Consistency results for the ROC curves of fused classifiers

The U.S. Air Force is researching the fusion of multiple sensors and classifiers. Given a finite collection of classifiers to be fused one seeks a new classifier with improved performance. An established performance quantifier is the Receiver Operating Characteristic (ROC) curve. This curve allows one to view the probability of detection versus probability of false alarm in one graph. In reality only finite data is available so only an approximate ROC curve can be constructed. Previous research shows that one does not have to perform an experiment for this new fused classifier to determine its ROC curve. If the ROC curve for each individual classifier has been determined, then formulas for the ROC curve of the fused classifier exist for certain fusion rules. This will be an enormous saving in time and money since the performance of many fused classifiers will be determined without having to perform tests on each one. But, again, these will be approximate ROC curves, since they are based on finite data. We show that if the individual approximate ROC curves are consistent then the approximate ROC curve for the fused classifier is also consistent under certain circumstances. We give the details for these circumstances, as well as some examples related to sensor fusion.