of classification performance, but the property that it increases with rebalancing has, as yet, no theoretical explanation. In this note, using Gaussian-based linear discriminant analysis (LDA) as the classifier, we demonstrate that, at least for LDA, there is an intrinsic, positive relationship between the rebalancing of class sizes and the improvement of AUC. We show that the largest improvement of AUC is achieved, asymptotically, when the two classes are fully rebalanced to be of equal sizes.