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A Bayesian On-Line Change Detection Algorithm with Process Monitoring Applications Pradipta Sarkar Iowa State University

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A Bayesian On-Line Change Detection Algorithm with Process Monitoring Applications Pradipta Sarkar Iowa State University Statistics Preprints Statistics 1998 A Bayesian on-line change detection algorithm with process monitoring applications Pradipta Sarkar Iowa State University William Q. Meeker Iowa State University, [email protected] Follow this and additional works at: http://lib.dr.iastate.edu/stat_las_preprints Part of the Statistics and Probability Commons Recommended Citation Sarkar, Pradipta and Meeker, William Q., "A Bayesian on-line change detection algorithm with process monitoring applications" (1998). Statistics Preprints. 1. http://lib.dr.iastate.edu/stat_las_preprints/1 This Article is brought to you for free and open access by the Statistics at Iowa State University Digital Repository. It has been accepted for inclusion in Statistics Preprints by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A Bayesian on-line change detection algorithm with process monitoring applications Abstract This work has been motivated by some issues of process monitoring in casting applications. In die casting, as time passes, material is deposited on the inside wall of the die. This foreign material can cause an unacceptable level of porosity in the output of the casting process. After accumulated material reaches a certain level, some of the output of the casting process no longer meets the specifications and the process must be stopped to clean the die. Sudden changes in output quality are also common in casting processes. The degree of porosity of the output of the casting process changes suddenly if, for example, one opens the door of the production facility thereby changing the laboratory temperature, causing an immediate larger accumulation of foreign material. Also day-to-day variation in weather, changes in humidity, and other factors can be responsible for this kind of change. Keywords Nondestructive testing, Nondestructive evaluation, Empirical and hierarchical Bayes, Rejection sampling, Trend Disciplines Statistics and Probability Comments This preprint has been published in Quality Engineering 10 (1998): 539- 549.. This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/stat_las_preprints/1 ABAYESIAN ON-LINE CHANGE DETECTION ALGORITHM WITH PROCESS MONITORING APPLICATIONS Pradipta Sarkar William Q. Meeker [email protected] [email protected] Center for NondestructiveEvaluation and Department of Statistics Iowa State University Ames, Iowa 50011 . 1 Intro duction 1.1 Motivation: Casting Example This work has b een motivated by some issues of pro cess monitoring in casting applications. In die casting, as time passes, material is dep osited on the inside wall of the die. This foreign material can cause an unacceptable level of p orosity in the output of the casting pro cess. After accumulated material reaches a certain level, some of the output of the casting pro cess no longer meets the sp eci cations and the pro cess must b e stopp ed to clean the die. Sudden changes in output quality are also common in casting pro cesses. The degree of p orosity of the output of the casting pro cess changes suddenly if, for example, one op ens the do or of the pro duction facility therebychanging the lab oratory temp erature, causing an immediate larger accumulation of foreign material. Also day-to-dayvariation in weather, changes in humidity, and other factors can b e resp onsible for this kind of change. 1.2 On-line vs. O -line Change Detection Algorithm Let y ;y ;::: b e a sequence of observed pro cess output values (such as the degree of p orosity) with 1 2 conditional density f (y jy ;:::;y ; ). Let the unknown time of change b e t . In on-line change- k k 1 1 0 detection algorithms, the ob jective is to detect a change as so on as p ossible, if not immediately after it o ccurs. Detection of the change-time is p erformed by considering all past data and stopping at the rst p oint in time at which there is sucient evidence that a change of imp ortance has o ccurred. 1 Mathematically, such a stopping-time rule can b e expressed as: t =minf (y ;:::;y ) g a n 1 n n where ; ;::: is a family of functions of n co-ordinates indexed by time and is a threshold 1 2 sp eci ed to control the probability of detection errors. For example, may b e the probability n of exceeding the critical level at the n th insp ection. Another alternativewould b e to de ne n n n as max log ; where is the likeliho o d ratio for the observations from time j to time n (The j j 1j n likeliho o d ratio is generally de ned as the ratio of the probability of the data maximized over all pro cess states to the probability when the pro cess is op erating correctly.) O -line algorithms, on the other hand, can b e lo oked up on as p ostmortem analyses. They are designed to collect data and test if there had b een anychange in the level of the pro cess sometime during the past. If it is suggested that there was a change, these algorithms try to estimate the unknown change time. In this pap er we will fo cus mainly on the on-line algorithms. 