Robustness of the EWMA control chart to non-normality

1
Robustness of the EWMA control chart to
non-normality

• Connie M Borror Douglas C Montgomery George C
  Runger
• Journal of Quality Technology Jul 1999 31, 3

2
Introduction

• Individual measurements occur frequently in the
  chemical and process industries.
• The traditional method of dealing with the case
  of n1 is to use the Shewhart individuals control
  chart to monitor the process mean.
• The individuals control chart has two
  widely-cited disadvantages
• (1) the chart is not very sensitive to small
  shifts in the process mean.
• (2) the performance of the chart can be adversely
  affected if the observations are not normally
  distribution.
• It is certainly true that non-normality of the
  process data is often not a significant concern
  if the X-bar control chart is used to monitor the
  mean.

3
Introduction

• In this paper, we show that the ARL performance
  of the Shewhart individuals control chart when
  the process is in control is very sensitive to
  the assumption of normality.
• We suggest the EWMA control chart as an
  alternative to the individuals chart for
  non-normal data.
• We show that, in the non-normal case, a properly
  designed EWMA control chart will have an
  in-control ARL that is reasonably close to the
  value of 370.4 for the individuals chart for
  normally distributed date.
• For all cases, the ARLs were computed using the
  Markov chain method.

4
Background Information-EWMA

• The EWMA is defined as
• Where xi is the current observation and ?,
  smoothing parameter, is a constant for 0???1
• The control limits for the EWMA control chart are
• Where L determines the width of the control
  limits

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Background Information-EWMA

• For large values of i , the steady-state EWMA
  control limits are
• If any point exceeds the control limits, the
  process is assumed to be out of control.

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Background Information-Skewed and symmetric
distribution

• To study the robustness of the EWMA control chart
  and the individuals control chart to normality
  assumption, both skewed and symmetric
  distribution were examined.
• Symmetric distributiont distribution
• Let k is degree of freedom
• The Mean is 0
• The Variance is k/(k-2)

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Various t distribution and normal distribution
with the same mean and variance
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Background Information-Skewed and symmetric
distribution

• Skewed distributionGamma distribution
• Let a0.5, 1, 2, 3, and 4,
• while holding ß1

9
Various Gamma distribution and normal
distribution with the same mean and variance
10
Results

• The normal-theory ARL for individuals control
  chart with 3s is known to be 370.4.
• For the EWMA, we can determine the values of ?
  and L to obtain approximately the same ARL of
  370.4.
• Value of 0.05, 0.1, and 0.2 were chosen for ?,
  with the corresponding value of 2.492, 2.703, and
  2.86, respectively, chosen for L.

