Optimal fusion rules for label fusion of dependent classification systems

A classification system with M possible output labels (or decisions) will have M(M-1) possible errors. The Receiver Operating Characteristic (ROC) manifold was created to quantify all of these errors. When multiple classification systems are fused, the assumption of independence is usually made in order to mathematically combine the individual ROC manifolds for each system into one ROC manifold. In this paper we will start with the independence assumption and then investigate fused statistically-dependent classification systems. Specifically, we will use label fusion (also called decision fusion) of multiple classification systems to combine these dependent systems and demonstrate the benefit in performance of incorporating the dependence effects into the fused classification system. We will derive the formula for the generalized AND rule for the resultant ROC manifold of the fused classification system which incorporates the individual dependent classification systems. We will also develop a method utilizing permutation matrices to generate formulas for other label-fusion rules. Examples will be given that demonstrate how the formulas are used.