Daumer, Martin; Bauer, C. und Lederer, C.
(1997):
Average Run Length and Mean Delay for Changepoint Detection: Robust Estimates for Threshold Alarms.
Sonderforschungsbereich 386, Discussion Paper 95

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Abstract
Online Monitoring is a rapidly expanding field in different areas such as quality control, finance and navigation. The automated detection of socalled changepoints is playing a prominent role in all these fields, be it the detection of sudden shifts of the mean of a continuously monitored quantity, the variance of stock quotes or the change of some characteristic features indicating the malfunctioning of one of the detectors used for navigation (the ``faulty sensor problem''). A prominent example for the application of advanced statistical methods for the detection of changepoints in biomedical time series is the multiprocess Kalman filter used by Smith and West [Smith 1983] to monitor renal transplants. However, despite the fact that the algorithm could be tuned in such a way that the computer could predict dangerous situations on the average one day before the human experts it has nevertheless become superfluous as soon as new diagnosic tools became available. Many of the automated monitoring systems which are widely used in practice are based on simple threshold alarms. Some upper and lower limits are chosen at the beginning of the monitoring session and an alarm is triggered whenever the measured values exceed the upper limit or fall below the lower limit. This is e.g. common practice for the monitoring of patients during surgery, where such thresholds are chosen for heart rate, blood pressure etc. by the anaesthesist. The fate of the multiprocess Kalman filter for monitoring renal transplants teaches two lessons: first, there is considerable power in statistical methods to improve conventional biomedical monitoring techniques. Second, if the statistical model and the methods are too refined they may never be used in practice. We shall suggest a stochastic model for changepoints which we have found to have the capacity to be very useful in practice, i.e. which is sufficiently complex to cover the important features of a changepoint system but simple enough to be understandable and adaptible. We focus our attention on the properties of the threshold alarm for different values of the parameters of the threshold alarm and the model. This will give us practically relevant estimates for this important class of alarm systems and moreover a benchmark for the evaluation of competing alternative algorithms. Note that virtually every algorithm designed to detect changepoints is based on a threshold alarm, the only difference being that the threshold alarm is not fed with the original data but by a transformation thereof, usually called ``residuum'' [Basseville 1993]. As a general measure for quality, we look on the one hand at the mean delay time $\tau$ between a changepoint and its detection and on the other hand at the mean waiting time for a false alarm, the socalled average run length ARL.