Application of Category Theory and Analysis of Receiver Operating Characteristics to Information Fusion

摘要

Multisensor data fusion is presented in a rigorous mathematical format, with definitions consistent with the desires of the data fusion community. A model of event-state fusion is developed and described. Definitions of fusion rules and fusors are introduced, along with the functor categories of which they are objects. Defining fusors and competing fusion rules involves the use of an objective function of the researcher's choice. One such objective function, a functional on families of classification systems, and in particular, receiver operating characteristics (ROCs), is introduced. Its use as an objective function is demonstrated in that the argument that minimizes it (a particular ROC) corresponds to the Bayes Optimal Threshold, given certain assumptions. This is proven using a calculus of variations approach using ROC curves as a constraint. This constraint is extended to ROC manifolds, in particular, topological subspaces of R(exp n). These optimal points can be found analytically if the closed form of the ROC manifold is known, or is calculated from the functional (as the minimizing argument) when a finite number of points are available for comparison in a family of classification systems. Under different data assumptions, the minimizing argument of the ROC functional is shown to be the point of an ROC manifold corresponding to the Neyman-Pearson criteria. A second functional, the l(sub2) norm, is shown to determine the min-max threshold. A more robust functional is developed from the offered functionals.