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1、Chapter 8Cumulative Sum and Exponentially Weighted Moving Average Control ChartsIntroduction to Statistical Quality Control,4th EditionIntroduction Chapters 4 through 6 focused on Shewhart control charts.Major disadvantage of Shewhart control charts is that it only uses the information about the pro
2、cess contained in the last plotted point.Two effective alternatives to the Shewhart control charts are the cumulative sum(CUSUM)control chart and the exponentially weighted moving average(EWMA)control chart.Especially useful when small shifts are desired to be detected.Introduction to Statistical Qu
3、ality Control,4th Edition8-1.The Cumulative-Sum Control Chart 8-1.1 Basic Principles:The Cusum Control Chart for Monitoring the Process MeanThe cusum chart incorporates all information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a targ
4、et value.If 0 is the target for the process mean,is the average of the jth sample,then the cumulative sum control chart is formed by plotting the quantityIntroduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean Let xi be the ith ob
5、servation on the processIf the process is in control thenAssume is known or can be estimated.Accumulate derivations from the target 0 above the target with one statistic,C+Accumulate derivations from the target 0 below the target with another statistic,CC+and C-are one-sided upper and lower cusums,r
6、espectively.Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean The statistics are computed as follows:The Tabular Cusumstarting values areK is the reference value(or allowance or slack value)If either statistic exceed a decis
7、ion interval H,the process is considered to be out of control.Often taken as a H=5Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean Selecting the reference value,KK is often chosen halfway between the target 0 and the out-of
8、-control value of the mean 1 that we are interested in detecting quickly.Shift is expressed in standard deviation units as 1=0+,then K is Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean Example 8-10=10,n=1,=1Interested in
9、detecting a shift of 1.0=1.0(1.0)=1.0Out-of-control value of the process mean:1=10+1=11K=and H=5=5(recommended,discussed in the next section)The equations for the statistics are then:Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Proc
10、ess Mean Example 8-1 Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean Example 8-1The cusum control chart indicates the process is out of control.The next step is to search for an assignable cause,take corrective action requ
11、ired,and reinitialize the cusum at zero.If an adjustment has to be made to the process,may be helpful to estimate the process mean following the shift.Introduction to Statistical Quality Control,4th Edition8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean Example 8-1If an adjust
12、ment has to be made to the process,may be helpful to estimate the process mean following the shift.The estimate can be computed from N+,N-are counters,indicating the number of consecutive periods that the cusums C+or C-have been nonzero.Introduction to Statistical Quality Control,4th Edition8-1.4 Th
13、e Standardized Cusums It may be of interest to standardize the variable xi.The standardized cusums are thenIntroduction to Statistical Quality Control,4th Edition8-1.5 Rational Subgroups Shewhart chart performance is improved with rational subgroupingCusum is not necessarily improved with rational s
14、ubgroupingOnly if there is significant economy of scale or some other reason for taking larger samples should rational subgrouping be considered with the cusumIntroduction to Statistical Quality Control,4th Edition8-1.6 Improving Cusum Responsiveness for Large Shifts Cusum control chart is not as ef
15、fective in detecting large shifts in the process mean as the Shewhart chart.An alternative is to use a combined cusum-Shewhart procedure for on-line control.The combined cusum-Shewhart procedure can improve cusum responsiveness to large shifts.Introduction to Statistical Quality Control,4th Edition8
16、-1.7 The Fast Initial Response or Headstart Feature These procedures were introduced to increase sensitivity of the cusum control chart upon start-up.The fast initial response(FIR)or headstart sets the starting values equal to some nonzero value,typically H/2.Setting the starting values to H/2 is ca
17、lled a 50 percent headstart.Introduction to Statistical Quality Control,4th Edition8-1.8 One-Sided Cusums There are practical situations where a single one-sided cusum is useful.If a shift in only one direction is of interest then a one-sided cusum would be applicable.Introduction to Statistical Qua
18、lity Control,4th Edition8-1.9 A Cusum for Monitoring Process Variability LetThe standardized value of xi isA new standardized quantity(Hawkins(1981)(1993)is given by Hawkins suggest that the i are sensitive to variance changes rather than mean changes.Introduction to Statistical Quality Control,4th
19、Edition8-1.9 A Cusum for Monitoring Process Variability I N(0,1),two one-sided standardized scale cusums areThe Scale Cusumwhereif either statistic exceeds h,the process is considered out of control.Introduction to Statistical Quality Control,4th Edition8-1.11 The V-Mask Procedure The V-mask procedu
20、re is an alternative to the tabular cusum.It is often strongly advised not to use the V-mask procedure for several reasons.1.The V-mask is a two-sided scheme;it is not very useful for one-sided process monitoring problems.2.The headstart feature,which is very useful in practice,cannot be implemented
21、 with the V-mask.3.It is sometimes difficult to determine how far backwards the arms of the V-mask should extend,thereby making interpretation difficult for the practitioner.4.Ambiguity associated with with and Introduction to Statistical Quality Control,4th Edition8-2.The Exponentially Weighted Mov
22、ing Average Control Chart The Exponentially Weighted Moving Average Control Chart Monitoring the Process MeanThe exponentially weighted moving average(EWMA)is defined aswhere 0 0.1Introduction to Statistical Quality Control,4th Edition8-2.4 Robustness of the EWMA to Non-normalityAs discussed in Chapter 5,the individuals control chart is sensitive to non-normality.A properly designed EWMA is less sensitive to the normality assumption.Introduction to Statistical Quality Control,4th Edition