Mathematics - Colloquium
Friday, April 26, 2013
11:00 AM-12:00 PM
ABSTRACT: We consider the problem of quickest detection in the presence of multiple random sources each driven by distinct sources of noise represented by a Brownian motion. We make the assumption that the driving noises are independent. The case described in this set-up corresponds to 0 correlation in the noise component of each source.
The first problem we will address is the one of detecting a change in the drift of Brownian motions received in parallel at the sensors of decentralized systems. We examine the performance of one shot schemes in decentralized detection in the case of many sensors with respect to appropriate criteria. One shot schemes are schemes in which the sensors communicate with the fusion center only once; when they must signal a detection. The communication is clearly asynchronous and we consider the case that the fusion center employs one of two strategies, the minimal and the maximal. According to the former strategy an alarm is issued at the fusion center the moment in which the first one of the sensors issues an alarm, whereas according to the latter strategy an alarm is issued when both sensors have reported a detection. In this work we derive closed form expressions for the expected delay of both the minimal and the maximal strategies in the case that CUSUM stopping rules are employed by the sensors. We prove asymptotic optimality of the above strategies in the case of across-sensor independence and specify the optimal threshold selection at the sensors. Moreover, we discuss extensions of this recent result in the presence of correlations as well.
We also consider the problem of a two-dimensional hypothesis test in which we attempt to distinguish amongst four states of a two-sensor system, each corresponding to a distinct hypothesis H_ij, where i=0 if no signal is received on the first sensor or i=1 if a signal is received, and likewise for j=0 or 1 on the second sensor. We set up the problem as a min-max optimization in which we wish to find a decision rule that minimizes the length of continuous observation time required to make a decision about the state of the system subject to Type I errors α_ij, the probability of falsely deciding that H_ij is the correct hypothesis (we refer to these as Type I/ij error). We assume that the noise in the two sources of observations is uncorrelated, and propose running in parallel two sequential probability ratio tests and compute their thresholds in terms of each of the error probabilities. We demonstrate asymptotic optimality of the proposed rule as the error probabilities tend to 0 and discuss the case of non-zero correlation across channels.
Suggested Audiences: Adult, College
Last Modified: April 3, 2013 at 1:07 PM