ABSTRACT
Consider the situation where a fusion center, in cooperation with multiple sensors, namely, L sensors, wishes to determine which one of the two or m possible hypotheses is true. In this chapter, we derive a set of fixed-point type equations that the optimal sensor rules must satisfy for the parallel network decision systems. Based on the discretization of the optimality equations, we obtain a finitely convergent method for the computation of a set of optimal sensor rules. First, we show that any given fusion rule can be formulated as a number of bivalued polynomial functions of the local compression rules. Then, under a given fusion rule, we present a fixed-point type necessary condition for the optimal local compression rules and propose an efficient discretized iterative algorithm and prove its finite convergence. After this, we consider the optimal fusion rule problem for a class of the systems with the fusion center building at the Lth sensor. For such a system with the fusion center receiving complete measurement y L , we show that any fusion rule of general form can be equivalently converted to a specific fusion rule consisting of a number of bivalued polynomial functions of the local compression rules. Therefore, any given fusion rule is a special case of the aforementioned specific fusion rule with some of the sensor rules fixed to be identical to zero or one. In this way, the optimization of both the fusion rule and sensor rules are unified to be a specific objective function of only sensor rules. Thus, to get a globally optimal decision performance of the system, what one needs to do is to only calculate optimal sensor rules. More importantly, this unified/optimal objective function does not depend on the statistical properties of the observational data or even on the decision criteria. Thus, in Section 2.3, the above results are extended to the Neyman–Pearson decision problems. Besides, we report some encouraging numerical results. It should be noted that the results in this chapter do not assume the conditional independence of the data vectors given the hypothesis. The independence assumption, though unrealistic in many practical situations, was made extensively in the previous research on this problem. For example, see Tenney and Sandell (1981), Chair and Varshney (1986), and Varshney (1997).
