Define the parameter vectors

The on-line change detection can be formulated as a GLR test using Eqs. (8.101-8.104). If the AR model Mo (with parameter vector 9°) for the no change hypothesis is not known, identify it with a recursive growing memory filter. For each possible change time r, identify the after change AR model Mi using data for the time window [r,k] and compute the log-likelihood ratio Sr • Maximize 5* over r. Simplifications for saving computation time and other distance measures between models Mo and M\ are discussed in

Parameter Change Detection (PCD) method for SPM of Strongly Autocor-related Processes. The model parameter estimation paradigm is a powerful change detection method for strongly autocorrelated processes. The SPM framework based on time series model forecasts introduced by Alwan and Roberts [16] is one of the most widely used approaches to handle the SPC of processes with autocorrelated data [402, 220]. In this framework, a time series model that describes the autocorrelated process behavior is determined from either some preliminary information or a data set collected when the process was in a state of statistical control. The residuab are generated from the difference of actual measurements and one-step-ahead predictions computed by using this model. The estimation of the one-step-ahead minimum variance forecasts can also be formulated by using a state-space form of the time series model, where the states are the one-step-ahead forecasts. The optimal estimation is given by a Kalman filter. In this approach, change detection in the autocorrelated signal is converted to a change detection in residuals that have suitable statistical properties such as iid which permit the use of standard SPC charts. Generalized likelihood ratio was also used to develop process monitoring schemes based on residuals [45, 677]. Monitoring of forecast residuals have not proven to be very useful SPC tools especially for highly positively correlated time series models. The ability to make correct decisions gets worse particularly when the AR part of the model has roots close to the unit circle [220]. This is frequently encountered in process variables that are under feedback control. The behavior of the controlled variable is dominated by the closed-loop dynamics that include the feedback controller. Usually the controller has an integrator by design in order to compensate for steady-state offset, and the integral action yields roots of magnitude one.

An alternative SPM framework can be developed by monitoring the variations in model parameters that are updated at each new measurement instant. Sastri [534] used such an approach together with the concept of discounted recursive least-squares also known as recursive weighted least-squares (RWLS). An extension of this approach was called parameter change detection (PCD) method for monitoring autocorrelated processes

[414]. The new features of the PCD method include use of recursive variable weighted least squares (RVWLS) with adaptive forgetting, and an implicit parametrization scheme that estimates the process level at each sampling instant. RVWLS parameter updating with adaptive forgetting provides better tracking of abrupt changes in the process parameters than the usual RWLS updating, and it reduces the number of false detections of change as well. The detection capabilities of PCD method are superior to methods based on forecast residuals for highly positively correlated processes. As autocorrelation increases, the improvement of PCD over residuals based SPM methods becomes more significant. The PCD method possesses several attractive features for on-line, real time operation: its computations are efficient, its implementation is easy, and the resulting charts are clearly interpretable. The implicit parametrization feature of PCD provides a statistic for the process level (mean) which is used to detect and distinguish between changes in level and eigenstructure of a time series. Model eigenstructure is determined by the roots (or eigenvalues) of a model. It is related to the order and parameter values of AR or ARMA models, and has a direct effect on the level (bias) of the variable described by the model and its variance (spread). Based on the values assigned by PCD to various indicators one can determine if an eigenstructure change has occurred and if so whether this involves a level change, a spread change, or both. The outcome of implicit parametrization confirms the existence or lack of a level change, and provides the magnitude of the level change [414, 411].

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