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Introduction to Statistica
Chapter 8
Cumulative Sum and
Exponentially Weighted Moving
Average Control Charts
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Introduction to Statistica
Introduction
• Chapters 4 through 6 focused on Shewhart controlcharts.
• Major disadantage of Shewhart control charts isthat it only uses the infor!ation a"out the processcontained in the last plotted point.
• #wo effectie alternaties to the Shewhart controlcharts are the cu!ulatie su! $C%S%M& controlchart and the e'ponentially weighted !oingaerage $E(M)& control chart. Especially usefulwhen s!all shifts are desired to "e detected.
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Introduction to Statistica
8-1. he Cumulative-Sum Control
Chart8-1.1 !asic "rinciples# he Cusum Control Chart $or
Monitoring the "rocess Mean
• #he cusu! chart incorporates all infor!ation in these*uence of sa!ple alues "y plotting the cumulative sums
of the deiations of the sa!ple alues fro! a target alue.
• If µ+ is the target for the process !ean, is the aerage of
the jth sa!ple, then the cu!ulatie su! control chart isfor!ed "y plotting the *uantity
∑
µ
=
i
j+ ji &'$C
j'
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess Mean• -et 'i "e the ith o"seration on the process
• If the process is in control then• )ssu!e σ is nown or can "e esti!ated.
• )ccu!ulate deriations fro! the target µ+ a"oe the targetwith one statistic, C/
• )ccu!ulate deriations fro! the target µ+ "elow the targetwith another statistic, C 0
• C/ and C11 are one1sided upper and lower cusu!s,respectiely.
&,$ 23' +i σ
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess Mean• #he statistics are co!puted as follows
he a&ular Cusum
starting alues are K is the re$erence value $or allowance or slac alue&
If either statistic e'ceed a decision interal H , the processis considered to "e out of control. 5ften taen as a H 7σ
ii+i
i+ii
C'& $,+!a'C
C& $',+!a'C
+CC ++ =
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess MeanSelecting the re$erence value' K
• K is often chosen halfway "etween the target µ+ and theout1of1control alue of the !ean µ that we are interested
in detecting *uicly.
• Shift is e'pressed in standard deiation units as µ µ+/δσ,
then K is
889
+ µ
==
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess MeanExample 8-1
∀ µ+ +, n , σ
• Interested in detecting a shift of .+σ .+$.+& .+
• 5ut1of1control alue of the process !ean µ + /
• K : and H 7σ 7 $reco!!ended, discussed in the
ne't section&
• #he e*uations for the statistics are then
iii
iii
C'7.+,+!a'C
C7.+',+!a'C
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess MeanExample 8-1
-5
0
5
-5
5
0 10 20 30
Subgroup Number
C
u m u l a t i v e S u m
Upper CUSUM
Lower CU SUM
CUSUM Chart for x
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess MeanExample 8-1
• #he cusu! control chart indicates the process is out of
control.
• #he ne't step is to search for an assigna"le cause, tae
correctie action re*uired, and reinitiali;e the cusu! at
;ero.
• If an adjust!ent has to "e !ade to the process, !ay "ehelpful to esti!ate the process !ean following the shift.
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Introduction to Statistica
8-1.% he a&ular or Algorithmic
Cusum $or Monitoring the "rocess MeanExample 8-1
• If an adjust!ent has to "e !ade to the process, !ay "e
helpful to esti!ate the process !ean following the shift.
• #he esti!ate can "e co!puted fro!
• 2/, 21 are counters, indicating the nu!"er of consecutie
periods that the cusu!s C/ or C1 hae "een non;ero.
>
>
=
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Introduction to Statistica
8-1.( he Standardi)ed Cusums
• It !ay "e of interest to standardi;e the aria"le 'i.
• #he standardi;ed cusu!s are then
σ
µ
=
+i
i
'y
iii
iii
Cy ,+!a'C
C y,+!a'C
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Introduction to Statistica
8-1.* +ational Su&groups
• Shewhart chart perfor!ance is i!proed with
rational su"grouping
• Cusu! is not necessarily i!proed with rational
su"grouping
• 5nly if there is significant econo!y of scale or
so!e other reason for taing larger sa!plesshould rational su"grouping "e considered with
the cusu!
