Change detection in autocorrelated processes

The problem of change detection is about quick detection of a change in a dynamic system or process at a low rate of false alarm by sequentially observing the system or process. It has important applications in quality control, signal processing and other areas. This thesis studies the problem in the context of autocorrelated processes. First, we systematically investigate the properties of process residuals (one-step ahead forecast errors) for change detection. We show that process residuals are statistically sufficient for the problem of change detection, and change detection can be done by using process residuals. We show that process residuals are mutually uncorrelated with zero means when there is no change to the process, that is, when the process is in-control. We develop a general procedure for specifically deriving the forms of process residuals. Using the procedure, we derive the forms of residuals of general autoregressive integrated moving average (ARIMA) processes and state space models, and obtain some specific properties of the residuals under several situations. Under the Gaussian assumption, for an ARIMA process or a steady-state state space model subject to a change in process mean level, the residuals are i.i.d. with zero means before the occurrence of change. After the occurrence of change, the residuals are still mutually independent with the same variance as before, but with time-varying and generally nonzero means. For an autocorrelated process subject to a change in process mean level, we find that the properties of residuals are independent of the feedback control applied to the process. We then concentrate on detection of a change in process mean level in an autocorrelated process by using the process residuals. Cumulative sum (CUSUM), exponentially weighted moving average (EWMA) and Shewhart control chart procedures are applied to the residuals. For computation of the average run lengths (ARLs) of the control chart procedures applied to the residuals whose means are time-varying after change, we derive an explicit formula for Shewhart, establish integral equations for CUSUM and EWMA, and develop efficient numerical procedures for solving the integral equations. Under the ARL criterion, we numerically study the performance of the control chart procedures applied to the residuals of autocorrelated processes under several situations. We study the likelihood ratio (LR) testing procedure applied to the process residuals. We extend the classical LR testing procedure by replacing its constant threshold value with a time-varying threshold sequence. We propose a combined CUSUM and Shewhart control chart procedure to approximate the extended LR testing procedure. We develop numerical procedures based on integral equations for computation of the ARLs of these change detection procedures, and numerically study their performance. We study the problem of optimal sequential testing on process mean levels. The problem is a special situation of change detection, an extension of the Wald's sequential probability ratio testing (SPRT) problem, and has wide application in signal processing and other areas. We formulate the problem as an optimal stopping problem, derive the optimal stopping rule, obtain some important properties of the optimal stopping boundaries, develop a numerical procedure for computation of the optimal stopping boundaries, and present a numerical example.