8_CUSUM

# 8_CUSUM - Spring 2008 EMSE 182, 282, Professor Jim Harris,...

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Spring 2008 EMSE 182, 282, Professor Jim Harris, CUSUM and EWMA Charts - 1 CUSUM AND EXPONENTIALLY WEIGHTED MOVING AVENGE CHARTS CUSUM CHARTS Motivation • Slight changes take a long time to detect for X Charts X~N(25,400), then in an X Chart with n=4, if 32 . Ä • X Charts are good for detecting shifts of >1.5 5 • If information of the whole sequence could be compounded,

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Spring 2008 EMSE 182, 282, Professor Jim Harris, CUSUM and EWMA Charts - 2 C ( x) ( ( 34 3 4 4œ" 4œ" 33 " œ œ ±± .. . 00 0 ( x ) C " . 0 where C 0 and is the process mean target, then reaction to smaller ! œ . 0 shifts in the process mean would be quicker (compound effect). •This is the basis for the CUSUM or CUMulative SUM Chart - a plot of C vs i. 3 • Note that CUSUM are better (faster or lower ARL) for detecting small shifts and are particularly useful for smaller sample sizes, especially n 1. œ • if the process is in control with mean then the sequence . ! {C } is a random walk with mean 0. 3 • a positive (negative) trend in {C } occurs if the process mean shifts 3 upwards (downwards).
Spring 2008 EMSE 182, 282, Professor Jim Harris, CUSUM and EWMA Charts - 3 Assumptions X ~ N ( , ) 3! # .5 • {X } iid. 3 known or well estimated 5 n 1 œ Method • Use a table to track or plot C max{0, x K) C } for upward shifts 3  3 œ Ð . 1 C max{0, K) x C } for downward shifts 33  !3 œÐ . 1 where C C 0. 0 0 œœ K reference value, allowance, or slack value K for detecting a shift in standard deviation units \$ .. 2 5 ll # 10 of , i.e. \$5 . . . !" ! Äœ • An assignable cause is assumed when C or C H the decision 3 3 interval. H is chosen with ARL in mind but H 5 is recommended. œ 5

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Spring 2008 EMSE 182, 282, Professor Jim Harris, CUSUM and EWMA Charts - 4 • Table + S H I F T S SHIFTS Period x x K) C N K) N 33 ! ! 3 3 3  Ð Ð .. + N (N ) tells the run number for positive C (C ) ++ • Example 10, 1, K .5, H 5 .5 ! œœ œ œ  SHIFTS SHIFTS Period x x 10.5 C N 9.5 x C N 1 9.45 1.05 0 0 0.05 0.05 1 2 7.99 2.52 0 0 1.51 1.56 2 3 9.29 1.21 0 0 0.21 1.77 3 4 11.66 1.16 1.16 1 2 3 3 3 + .16 0 0 5 12.16 1.66 2.82 2 2.66 0 0 6 10.18 0.32 2.50 3 0.68 0 0 7 8.04 2.46 0.04 4 1.46 1.46 1 8 11.46 0.96 1.00 5 1.96 0 0 9 9.20 1.30 0. 0 0.30 0.30 1
Spring 2008 EMSE 182, 282, Professor Jim Harris, CUSUM and EWMA Charts - 5 • To plot use CUSUM Status Charts

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## This note was uploaded on 01/02/2010 for the course EMSE 282 taught by Professor Harris during the Spring '07 term at GWU.

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8_CUSUM - Spring 2008 EMSE 182, 282, Professor Jim Harris,...

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