# 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

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 )

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

3 Changsoon Park IMA(0,1,1) with A Step Shift

4 If, then adjust the process Control Procedure 1. Bounded Adjustments Changsoon Park : one-step ahead forecast : total output compensation : predicted deviation : observed deviation

7 Changsoon Park : adjustment time immediately before Random shock representation of and : no. of adjustments before a special cause occurs

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

9 Changsoon Park EWMA Monitoring

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

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

12 2.2 Partition of : no adjustment interval points :, weights :, (odd), Changsoon Park weight subintervalpoints

13 3. Transition Matrix Representation in each period Changsoon Park bivariate process states Transition matrix

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

15 3.1 Period classes classstateno. of states no action adjustment only false signal only false signal & adjustment whole states Changsoon Park

16 decomposition of the total transition matrix Changsoon Park

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 )

18 : starting state vector of period where Changsoon Park : time that a false signal occurs Average no. of false signal false signal

20 where : adjustment interval length given Changsoon Park : SSD in an action(adjustment) interval given

21 Changsoon Park Average period length Expected SSD start special cause period : average no. of visits to &

22 Changsoon Park Probability of - the last adjustment time before a special cause occurs

23 merge into 3.2 Period Changsoon Park For : starting state vector of period class of no action and false signal only keep,

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

25 classes after classstateno. of states no action adjustment signal whole states Changsoon Park

26 decomposition of the total transition matrix : absorbing states Changsoon Park Define

27 period Changsoon Park : action(adjustment or siganl) interval length

28 Changsoon Park Average no. of adjustments : length of period : no. of adjustment in period Average period length

29 Changsoon Park Expected SSD

30 Changsoon Park Final state probability of period where

31 3.3 Period decomposition of the total transition matrix : absorbing state : starting state vector of period For Changsoon Park

32 Define Average period length Average no. of adjustments Changsoon Park

33 Expected SSD For Changsoon Park

34 Expected Cost Per Unit Time (ECU) : cost per monitoring Changsoon Park : cost per adjustment : off-target cost per SSD : cost per false signal

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