Control charting normal variance reflections, curiosities, and recommendations

Transcription

1 Control charting normal variance reflections, curiosities, and recommendations Sven Knoth September 2007

2 Outline 1 Introduction 2 Modelling 3 Two-sided EWMA charts for variance 4 Conclusions

3 Introduction Aim of control charting is to detect deviations from stability as fast as possible without too many false alarms. Parameters characterizing stability are mean level, scale (uniformity, variance, repeatability),...

4 Why variance? Ensure appropriate control limits for mean chart. Detect detoriated uniformity. Woodall & Montgomery (1999) demanded it ;-)

5 Why variance? Ensure appropriate control limits for mean chart. Detect detoriated uniformity. Woodall & Montgomery (1999) demanded it ;-)

6 Two examples from a Mask Shop 1 CD (critical dimension) uniformity: Measure a certain number ( ) of, e. g., lines of nominal size 200 nm on a single plate, calculate sample mean CD and standard deviation S CD, chart both. 2 Gauge repeatibility CD-SEM (scanning electron microscope): Repeat a few times (e. g., 5) the measurement of one given line, calculate standard deviation S R, chart it.

7 Two examples from a Mask Shop 1 CD (critical dimension) uniformity: Measure a certain number ( ) of, e. g., lines of nominal size 200 nm on a single plate, calculate sample mean CD and standard deviation S CD, chart both. 2 Gauge repeatibility CD-SEM (scanning electron microscope): Repeat a few times (e. g., 5) the measurement of one given line, calculate standard deviation S R, chart it.

8 Some remarks about variance monitoring Variance components: Yashchin (1994), Woodall & Thomas (1995), Srivastava (1997), individual measurements, fixed or choosable batch sizes, small or large batch sizes, Focus: Small batch sizes larger 1, one variance component only.

9 Some remarks about variance monitoring Variance components: Yashchin (1994), Woodall & Thomas (1995), Srivastava (1997), individual measurements, fixed or choosable batch sizes, small or large batch sizes, Focus: Small batch sizes larger 1, one variance component only.

10 Some remarks about variance monitoring Variance components: Yashchin (1994), Woodall & Thomas (1995), Srivastava (1997), individual measurements, fixed or choosable batch sizes, small or large batch sizes, Focus: Small batch sizes larger 1, one variance component only.

11 Some remarks about variance monitoring Variance components: Yashchin (1994), Woodall & Thomas (1995), Srivastava (1997), individual measurements, fixed or choosable batch sizes, small or large batch sizes, Focus: Small batch sizes larger 1, one variance component only.

12 Modelling Sequence {X ij }, i = 1, 2,... and j = 1, 2,..., n > 1 with X ij N (µ, σ 2 ), independence. The change-point model: For a certain unknown m σ 2 = { σ0 2 = 1, i < m σ1 2 σ2 0, i m.

13 Modelling Sequence {X ij }, i = 1, 2,... and j = 1, 2,..., n > 1 with X ij N (µ, σ 2 ), independence. The change-point model: For a certain unknown m σ 2 = { σ0 2 = 1, i < m σ1 2 σ2 0, i m.

14 Pre-processing of batch data In order to monitor σ the usual suspects are R i = max X ij min X ij, j j Si 2 = 1 n ( Xij n 1 X ) 2 i, Xi = 1 n j=1 S i = Si 2, n X ij, j=1 ls 2 i = log S 2 i, abcs 2 i = a + b log(s 2 i + c).

15 Pre-processing of batch data In order to monitor σ the usual suspects are R i = max X ij min X ij, j j Si 2 = 1 n ( Xij n 1 X ) 2 i, Xi = 1 n j=1 S i = Si 2, n X ij, j=1 ls 2 i = log S 2 i, abcs 2 i = a + b log(s 2 i + c).

16 Why log? 1 Box, Hunter & Hunter (1978) recommended it. 2 It transforms scale-change into level change. 3 The variance of log S 2 does not depend on σ. 4 New statistic behaves nearly normally and 5 is, of course, more symmetric. Is this reasonable?

17 Why log? 1 Box, Hunter & Hunter (1978) recommended it. 2 It transforms scale-change into level change. 3 The variance of log S 2 does not depend on σ. 4 New statistic behaves nearly normally and 5 is, of course, more symmetric. Is this reasonable?

18 a + b log(s 2 + c) log S 2 Figure 1: Normalized density plots solid line in control model, σ= 1 dashed lines out of control models, left σ=0.75, right σ= 1.25 density density x x S 2 S R density density density x x x

19 Who is who in log S 2 -SPC Crowder & Hamilton (1992), EWMA, Chang & Gan (1994), EWMA, Chang & Gan (1995), CUSUM, Amin & Wilde (2000), Crosier-type CUSUM, Castagliola (2005), a + b log(s 2 + c) EWMA,...

