Unstructured sequential testing in sensor networks

We consider the problem of quickly detecting a signal in a sensor network when the subset of sensors in which signal may be present is completely unknown. We formulate this problem as a sequential hypothesis testing problem with a simple null (signal is absent everywhere) and a composite alternative (signal is present somewhere). We introduce a novel class of scalable sequential tests which, for any subset of affected sensors, minimize the expected sample size for a decision asymptotically, that is as the error probabilities go to 0. Moreover, we propose sequential tests that require minimal transmission activity from the sensors to the fusion center, while preserving this asymptotic optimality property.Comment: 6 two-column pages, To appear in the Proceedings 2013 IEEE Conference on Decision and Control, Firenze, Italy, December 201

Change Acceleration and Detection

A novel sequential change detection problem is proposed, in which the change should be not only detected but also accelerated. Specifically, it is assumed that the sequentially collected observations are responses to treatments selected in real time. The assigned treatments not only determine the pre-change and post-change distributions of the responses, but also influence when the change happens. The problem is to find a treatment assignment rule and a stopping rule that minimize the expected total number of observations subject to a user-specified bound on the false alarm probability. The optimal solution to this problem is obtained under a general Markovian change-point model. Moreover, an alternative procedure is proposed, whose applicability is not restricted to Markovian change-point models and whose design requires minimal computation. For a large class of change-point models, the proposed procedure is shown to achieve the optimal performance in an asymptotic sense. Finally, its performance is found in two simulation studies to be close to the optimal, uniformly with respect to the error probability

Signal Recovery With Multistage Tests And Without Sparsity Constraints

A signal recovery problem is considered, where the same binary testing problem is posed over multiple, independent data streams. The goal is to identify all signals, i.e., streams where the alternative hypothesis is correct, and noises, i.e., streams where the null hypothesis is correct, subject to prescribed bounds on the classical or generalized familywise error probabilities. It is not required that the exact number of signals be a priori known, only upper bounds on the number of signals and noises are assumed instead. A decentralized formulation is adopted, according to which the sample size and the decision for each testing problem must be based only on observations from the corresponding data stream. A novel multistage testing procedure is proposed for this problem and is shown to enjoy a high-dimensional asymptotic optimality property. Specifically, it achieves the optimal, average over all streams, expected sample size, uniformly in the true number of signals, as the maximum possible numbers of signals and noises go to infinity at arbitrary rates, in the class of all sequential tests with the same global error control. In contrast, existing multistage tests in the literature are shown to achieve this high-dimensional asymptotic optimality property only under additional sparsity or symmetry conditions. These results are based on an asymptotic analysis for the fundamental binary testing problem as the two error probabilities go to zero. For this problem, unlike existing multistage tests in the literature, the proposed test achieves the optimal expected sample size under both hypotheses, in the class of all sequential tests with the same error control, as the two error probabilities go to zero at arbitrary rates. These results are further supported by simulation studies and extended to problems with non-iid data and composite hypotheses.Comment: 58 pages, 12 figure