### Bayesian Statistical Process Control

Organizer and Session Chair: Panagiotis Tsiamyrtzis, Dept. of Statistics, Athens University of Economics & Business.**A Bayesian Approach to Monitoring the Mean of a Multivariate Normal Process**

Frank B. Alt

^{1}and Scott D. Grimshaw

^{2}

^{1}University of Maryland College Park

^{2}Brigham Young University

Abstract: Several Bayesian multivariate quality control procedures for monitoring the process mean vector are introduced and compared with existing multivariate procedures. The first procedure, denoted as IMBP, performed significantly better for detecting small shifts than Crosier’s multivariate CUSUM procedure (CMCUSUM) and Lowry et al’s multivariate exponentially weighted moving average (MEWMAE) procedure with exact covariance matrix. The IMBP was revised (RMBP) so that it could be designed without resorting to simulation. The RMBP also performed better than CMCUSUM and MEWMAE for detecting moderate and larger shifts. The RMBP was further revised to overcome the problem of inertia. Furthermore, it appears that the RMBPs are quite robust to departures from the assumption of multivariate normality. The robustness study also indicated that the MBPs are able to detect increased variability as well as a shift in the process mean vector.

**A Bayesian Approach to Change Point Estimation in Multivariate SPC**

Rong Pan

^{1}and Steven E. Rigdon

^{2}

^{1}Arizona State University

^{2}Southern Illinois University at Edwardsville.

Abstract: A multivariate control chart is used to monitor more than one process variable simultaneously. When a control chart signals an out-of-control condition, an investigation should be undertaken to determine when the change happened and which variable actually caused the change. A Bayesian method can be applied to this change point problem. The posterior distribution can be obtained for the point at which the change occurred. We consider three cases: (1) when the process parameters before and after the change are known, (2) the process parameters before the change are known, but the parameters after the change are unknown, and (3) the process parameters, both before and after the change are unknown. This Bayesian approach will help users identify the most likely change points and the (posterior) probabilities of a change at each time that a subgroup was taken.

**Recursive Sequential Bayesian Algorithm for Bio-surveillance**

Gideon Zamba

University of Iowa.

Abstract: We develop a serial signal detection algorithm to monitor medical diagnosis data sets that pertain to bio-surveillance. The method is a three-state sequential algorithm based on bayesian thinking, to account for stationarity, irregularity, seasonality, and to capture an epidemic’s serial structural detail. We set a prior distribution for the time indexed epidemic readings, find a posterior based on the observed data, assess the state of the epidemic and use the posterior state probabilities as priors for the next stage. The control scheme is based on probability surges in the trichotomous variable governing the states of the epidemic. We provide analytical formulas for the predictive distribution for error management, and conduct sensitivity analyses as means for validation.