CUSUM Chart

8/19/2019 CUSUM Chart

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Introduction to Statistica

Chapter 8

Cumulative Sum and

Exponentially Weighted Moving

Average Control Charts


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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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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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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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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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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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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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-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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• #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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• 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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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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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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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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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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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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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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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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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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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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8-%.1 he Exponentially Weighted

Moving Average Control Chart


• #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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8-%.1 he Exponentially Weighted

Moving Average Control Chart


• )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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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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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.

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