Practical Aspects of False Alarm Control for Change Point Detection: Beyond Average Run Length

Abstract

A popular method for detecting changes in the probability distribution of a sequence of observations is CUSUM, which proceeds by sequentially evaluating a loglikelihood ratio test statistic and comparing it to a predefined threshold; a change point is detected as soon as the threshold is exceeded. It is desirable to choose the threshold such that the number of false alarms is kept to a specified level. Traditionally, the number of false alarms is measured by the average run length-the expected stopping time until the first false alarm. However, this is does not in general allow one to control the number of false alarms at every particular time instance. Thus, in this paper two stronger false alarm criteria are considered, for which approximation methods are investigated to facilitate the selection of a threshold.

Controlling the exponentially weighted moving average S2 control chart false alarm behavior when the in‐control variance level must be estimated

“…For example,the authors in Reference 34 wrote: “Though ARL is probably meaningful in the off‐target situation, it can be highly misleading when the on‐target case is under study (primarily because the set of possible CUSUM paths includes ‘too many’ extremely ‘short’ members)” . For a more recent critique, see Reference 35 or 36 and the references therein. Reference 34 also reported (for the competing CUSUM control chart): “In general, the user of a CUSUM scheme probably feels uneasy about specifying a particular ARL for the on‐target situation; what he typically wants is that the scheme will not generate a false alarm within a certain period of time (say, a shift) with probability of at least, say, 0.99.” This last passage refers to our design principle.…”