Sequential Decision Making in Two-Dimensional Hypothesis Testing

In this work, we consider the problem of sequential decision making on the state of a two-sensor system with correlated noise. Each of the sensors is either receiving or not receiving a signal obstructed by noise, which gives rise to four possibilities: (noise, noise), (signal, noise), (noise, signal), (signal, signal). We set up the problem as a min-max optimization in which we devise a decision rule that minimizes the length of continuous observation time required to make a decision about the state of the system subject to error probabilities. We first assume that the noise in the two sources of observations is uncorrelated, and propose running in parallel two sequential probability ratio tests, each involving two thresholds. We compute these thresholds in terms of the error probabilities of the system. We demonstrate asymptotic optimality of the proposed rule as the error probabilities decrease without bound. We then analyze the performance of the proposed rule in the presence of correlation and discuss the degenerate cases of perfect positive or negative correlation. Finally, we purport the benefits of our proposed rule in a decentralized sensor system versus one in constant communication with a fusion center.