On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves
This work addresses the challenge of interpreting and applying binary classification metrics for researchers and practitioners, offering a geometric framework to improve classifier deployment, but it is incremental as it builds on existing statistical foundations.
The paper tackles the problem of understanding and optimizing binary classification metrics by showing that common metrics like ROC and PR curves are functions of a composition of class-conditional cumulative distribution functions, providing tools for selecting operating points and comparing classifiers. It explores conditions for classifier dominance and links the geometry to practical considerations like model calibration and cost-sensitive optimization.
We study the geometry of Receiver Operating Characteristic (ROC) and Precision-Recall (PR) curves in binary classification problems. The key finding is that many of the most commonly used binary classification metrics are merely functions of the composition function $G := F_p \circ F_n^{-1}$, where $F_p(\cdot)$ and $F_n(\cdot)$ are the class-conditional cumulative distribution functions of the classifier scores in the positive and negative classes, respectively. This geometric perspective facilitates the selection of operating points, understanding the effect of decision thresholds, and comparison between classifiers. It also helps explain how the shapes and geometry of ROC/PR curves reflect classifier behavior, providing objective tools for building classifiers optimized for specific applications with context-specific constraints. We further explore the conditions for classifier dominance, present analytical and numerical examples demonstrating the effects of class separability and variance on ROC and PR geometries, and derive a link between the positive-to-negative class leakage function $G(\cdot)$ and the Kullback--Leibler divergence. The framework highlights practical considerations, such as model calibration, cost-sensitive optimization, and operating point selection under real-world capacity constraints, enabling more informed approaches to classifier deployment and decision-making.