Abstract
A popular method for detecting changes in the probability distribution of a sequence of observations is CUSUM, which proceeds by sequentially evaluating a log-likelihood 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.
Highlights
False alarm control for change point detection procedures is an important problem in many application domains; see (Tartakovsky et al 2014, Section 1.3)
We show that the test statistic of a large class of change point detection procedures can be expressed in form of a first order vector autoregressive process (VAR(1))
In this paper we considered two false alarm criteria derived from the maximal local false alarm probability (MLFA)
Summary
False alarm control for change point detection procedures is an important problem in many application domains; see (Tartakovsky et al 2014, Section 1.3). In view of the above, one wishes for further understanding of the distribution of the stopping time as well as simple but effective methods for selecting the threshold such that the probability of raising a false alarm is kept low in a stronger sense than allowed by the ARL criterion. We check that CUSUM (with or without windows) is asymptotically optimal under this modified false alarm criterion, and investigate methods for selecting the threshold such that it is satisfied To do this exactly, one would need closed form expressions for the distribution of the stopping time. In the second part of the paper, we focus on the criterion sup P0(T = n | T ≥ n − 1) ≤ α Note that this implies that the false alarm probability is limited at any given time n.
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