3.4 3D ROC Analysis

From the hypothesis testing problem specified by (3.1), a detector makes a binary hard decision by thresholding a real-valued LRT, Λ(r), via a threshold τ (see (3.5)). Accordingly, the detector performance is determined by two parameters, Λ(r) and τ, both of which are real values. As a result, the detection rate, PD, in (3.2) and (3.6) and the false alarm probability/rate PF in (3.3) and (3.7) are indeed functions of Λ(r) and threshold τ. However, in the Neyman–Pearson detection theory the cost function img and prior probabilities img are assumed to be not known, nor is τ. In this case, the false alarm rate PF is used as a cost function and the threshold τ becomes a dependent function of PF via (3.7) by setting PF = β in (3.4). This is contradictory to the original detection problem where PF = β is actually obtained by a specific value of the threshold τ. Therefore, when an ROC curve is plotted in Figure 3.4 based on PD versus PF, the threshold τ is implicitly absorbed in PF and there is no way to show how the threshold τ specifies PF as the way it should be in Bayes detection theory in (3.1). To resolve this issue, this section develops a new approach to ROC analysis, referred as 3D ROC analysis, which extends the traditional 2D ROC analysis in Section 3.3 by including ...

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