Efficient performance evaluation of the generalized Shiryaev–Roberts detection procedure in a multi‐cyclic setup

Abstract

We propose a numerical method to evaluate the performance of the emerging Generalized Shiryaev-Roberts (GSR) change-point detection 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 * Address correspondence to A.S. Polunchenko, characteristics of the GSR procedure and aid a practitioner to design the GSR procedure as needed while fully utilizing its potential. Keywords:Sequential analysis, Sequential change-point detection, Shiryaev-Roberts procedure, Generalized Shiryaev-Roberts procedure IntroductionSequential (quickest) change-point detection is concerned with the design and analysis of statistical machinery for "on-the-go" detection of unanticipated changes that may occur in the characteristics of a ongoing (random) process. Specifically, the process is assumed to be continuously monitored through sequentially made observations (e.g., measurements), and should their behavior suggest the process may have statistically changed, the aim is to conclude so within the fewest observations possible, subject to a tolerable level of the false detection risk [1-3]. The subject's areas of application are diverse and virtually unlimited, and include industrial quality and process control [4-8], biostatistics, economics, seismology [2], forensics, navigation [2], cybersecurity [9-12], communication systems [2], and many more. A sequential change-point detection procedure-a rule whereby one stops and declares that (apparently) a change is in effect-is defined as a stopping time, T , that is adapted to the observed data, {X n } n 1 .This work's focus is on the multi-cyclic change-point detection problem. It was first addressed in [13,14] in continuous time; see also, e.g., [15,16]. We consider the basic discrete-time case [17,18] which assumes the observations are independent t...

A note on efficient performance evaluation of the Cumulative Sum chart and the Sequential Probability Ratio Test

“…Copyright The foregoing lemma is the main result of this paper. It is an obvious extension of the results obtained previously in [23,24] for the Generalized Shiryaev-Roberts procedure. As simple as it may seem, the established connection between K 1 (z) and K 0 (z) has far-reaching consequences.…”

“…whence dP Λ 1 (t) = t dP Λ 0 (t), t ⩾ 0, as needed; cf. [23,24] and [48,Chapter 3]. As an immediate implication of Lemma 3.1, observe that because…”

“…More recently, however, in , the technique was put to a different, more applicative use: to improve the accuracy and efficiency of the numerical method the authors of these papers developed to compute the performance of the so‐called Generalized Shiryaev–Roberts detection procedure; the Generalized Shiryaev–Roberts procedure was proposed in as a head‐started version of the classical Shiryaev–Roberts procedure, and the motivation to head start the latter was drawn from the seminal work of Lucas and Crosier where it was proposed to head start the CUSUM chart. The aim of this work is to extend the ideas laid out in beyond the Generalized Shiryaev–Roberts procedure, namely, to the CUSUM chart and the SPRT; the possibility of such an extension was previously entertained in , Section 5]. Specifically, in this work, we employ Wald's LR identity and establish a connection between a host of in‐control characteristics of the CUSUM Run Length and their out‐of‐control counterparts.…”

Comparative performance analysis of the Cumulative Sum chart and the Shiryaev‐Roberts procedure for detecting changes in autocorrelated data

“…The framework here is a build‐up to the one previously offered and applied in the works of Moustakides et al for the i.i.d. model (with no cross‐dependence in the observed data); see also, eg, the works of Polunchenko et al Accordingly, just as the prototype framework of other works, our framework is also developed in two stages: we first derive a renewal integral equation for each performance metric involved, and then, as neither one of the obtained equations can be solved analytically, we supply a numerical method to do so and carry out an analysis of the method's accuracy. What is new in the setting considered here is that the integral equations are not one but two dimensional, and (therefore) the numerical method is not deterministic but rather a Monte Carlo–type estimation technique of prescribed proportional closeness , a criterion considered, eg, in the work of Ehrenfeld and Littauer72(p339) in other works …”

On robustness of the Shiryaev–Roberts change-point detection procedure under parameter misspecification in the post-change distribution

“…This allowed Moustakides et al (2011) to derive an integral (renewal) equation directly on the entire sum without any truncation, and then develop a numerical method to treat the integral equation; their method was recently extended by Polunchenko et al (2014a). This is precisely how we intend to evaluate STADD(S A ) in our case study offered in the next section: STADD(S A ) will be computed numerically as RIADD(S A ) with the aid of the numerical methods of Moustakides et al (2011) and Polunchenko et al (2014a). We note that these numerical methods do not require any truncation of either the infinite sum appearing in the definition of RIADD(T ) or the limit appearing in the definition of STADD(T ).…”