Comprehensive Analysis of Receiver Operating Characteristic (ROC) Curves for Hyperspectral Anomaly Detection
Chein‐I Chang
Abstract
The receiver operating characteristic (ROC) curve of detection probability (PD) versus the false alarm probability (PF), referred to as 2D ROC curve, has been widely used to evaluate hyperspectral anomaly detection (AD) performance. This article explores several fundamental and conceptual issues of a 2D ROC curve used for AD, which has been overlooked and never investigated in the past. How can a Neyman–Pearson (NP) detector work for AD? How is an ROC curve plotted without probability distributions? Why is a 2D ROC curve reported in the literature as a step function and later linearly interpolated as a linear piecewise function? How can an ROC curve be used to evaluate background suppression (BS)? To address all these issues, a mathematical theory of 2D ROC curve is rederived by a random Neyman–Pearson detector (RNPD) via a threshold parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> , which actually determines PD and PF, and its detailed theoretical proofs along with comprehensive analysis are also provided. Specifically, a binary communication channel example is included to illustrate how RNPD works. This threshold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -driven approach, indeed, paves a way for deriving a 3D ROC curve as a function of three parameters: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> , PD, and PF. By virtue of a 3D ROC curve, three 2D ROC curves of (PD,PF), (PD, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau$ </tex-math></inline-formula> ), and (PF, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau$ </tex-math></inline-formula> ) can be derived to perform AD performance analysis effectively in terms of anomaly detectability and BS. Experiments demonstrate that many anomaly detectors that claim to perform well on AD using 2D ROC curves are actually performed very poorly in BS.