The Geometric Moving AverageThe Geometric Moving Average Control Chart: A Full-Purpose

Process-Control ToolASQ-Baltimore Section Meeting

December 10, 2002December 10, 2002Melvin T. Alexander

P t Ch i ASQ H lth C Di i iPast Chair – ASQ Health Care Division

Focus: SPC Forecasting Control ChartChart

liOutline

• Introduction• Classical vs Modern viewsClassical vs. Modern views• Comparison between charts

C i GMA h• Constructing GMA charts• Example• Conclusions

SPC Popular Due to

• Successful in Pre-1940s Technologygy• Japanese Advances in Quality• Regain Competitve Edge in 1980s• Regain Competitve Edge in 1980s

In Catching Up We Found

• Automated Systems Expanded• “Older Tools” Inadequate• Modern Tools Not Fully Usedy• Modern Methods Adopted/Adapted

Classical SPC “Passive”Classical SPC Passive

• Describes What Happened• Describes What Happened• Curative in Approach• Monitors Stability

Modern SPC “Active”

• Analyzes Causes & EffectP di F• Predicts Future

• Gives Dynamic Feedback/Feedfront yBasis for Action

Standard Control Chart Schemes

• Shewhart• Individuals, Moving Range, &

Moving Averageg g• CUSUM• GMA (EWMA)• GMA (EWMA)

Sh h C l ChShewhart Control Charts

• Developed by Dr. W.A. Shewhart (1924)• Monitors Process Stability with ControlMonitors Process Stability with Control Limits

• Points Outside Limits Signal Instability• Points Outside Limits Signal Instability• Places “Weight” on Latest Observation

Shewhart Charts ProsShewhart Charts Pros

E C• Easy to Construct• Detects Large Process

ShiftsShifts• Helps find process

change caused sochange caused so that countermeasures against adverse effects gmay be implemented

Shewhart Chart Cons

• Ignores History• Hard to Detect Small

Process shifts• Hard to Predict In/Out-of-

Control ProcessesControl Processes• Really is not Process

CONTROL, but is usedCONTROL, but is usedfor RESEARCH Control

Individuals, Moving Range & M i A ChMoving Average Charts

• Special-Case Shewhart Charts• Single Observations• Single Observations• Control signal Like Shewhart

Individual (I) Charts

• Few Units Made• Automated Testing on Each Unit• Automated Testing on Each Unit• Expensive Measurements• Slow Forming Data• Periodic (Weekly, Monthly) Adminstrative Data

Moving Range (MR) Charts

• Correspond to Shewhart Range h tcharts

• Used with Individuals • Plot Absolute Differences of Successive Pairs (Lags)( g )

Moving Average (MA) Charts

• Running Average of (=4,5) ObservationsObservations

• New MA Drops “Oldest” Valuei h l lik h h• Weight Places like Shewhart

(n=2 for MR; n=4,5 in MA;( ; , ;0 Otherwise)

I/MA/MR PI/MA/MR Pros

ll• Detects Small Process Shifts

• Uses Limited HistoryHistory

I/MA/MR CI/MA/MR Cons• Auto(Serial)

correlated Data Mislead Process StabilitySh N d• Show Nonrandom Patterns with R d D tRandom Data

Cusum Control Charts

• Introduced by E.S. Page (1954)• Cumulative Sums single Observations• Points Outside V-Mask limbs• Put Equal Weight on Observations

Cusum Chart Pros

D t t “S ll”• Detect “Small” Process Shifts

• Uses History

Cusum Chart Cons

• Fail to DiagnoseFail to Diagnose Out-of-control PatternsPatterns

• Hard to Construct V MaskV-Mask

Geometric (Exponetially Weighted) Moving Average (GMA/EWMA)Control Charts

• Introduced by S.W. Roberts (1959)t oduced by S.W. obe ts ( 959)Time Series

• Points Fall Outside Forecast Control o s Ou s de o ec s Co oLimits

• Weights (w) Assign Degrees of g ( ) g gImportance

• Resembles Shewhart & Cusum SchemesResembles Shewhart & Cusum Schemes

GMA(EWMA) Chart Pros

• Regular Use of iHistory

• Approaches Sh h if 1Shewhart if w=1,

• Approaches Cusum if 0if w=0

GMA (EWMA) Control chart Cons• Late in Catching TurningLate in Catching Turning

Points• Inaccurate for Auto(Serial) ( )

Correlated Data• Only One-ahead Forecasts

P iblPossible• Hard to Monitor Changes in

Weight (w)Weight (w)• Allow Only One Variation

Componentp

GMA Forecasting ModelF A F+ 1( ) * ( ) * ( )F w w A w F wt t t+ = + −1 1( ) * ( ) * ( )

= + −F w w A F wt t t( ) * ( ( ))t t t( ) ( ( ))

= +F w w e wt t( ) * ( )where

At : Actual Observation at time ttFt, t+1(w): Forecast at time t, t+1 using weight factor w

: Observed error ( ) at time te wt ( ) )(wFA tt −=

The Minimum-error Weight (w’) is ChosenThe Minimum-error Weight (w ) is Chosen from {w1, w2, … , wk} ≡ {1/T, 2/T, …, k/T}, S h th tSuch that

MiT T T T

2 2 2 2∑ ∑ ∑ ∑( ') { ( ) ( ) ( )}e w Min e w e w e wtt

tt

tt

tt

k2 2

12

22∑ ∑ ∑ ∑=( ') { ( ), ( ),..., ( )}

where 0 ≤ w1 ≤ wk ≤ 1 for k=T-1.

Sigmas (σ, σgma )

∑T

2 ( )σ = =

∑ e wT

tt

2

11

( )( )−T 1( )

T N f S lT = No. of Samples

The k-th Lag Error Autocorrelation ( ) :)ρ( ) :ρk

− −∑ [ ( ) ][ ( ) ]e w e e w ek

T

)ρk =−

= +∑

∑

[ ( ) ][ ( ) ]

[ ( ) ]

e w e e w e

e w e

t t kt k

T1

2−−= +∑ [ ( ) ]e w et k

t k 1T

∑For k = 0, 1, 2, …, T , e = e w

T

tt=∑

1( )

T

1k

If)ρk < 2* ( )1 2 2

1

+−

∑ )ρt

k

ρk 1=tT

)ρkThen ≈ 0

Otherwise, Estimates the Non-zero Error Autocorrelation

)ρk

If the First-Lag Autocorrelation Exists (i e ≠ 0) Then a “Corrected” sigma (σ )

)ρ(i.e., ≠ 0), Then a Corrected sigma (σc) becomes:

ρ1

σc = [ * ( ) * / ]* /1 2 1 12+ −T T T)ρ σ

From Equation 18.3 of Box-Hunter-Hunter (1978 p 588)(1978, p. 588)

σgma = ])1(1[2

2tww

w−−

−σ

as t≅ wσ ∞→as t

Note: σ replaces σ if ≠ 0

w−2σ →

)ρ1Note: σc replaces σ if ≠ 0ρ1

Bayes Theorem cont.

Kalman filtering (Graves et al., 2001) and Abraham and Kartha (ASQC Annual TechnicalAbraham and Kartha (ASQC Annual Technical Conf. Trans., 1979) described ways to check weight stability.g y

Bayes Theorem cont.

H t l h t l t f thHowever, control chart plots of the errors help detect weight changes (i.e., errors h ld hibit d ttshould exhibit a random pattern,

if not, change the weight or th f t th d)the forecast method)