Non-restarting cumulative sum charts and control of the false discovery rate

Cumulative sum or cusum charts are typically used to detect a change in the distribution of a sequence of observations, e.g., shifts in the mean. Usually, after signalling, the chart is restarted by setting it to some value below the signalling threshold. We propose a non-restarting cusum chart which is able to detect periods during which the stream is out of control. Further, we advocate an upper boundary to prevent the cusum chart rising too high, which helps to detect a change back into control. We present an algorithm to control the false discovery rate when considering cusum charts based on multiple streams of data. We consider two definitions of a false discovery: signalling out-of-control when the observations have been in control since the start and signalling out-of-control when the observations have been in control since the last time the chart was at zero. We prove that the false discovery rate is controlled under both these definitions simultaneously. Simulations reveal the difference in false discovery rate control when using these and other desirable definitions of a false discovery.