A Transition Matrix Representation of the Algorithmic Statistical Process Control Procedure with Bounded Adjustments and Monitoring Changsoon Park Department.

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A Transition Matrix Representation of the Algorithmic Statistical Process Control Procedure with Bounded Adjustments and Monitoring Changsoon Park Department of Statistics Chung-Ang University Seoul, Korea Slide 2 1 Changsoon Park Algorithmic Statistical Process Control (ASPC) - Vander Wiel, Tucker, Faltin, Doganaksoy(1992) - Integrated approach to quality improvement - An approach that realizes quality gains through process adjustment & process monitoring Process adjustment ; manipulate the compensating variables of a process to achieve the desired process behavior ( e.g., output close to a target ) - adjustment scheme ( feedforward, feedback ) ( e.g. repeated adjustment, bounded adjustment ) Process mornitoring ; monitor a process so as to detect and remove root causes of variability - control chart ( e.g. Shewhart, CUSUM, EWMA ) Slide 3 2 We consider Changsoon Park Disturbance Model IMA(0,1,1) with a step shift IMA(0,1,1) due to noise Step shift due to special cause ~ ASPC procedure Bounded Adjustments & EWMA Monitoring Derive properties by a transition matrix representation Slide 4 3 Changsoon Park IMA(0,1,1) with A Step Shift Slide 5 4 If, then adjust the process Control Procedure 1. Bounded Adjustments Changsoon Park : one-step ahead forecast : total output compensation : predicted deviation : observed deviation Slide 6 5 Changsoon Park : adjustment time : one-step output compensation If, adjust by, i.e. Restart with Recurrence relation observed deviation : Slide 7 6 Changsoon Park Bounded Adjustments Slide 8 7 Changsoon Park : adjustment time immediately before Random shock representation of and : no. of adjustments before a special cause occurs Slide 9 8 2. EWMA Monitoring When a signal is false, restart with : Bivariate process control statistic Changsoon Park : Forecast error adjustment, true signal : action EWMA statistic : If, signal : action time Slide 10 9 Changsoon Park EWMA Monitoring Slide 11 10 Transition Matrix Representation Calculate properties of ASPC procedure (no. of false signals, no. of adjustments, sum of squared deviations) 1. A cycle startend special cause period : false signal : adjustment : true signal Changsoon Park signal start special cause period signal end Slide 12 11 2. Representation of by a finite states - use of Gaussian quadrature points and weights 2.1 Partition of : no signal interval points :, weights :, (odd), weight subintervalpoints Changsoon Park Slide 13 12 2.2 Partition of : no adjustment interval points :, weights :, (odd), Changsoon Park weight subintervalpoints Slide 14 13 3. Transition Matrix Representation in each period Changsoon Park bivariate process states Transition matrix Slide 15 14 : one vector of dimension : zero vector of dimension : vector of dimension whose elements corresponding to class are all 1s and all the rests are. : vector of dimension whose element corresponding to the state is 1 and all the rests are. Partition the whole states into classes according to the action Denote each class by a character We are interested in each action, not in each state. We can identify the classes involved by the dimension of the matrix or vector. Changsoon Park Slide 16 15 3.1 Period classes classstateno. of states no action adjustment only false signal only false signal & adjustment whole states Changsoon Park Slide 17 16 decomposition of the total transition matrix Changsoon Park Slide 18 17 Define transition matrix until - rather than only in period - state of a special cause (occurrence or not) is added : transient state transition matrix Changsoon Park Occurrence of a special cause : absorbing state (there is no absorbing state in period ) Slide 19 18 : starting state vector of period where Changsoon Park : time that a false signal occurs Average no. of false signal false signal Slide 20 19 : time that an adjustment occurs Changsoon Park Average no. of adjustments adjustment Slide 21 20 where : adjustment interval length given Changsoon Park : SSD in an action(adjustment) interval given Slide 22 21 Changsoon Park Average period length Expected SSD start special cause period : average no. of visits to & Slide 23 22 Changsoon Park Probability of - the last adjustment time before a special cause occurs Slide 24 23 merge into 3.2 Period Changsoon Park For : starting state vector of period class of no action and false signal only keep, Slide 25 24 : the time, counted from the beginning of period, that a special cause occurs start special cause period Changsoon Park adjustment or signal : state vector at immediately before time Slide 26 25 classes after classstateno. of states no action adjustment signal whole states Changsoon Park Slide 27 26 decomposition of the total transition matrix : absorbing states Changsoon Park Define Slide 28 27 period Changsoon Park : action(adjustment or siganl) interval length Slide 29 28 Changsoon Park Average no. of adjustments : length of period : no. of adjustment in period Average period length Slide 30 29 Changsoon Park Expected SSD Slide 31 30 Changsoon Park Final state probability of period where Slide 32 31 3.3 Period decomposition of the total transition matrix : absorbing state : starting state vector of period For Changsoon Park Slide 33 32 Define Average period length Average no. of adjustments Changsoon Park Slide 34 33 Expected SSD For Changsoon Park Slide 35 34 Expected Cost Per Unit Time (ECU) : cost per monitoring Changsoon Park : cost per adjustment : off-target cost per SSD : cost per false signal