Bayesian Detection of a Change in a Random Sequence with Unknown Initial and Final Distributions
Quickest detection is a class of detection problems whereby the objective is to identify a change in distribution of an observed sequence of random variables as quickly as possible. Quickest detection has been applied to a wide range of applications, such as process monitoring, quality control, and disaster detection. In each of these applications, the initial state of the observed sequence is generally known. Considering these applications, there is an abundance of literature considering formulations of the quickest detection problem where the initial state of the sequence is assumed to be known. However, in some applications, the assumption of knowledge of the initial state of the observed sequence is not valid in general. Recently, spectrum sensing, the process of identifying wireless channel characteristics for the application of cognitive radio, has been cast as a quickest detection problem. Upon, first observation, the radio performing spectrum sensing would not know the initial state of the channel, rendering previous formulations of the quickest detection problem unusable here. In this thesis, an alternative formulation of the quickest detection problem is considered where the initial state of the observed sequence is assumed to be unknown. The problem is formulated as an optimal stopping problem, and a quickest detection scheme is developed based on Bayesian hypothesis testing and an assumed set of costs. The proposed sequential change detector tracks the minimum-risk hypotheses using a time-recursive algorithm which achieves constant computational complexity. It is shown analytically and via simulations that (i) the probability of detecting a change from an incorrect initial distribution asymptotically vanishes over time under suitable parameter choices, (ii) cost parameter choices trade off the probability of early detection of a change (false alarm) against the average delay to detection of a change, and (iii) cost parameter choices determine the certainty with which the initial distribution of the sequence is identified, trading off the probability of detecting a change from an incorrect initial distribution with the ability to detect early changes.
URI for this recordhttp://hdl.handle.net/1974/15747
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