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Introduction to Statistica
8-1., Improving Cusum
+esponsiveness $or arge Shi$ts• Cusu! control chart is not as effectie in
detecting large shifts in the process !ean as theShewhart chart.
• )n alternatie is to use a com&ined cusum-
Shehart procedure for on1line control.
• #he co!"ined cusu!1Shewhart procedure can
i!proe cusu! responsieness to large shifts.
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Introduction to Statistica
8-1./ he 0ast Initial +esponse or
eadstart 0eature
• #hese procedures were introduced to increase
sensitiity of the cusu! control chart upon start1up.
• #he fast initial response $>I?& or headstart sets
the starting alues e*ual to so!e non;ero
alue, typically
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Introduction to Statistica
8-1.8 2ne-Sided Cusums
• #here are practical situations where a
single one1sided cusu! is useful.
• If a shift in only one direction is of
interest then a one1sided cusu! would "e
applica"le.
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Introduction to Statistica
8-1.3 A Cusum $or Monitoring
"rocess 4aria&ility• -et
• #he standardi;ed alue of 'i is
• ) new standardi;ed *uantity $
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Introduction to Statistica
8-1.3 A Cusum $or Monitoring
"rocess 4aria&ility∀ νI 3 2$+, &, two one1sided standardi;ed scale cusu!s
are
he Scale Cusum
where
if either statistic e'ceeds h, the process is considered outof control.
iii
iii
S ,+!a'S
S ,+!a'S
+SS ii =
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Introduction to Statistica
8-1.11 he 4-Mas5 "rocedure
• #he D1!as procedure is an alternatie to the ta"ular
cusu!.
• It is often strongly adised not to use the D1!as
procedure for seeral reasons.. #he D1!as is a two1sided sche!e it is not ery useful for one1sided process !onitoring pro"le!s.
8. #he headstart feature, which is ery useful in practice, cannot "e
i!ple!ented with the D1!as.
. It is so!eti!es difficult to deter!ine how far "acwards thear!s of the D1!as should e'tend, there"y !aing interpretation
difficult for the practitioner.
4. )!"iguity associated with with α and β
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Introduction to Statistica
8-%. he Exponentially Weighted
Moving Average Control Charthe Exponentially Weighted Moving Average Control
Chart Monitoring the "rocess Mean
• #he e'ponentially weighted !oing aerage $E(M)& isdefined as
where + F λ ≤ is a constant.
;+ µ+ $so!eti!es ;+ &
iii ;&$';
'
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Introduction to Statistica
8-%.1 he Exponentially Weighted
Moving Average Control Chart
Monitoring the "rocess Mean
• #he control li!its for the E(M) control chart are
where L is the width of the control li!its.
i8
+
+
i8+
&$&8$--C-
C-
&$&8$
-%C-
λλ
λ
σ
µ
λ
λ
λσ
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Introduction to Statistica
8-%.1 he Exponentially Weighted
Moving Average Control Chart
Monitoring the "rocess Mean
• )s i gets larger, the ter! G1 $ 1 λ&8iH approachesinfinity.
• #his indicates that after the E(M) control chart has "een running for seeral ti!e periods, the control li!its
will approach steady-state alues gien "y
&8$--C-
C-&8$
-%C-
+
+
+
λ
λ
σ
µ
λ
λ
σ
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Introduction to Statistica
8-%.% 6esign o$ an EWMA Control
Chart• #he design para!eters of the chart are L and λ.• #he para!eters can "e chosen to gie desired )?-
perfor!ance.
• In general, +.+7 ≤ λ ≤ +.87 wors well in practice.• L wors reasona"ly well $especially with the largeralue of λ.
• L "etween 8.6 and 8.B is useful when λ ≤ +.
• Si!ilar to the cusu!, the E(M) perfor!s well against small shifts "ut does not react to large shifts as *uicly asthe Shewhart chart.
• E(M) is often superior to the cusu! for larger shifts particularly if λ +.
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Introduction to Statistica
8-%.( +o&ustness o$ the EWMA to
7on-normality
• )s discussed in Chapter 7, the individuals
control chart is sensitie to non1nor!ality.
• ) properly designed E(M) is less
sensitie to the nor!ality assu!ption.