Relationship between identification metrics: expected confusion and area under a ROC curve

The mathematical relationship between the expected-confusion metric and the area under a receiver operating characteristic (ROC) curve is derived. Given a limited database of subjects and an identification technique that generates a feature vector per subject, expected confusion is used to predict how well the feature vector will filter identity in a larger population. Related is the area under a ROC curve that can be used to determine the probability of correctly discriminating between subjects given the feature vector. These two measures have different connotations, but we show mathematically and verify experimentally that a simple transformation can be applied to the expected confusion to find the probability of incorrectly discriminating between subjects, which is the complement of the area under a ROC curve. Furthermore, we show that as a function of the number of subjects, this transformed expected-confusion measure converges more quickly than direct calculation of the area under a ROC curve.