Quickest change detection under transient dynamics: Theory and asymptotic analysis

The problem of quickest change detection under transient dynamics is studied, where the change from the initial distribution to the final persistent distribution does not happen instantaneously, but after a series of transient phases. The observations within the different phases are generated by different distributions. The objective is to detect the change as quickly as possible, while controlling the average run length (ARL) to false alarm, when the durations of the transient phases are completely unknown. Two algorithms are considered: the dynamic Cumulative Sum (CuSum) algorithm, proposed in earlier work, and a newly constructed weighted dynamic CuSum algorithm. Both algorithms admit recursions that facilitate their practical implementation, and they are adaptive to the unknown transient durations. Specifically, their asymptotic optimality is established with respect to both Lorden's and Pollak's criteria as the ARL to false alarm and the durations of the transient phases go to infinity at any relative rate. Numerical results are provided to demonstrate the adaptivity of the proposed algorithms and to validate the theoretical results.