Optimal Fusion Rules for Multi-label Fusion of Independent Classification System Families

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 combine the individual ROC manifolds for each system into one ROC manifold. This paper will investigate the label fusion (also called decision fusion) of multiple classification system families (CSF) that have the same number of output labels. Boolean rules do not exist for multiple symbols, thus, we will derive Boolean-like rules as well as other rules that will yield label fusion rules. An M-label system will have M! consistent rules. The formula for the resultant ROC manifold of the fused classification system family which incorporates the individual classification system families will be derived. Specifically, given a label rule and two classification system families, the ROC manifold for the fused family is produced. We generate the formula for the Boolean-like AND ruled and give the resultant ROC manifold for the fused CSF. We show how the set of permutations of the label set is used to generate all of the consistent rules and how the permutation matrix is incorporated into a single formula for the ROC manifold. Examples will be given that demonstrate how each formula is used.