Hyperspectral Data Processing: Algorithm Design and Analysis

3.3 ROC Analysis

Interestingly, according to (2.8), (2.11), and (3.5), no matter what the type of detector is, its detector statistics always turns out to be the likelihood ratio test (LRT). In other words, all the Bayes, minimax, or Neyman–Pearson detectors can be shown to have the same functional form determined by LRT. Nevertheless, NP detectors are the most practical detectors in real applications since they do not require prior knowledge of the cost function and probabilities of hypotheses that are generally unknown or difficult to obtain in practice. Instead, the performance is evaluated based on the four decisions described in Section 3.1 but with general descriptive terms given in the following:

1. When “H₀” is true, “H₀” is also true.

In this case, a correct decision is made and the decision is called “true negative” (TN). However, it should be noted that there is no detection term corresponding to this decision since nothing is detected other than noise.

2. When “H₀” is true, “H₁” is true.

In this case, an incorrect decision is made and the decision is called “false alarm” (FA) or “false positive” (FP) in medical diagnosis.

3. When “H₁” is true, “H₀” is also true.

In this case, an incorrect decision is made and the decision is called “miss” or “false negative” (FN).

4. When “H₁” is true, “H₁” is also true.

In this case, a correct decision is made and the decision is called “detection probability, rate or power” or “true positive” (TP).

As per the above decisions, ...