1.3 Overview Section 2 provides a brief survey of literature and describ es the basic problem of detecting a change of known magnitude. This basic problem is imp ortant for understanding the subsequentdevelopments in the following sections. In section 3 we will generalize to more realistic situations of multiple changes of unknown magnitude, and trend in the pro cess mean. Section 4 will b e devoted to the discussion of rejection sampling and its use in Bayesian up dating of the distributions used in making decisions. In section 5 we will use a simulated example to illustrate the b ehavior of the metho dologies develop ed in sections 2 and 3. In section 6 we will formulate a decision theory-based metho d to determine an optimum insp ection interval. Section 7 illustrates the metho dology of section 6 with an example. Finally, in section 8, we discuss the p ossibility of extending the metho dology describ ed in this pap er to a hierarchical Bayes formulation and provide some concluding remarks. 2 Brief Review of Literature 2.1 Bayes Typ e Algorithm for Detecting Changes in Pro cess Monitoring ABayesian change detection approachhasseveral advantages over non-Bayesian approaches. It is technically simple and is easier to implement and allows incorp oration of available engineering information. There is a large amount of literature on Bayesian changep oint detection problems. Smith (1975) presents the Bayesian formulation for a nite sequence of indep endent observations. In particular he gives details for binomial and normal distribution mo dels. Some other works in this area include Changep oint in univariate time series : Bo oth and Smith (1982), West and Harrison (1986). Gammatyp e random variables and Poisson pro cesses : Diaz (1982), Hsu(1982), Raftery and Akman (1986). 2 process level -2024681012 0 20 40 60 80 100 time Figure 1: Increase in pro cess mean, pro cess variance remaining the same . Changep oint in linear mo dels : Bacon and Watts (1971), Ferreira (1975), Holb ert and Bro emel- ing (1977), Choy and Bro emeling (1980), Mo en, Salazar and Bro emeling (1985), Smith and Co ok (1990). Hierarchical Bayes formulation of changep oint mo del : Carlin, Gelfand and Smith (1992). Most of these references provide metho dology for o -line problems. We are interested in pro cess control applications where on-line algorithms are needed. The next subsection describ es the simplest p ossible on-line scenerio. 2.2 Single Jump - Before and After-Change Parameter Values Known In the simplest case, the parameter values (e.g., mean pro cess level) b efore and after the change, = and = resp ectively, are b oth known. Figure 1 shows a pro cess where the mean level 0 1 changes after 50 cycles, but pro cess variance remains the same. Figure 2 shows a pro cess where the mean is constantbutvariance increases after 50 cycles. These kinds of changes are typical in many industrial pro cesses. Detection of suchchanges has b een discussed in detail by several authors in di erent contexts. Here we describ e brie y the solution to this problem to motivate our following developments. Some of the pioneering work for this mo del is due to Girshick and Rubin(1952) and Shirayaev(1961) and is describ ed nicely in Basseville and Nikiforov(1993). Our notation will follow closely that in Basseville and Nikiforov(1993). Assume that the prior distribution of (discrete) change time t is Geometric() and is given by, 0 k 1 P (t = k )=(1 ) ;k =1; 2;::: 0 3 process quality -4-20246 0 20 40 60 80 100 time Figure 2: Increase in pro cess variance, pro cess mean remaining the same . porosity -1 0 1 2 5101520 time Bayesian Updating rho=0.10 rho=0.25 rho=0.50 Posterior probability 0.0 0.4 0.8 5101520 Inspection time Figure 3: Bayesian up dating when pro cess mean changes. Top : Porositylevel, Bottom: Up dated probability that the pro cess parameter has changed 4 porosity -1.0 0.0 1.0 2.0 5101520 time Bayesian Updating rho=0.10 rho=0.25 rho=0.50 Posterior probability 0.0 0.2 0.4 0.6 0.8 0 5 10 15 20 Inspection time Figure 4: Bayesian up dating when there is no change in the pro cess mean.Top : Porositylevel, Bottom: Up dated probability that the pro cess parameter has changed where is the probabilityofchange in each pro duction cycle, an estimate of whichisavailable from pro cess history.
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