Efficient Performance Evaluation of the Generalized Shiryaev-Roberts Detection Procedure in the Multi-Cyclic Setup
A mathematical lecture with Aleksey S. Polunchenko from the State University of New York, Binghamton.
Description:Aleksey S. Polunchenko
State University of New York, at Binghamton
Abstract: Quickest change-point detection is a branch of statistics concerned with the design and analysis of reliable statistical machinery for rapid anomaly detection in "live" monitored data. The subject's current state-of-the-art detection procedure is the recently proposed Generalized Shiryaev--Roberts (GSR) procedure (it was proposed in 2008, but the paper was published only in 2011). Notwithstanding its "young age", the GSR procedure has already been shown to have very strong optimality properties not exhibited by such well-known mainstream procedures as the Cumulative Sum "inspection scheme" and the Exponentially Weighted Moving Average (EWMA) chart. To foster and facilitate further research on the GSR procedure we propose a numerical method to evaluate the performance of the GSR procedure in a "minimax-ish" multi-cyclic setup where the procedure of choice is applied repetitively (cyclically) and the change is assumed to take place at an unknown time moment in a distant-future stationary regime. Specifically, the proposed method is based on the integral-equations approach and uses the collocation technique with the basis functions chosen so as to exploit a certain change-of-measure identity and the GSR detection statistic's unique martingale property. As a result, the method's accuracy and robustness improve, as does its efficiency since using the change-of-measure ploy the Average Run Length (ARL) to false alarm and the Stationary Average Detection Delay (STADD) are computed simultaneously. We show that the method's rate of convergence is quadratic and supply a tight upperbound on its error. We conclude with a case study and confirm experimentally that the proposed method's accuracy and rate of convergence are robust with respect to three factors: a) partition fineness (coarse vs. fine), b) change magnitude (faint vs. contrast), and c) the level of the ARL to false alarm (low vs. high). Since the method is designed not restricted to a particular data distribution or to a specific value of the GSR detection statistic's headstart, this work may help gain greater insight into the characteristics of the GSR procedure and aid a practitioner to design the GSR procedure as needed while fully utilizing its potential.
This is joint work with Grigory Sokolov (Department of Mathematics, U. of Southern California) and Wenyu Du (Department of Mathematical Sciences, SUNY Binghamton).