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In-Control ARLs for EWMA-Gamma
EWMA EWMA EWMA EWMA Shewhart
? 0.05 0.1 0.2 1
L 2.492 2.703 2.86 3
Normal 370.4 370.8 370.5 370.4
Gam(4,1) 372 341 259 97
Gam(3,1) 372 332 238 58
Gam(2,1) 372 315 208 71
Gam(1,1) 369 274 163 55
Gam(0.5,1) 357 229 131 45
The Best Case
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Out-of-control ARLs for the EWMA-Gamma
Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations)
0.5 1 1.5 2 2.5 3
EWMA ?0.05 L2.492 Normal 26.5 10.8 6.8 5 4 3.4
EWMA ?0.05 L2.492 Gam(4,1) 26.4 11 6.9 5.1 4.1 3.4
EWMA ?0.05 L2.492 Gam(3,1) 26.4 11 7 5.1 4.1 3.5
EWMA ?0.05 L2.492 Gam(2,1) 26.4 11.1 7 5.2 4.1 3.5
EWMA ?0.05 L2.492 Gam(1,1) 26.4 11.2 7.1 5.3 4.2 3.5
EWMA ?0.05 L2.492 Gam(0.5,1) 26.6 11.4 7.3 5.4 4.3 3.6
EWMA ?0.1 L2.703 Normal 28.3 9.8 5.8 4.2 3.3 2.8
EWMA ?0.1 L2.703 Gam(4,1) 26.5 9.9 6 4.3 3.4 2.9
EWMA ?0.1 L2.703 Gam(3,1) 26.3 9.9 6 4.4 3.5 2.9
EWMA ?0.1 L2.703 Gam(2,1) 26 10 6.1 4.4 3.5 2.9
EWMA ?0.1 L2.703 Gam(1,1) 25.5 10.1 6.2 4.5 3.6 3
EWMA ?0.1 L2.703 Gam(0.5,1) 25.1 10.2 6.3 4.6 3.7 3.1
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Out-of-control ARLs for the EWMA-Gamma
Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations) Shift (Number of Standard Deviations)
0.5 1 1.5 2 2.5 3
EWMA ?0.2 L2.86 Normal 36.2 9.8 5.2 3.6 2.8 2.3
EWMA ?0.2 L2.86 Gam(4,1) 28 9.6 5.4 3.8 2.9 2.4
EWMA ?0.2 L2.86 Gam(3,1) 27.3 9.5 5.4 3.8 2.9 2.4
EWMA ?0.2 L2.86 Gam(2,1) 26.3 9.5 5.4 3.8 3 2.5
EWMA ?0.2 L2.86 Gam(1,1) 24.7 9.5 5.5 3.9 3 2.5
EWMA ?0.2 L2.86 Gam(0.5,1) 23.3 9.5 5.7 4 3.2 2.6
Shewhart Normal 155.2 44 15 6.3 3 2
Shewhart Gam(4,1) 34.2 15 7.7 4.5 3 2.2
Shewhart Gam(3,1) 31 14 7.4 4.5 3 2.2
Shewhart Gam(2,1) 27 12.6 7 4.4 3 2.2
Shewhart Gam(1,1) 21.7 11 6.4 4.2 3 2.3
Shewhart Gam(0.5,1) 18.3 9.7 6 4.1 3 2.4
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In-Control ARLs for EWMA-t
EWMA EWMA EWMA EWMA Shewhart
? 0.05 0.1 0.2 1
L 2.492 2.703 2.86 3
Normal 370.4 370.8 370.5 370.4
t50 369 365 353 283
t40 369 363 348 266
t30 368 361 341 242
t20 367 355 325 204
t15 365 349 310 176
t10 361 335 280 137
t8 358 324 259 117
t6 351 305 229 96
t4 343 274 188 76
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Out-of-control ARLs for the EWMA-t
Shift EWMA ?0.05 L2.492 EWMA ?0.1 L2.703 EWMA ?0.2 L2.86 Shewhart
0.5 t5010(26) N(26.5) t84(27) N?t5040(28.3) t304(28.430 ) N(36.2) t(3641 ) N(155.2) t506(13773 ) t4(63)
1 N(10.8) t(11) N?t(9.8) N?t5020(9.8) t154(9.910.3) N(44) t508(4339 ) t64(38)
1.5 N(6.8) t(6.7) N?t(5.8) N?t(5.2) N?t5020(15) t154(1619 )
2 N?t(5) N?t(4.2) N?t(3.6) N(6.3) t504(6.49 )
2.5 N?t(4) N?t(3.3) N?t(2.6) N(3) t504(3.34 )
3 N(3.4) t(3.3) N?t(2.8) N?t(2.3) N?t(2)
EWMA is better than Shewhart
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Comparing three EWMA control chart designs

• There have been many suggestion in the literature
  for designing an EWMA control chart.
• The table compares three EWMA control chart
  designs.
• 1st column ?0.1 and L2.7 (Montgomery, 1996)
• 2rd column ?0.1 and L3 (computer packages)
• 3th column ?0.4 and L3 (Hunter, 1989)

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Comparing three EWMA control chart designs
?0.1 ?0.1 ?0.4
L2.7 L3 L3
Normal 368 838 421
t50 362 815 368
t40 361 808 355
t30 358 798 336
t20 346 775 301
t15 346 751 271
t10 333 698 223
t8 321 655 195
t6 303 582 161
t4 272 461 124
In-Control
?0.1 ?0.1 ?0.4
L2.7 L3 L3
Normal 368 838 421
Gam(4,1) 339 648 173
Gam(3,1) 329 605 153
Gam(2,1) 313 538 128
Gam(1,1) 272 422 96
Gam(0.5,1) 228 328 76
For ?0.1 and L3, the ARLs are too large. For
?0.4 and L3, the ARLs are smaller than the
normal-theory value.
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Conclusions

• ?In control????,?0.05 and L2.492 EWMA
  ???????ARL????????ARL??????8???(????340.76)?
• ?In control????,?????????????(t6?t4?Gam1,1?Gam0.5,
  1),?0.1 and L2.703 EWMA ????????ARL????ARL????15
  ???(????315)?
• ????????????,EWMA??????????????????