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Unformatted text preview: On Sequential Detection for Galton-Watson Processes Abstract A sequential detection problem in which the observation sequence is a realization of a Galton- Watson process is considered. A Bayesian framework for this problem is introduced. Based on a dynamic programming argument, an optimal decision rule is obtained. The optimal sequential detector is shown to have a stationary structure analogous to the standard sequential probability ratio test. A detailed analysis of this problem for various cost assignments is presented along with some numerical examples. 1 Introduction The sequential detection problem for independent and identically distributed (i.i.d.) observations has been discussed extensively in the literature (see e.g. ). Pioneering work in the area is due to Wald . Various extensions of this problem have already been considered, many of which fall under the class of optimal stopping problems . This paper considers sequential detection for Galton-Watson processes, a popular model for population growth and cell kinetics. This problem is fundamentally different from the standard sequential detection problem in which observations are i.i.d. random variables. Indeed, the iterative structure of Galton-Watson processes introduces a strong time dependence among observations. In general, sequential detection for Galton-Watson processes cannot be reduced to the simpler detection problem where observations are i.i.d. random variables. We consider the instance in which the detector has to choose between two hypotheses, H and H 1 . We introduce a Bayesian framework for the sequential detection problem. The optimal solution is obtained as a limiting case of the truncated sequential detection problem in which a decision must be made by time T . The population size together with the probability of hypothesis H conditioned on past observations form a sufficient statistic for a dynamic programming solution to the sequential detection problem. The optimal detector is shown to have a stationary structure analogous to the standard sequential probability ratio test, albeit more complex. The remainder of this paper is organized as follows: In Section 2, we review the definition of Galton-Watson processes and introduce a performance metric against which sequential detectors are evaluated. In Section 3, we show that the problem of selecting an optimal terminal decision rule presents no particular difficulties. Indeed, the problem of optimal sequential detection lies with the selection of an optimal stopping rule. We derive an optimal solution to the truncated sequen- tial detection problem and to the infinite horizon sequential detection problem in Section 4 & 5, respectively. Finally, we include concluding remarks in Section 6. Some numerical examples can be found in Appendix A; while Appendix C contains proofs for all the Propositions and Lemmas....
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This note was uploaded on 03/30/2010 for the course ECEN 689 taught by Professor Enjeti during the Fall '07 term at Texas A&M.
- Fall '07