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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

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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 )

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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

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

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

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5 Changsoon Park : adjustment time : one-step output compensation If, adjust by, i.e. Restart with Recurrence relation observed deviation :

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6 Changsoon Park Bounded Adjustments

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

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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

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9 Changsoon Park EWMA Monitoring

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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

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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

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

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

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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

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

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16 decomposition of the total transition matrix Changsoon Park

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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 )

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

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19 : time that an adjustment occurs Changsoon Park Average no. of adjustments adjustment

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

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

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

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

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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

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

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

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

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

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29 Changsoon Park Expected SSD

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30 Changsoon Park Final state probability of period where

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

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

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33 Expected SSD For Changsoon Park

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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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