Performance study of change‐point detection thresholds for cumulative sum statistic in a sequential context

Abstract

Sequential detection is based on a recursive statistic and a threshold it must reach to report a change. In this paper, we consider the score‐based cumulative sum statistic and propose to evaluate the detection performance of some thresholds on simulated data. Three thresholds come from the literature: the Wald constant, the empirical constant, and the conditional empirical instantaneous threshold (the latter two are built by a simulation‐based procedure). Two new thresholds are built by a simulation‐based procedure: the first one is instantaneous, the second is a dynamical version of the previous one. The thresholds' performance measured by an estimation of the mean time between false alarm (MTBFA) and the average detection delay (ADD) are evaluated on independent and autocorrelated data for several scenario, according to the detection objective and the real change in the data. The simulations allow us to compare the difference between the thresholds' results and to see that their performances prove to be robust when a parameter of the prechange regime is misestimated or when the data independence assumption is violated. We found also that the conditional empirical threshold is the best at minimizing the detection delay while maintaining the given false alarm rate. However, on real data, we suggest to use the dynamic instantaneous threshold because it is the easiest to build for practical implementation.

Transferring biological sequence analysis tools to break‐point detection for on‐line monitoring: A control chart based on the local score

“…It may be due to the fact that we have to manage with survival data here. Table 1 also shows that the statistic Q has a better performance to detect the change (see the smaller values of ARL 1 ), which has also been highlighted 25 . But both ARL 0 and ARL 1 must be taken into account.…”

“…This choice is motivated to have a geometric stopping time, so the ARL 0 , also called the mean time between two false alarms (MTBFAs), corresponds to 1/ α as for the Shewhart control chart. Sahki et al 25 also propose to use the maximum of the CUSUM process . They propose a stopping time based on a constant threshold defined as the empirical percentile of order (1− Iα ) of the LS of sequence of a given length I , with .…”

“…This threshold is then used for every . Of course, as said by the authors, 25 for large I , this is not possible to achieve. They also propose to use the CUSUM process itself and a threshold depending on i , called the empirical instantaneous threshold, which corresponds to an empirical percentile or order α of the CUSUM process at time i .…”

“…Sahki et al 25 also propose a similar statistic as Q , with a threshold called dynamic empirical instantaneous threshold. Their proposed stopping time is based on a threshold that corresponds to the percentile or order 1− α of the statistic Q established by Monte Carlo simulation.…”

“…Among all the proposed charts, LS chart with α = 5% outperformed the others. In Sakhi et al 25 they estimate the ARL 0 values in a totally different manner. The authors make the hypothesis that the false alarm rate, which we denote in this sequel to avoid confusion with the nonconditional nominal value α of the test at time i without considering the test issues before i , is constant.…”