Second-order measures of quality for binary classification: A critical overview and their use for nonlinear receiver design

When deriving a detector, we are often led to consider design criteria such as second-order measures of quality. The aim of this paper is to provide a critical overview of these criteria. We first consider the case of deriving unconstrained detectors. We show that second-order criteria must satisfy a non-trivial condition to yield Bayes-optimal receivers, to be considered as relevant criteria for detector design. Next, we address the case where constraints are imposed on the detection structure, leading us to consider some set D of admissible detectors. In these conditions we prove that even if there exists a monotonic function of the likelihood ratio in D obtaining this statistic via the optimization of a second-order criterion, relevant or not, is not guaranteed. Results are illustrated by simulation examples. Finally, in order to derive nonlinear discriminants via optimization of second-order criteria, we propose a method based on the kernel trick used in the implementation of the well-known support vector machine method. The new method is tested on a number of real data sets.