20 Short list of comparison papers Tuprah & Ncube (1987), Srivastava & Chow (1992), Lowry, Champ & Woodall (1995), Mittag, Stemann & Tewes (1998), Acosta-Mejía, Pignatiello Jr. & Rao (1999), Poetrodjojo, Abdollahian & Debnath (2002),...

21 Further transformations Hawkins (1981), (X µ 0 )/σ 0 1/ , [ ( APR (1999), Φ 1 F (n 1)S 2 χ 2 n 1 APR (1999),... σ 2 0 )], [ (S ( )] 2 /σ0) 2 1/3 / (n 1) 9(n 1),

22 Further transformations Hawkins (1981), (X µ 0 )/σ 0 1/ , [ ( APR (1999), Φ 1 F (n 1)S 2 χ 2 n 1 APR (1999),... σ 2 0 )], [ (S ( )] 2 /σ0) 2 1/3 / (n 1) 9(n 1),

23 Objects of this talk are two-sided EWMA (exponentially weighted moving average) charts, in order to validate the symmetry story.

24 Objects of this talk are two-sided EWMA (exponentially weighted moving average) charts, in order to validate the symmetry story.

25 Two-sided EWMA charts for variance V i { Si 2, S i, R i, log Si 2, a + b log(si 2 + c) }, Z 0 = z 0 = E (V i ), Z i = (1 λ) Z i 1 + λ V i, i 1, L = min { i N : Z i / [c l, c u ] }. Var(Z i ) = Z i = (1 λ) z 0 + λ i (1 λ) i j V j, j=1 λ ( 1 (1 λ) 2i ) Var(V i ). 2 λ

26 Two-sided EWMA charts for variance V i { Si 2, S i, R i, log Si 2, a + b log(si 2 + c) }, Z 0 = z 0 = E (V i ), Z i = (1 λ) Z i 1 + λ V i, i 1, L = min { i N : Z i / [c l, c u ] }. Var(Z i ) = Z i = (1 λ) z 0 + λ i (1 λ) i j V j, j=1 λ ( 1 (1 λ) 2i ) Var(V i ). 2 λ

27 Two-sided EWMA charts for variance V i { Si 2, S i, R i, log Si 2, a + b log(si 2 + c) }, Z 0 = z 0 = E (V i ), Z i = (1 λ) Z i 1 + λ V i, i 1, L = min { i N : Z i / [c l, c u ] }. Var(Z i ) = Z i = (1 λ) z 0 + λ i (1 λ) i j V j, j=1 λ ( 1 (1 λ) 2i ) Var(V i ). 2 λ

28 1 Calibrate all schemes to give E (L) = 500. Comparison study 2 Deploy ARL-unbiased designs (see APR (1999)). 3 Look for optimal λ, that is, minimize L L 1.25 and L L 1.5, respectively, among λ {0.02, 0.03,..., 0.99, 1.00}. 4 Optimal values for λ are: case statistic R S 2 S ls 2 abcs 2 L L L L

29 1 Calibrate all schemes to give E (L) = 500. Comparison study 2 Deploy ARL-unbiased designs (see APR (1999)). 3 Look for optimal λ, that is, minimize L L 1.25 and L L 1.5, respectively, among λ {0.02, 0.03,..., 0.99, 1.00}. 4 Optimal values for λ are: case statistic R S 2 S ls 2 abcs 2 L L L L

30 1 Calibrate all schemes to give E (L) = 500. Comparison study 2 Deploy ARL-unbiased designs (see APR (1999)). 3 Look for optimal λ, that is, minimize L L 1.25 and L L 1.5, respectively, among λ {0.02, 0.03,..., 0.99, 1.00}. 4 Optimal values for λ are: case statistic R S 2 S ls 2 abcs 2 L L L L

31 1 Calibrate all schemes to give E (L) = 500. Comparison study 2 Deploy ARL-unbiased designs (see APR (1999)). 3 Look for optimal λ, that is, minimize L L 1.25 and L L 1.5, respectively, among λ {0.02, 0.03,..., 0.99, 1.00}. 4 Optimal values for λ are: case statistic R S 2 S ls 2 abcs 2 L L L L

32 Illustration for S 2 EWMA ARL λ σ

33 Competition for minimal L L 1.25 ARL log S 2 a + b log(s 2 + c) S 2 log S 2 a + b log(s 2 + c) S 2 S R σ

34 Competition for minimal L L 1.25 II ARL difference log S 2 a + b log(s 2 + c) S 2 S R log S 2 a + b log(s 2 + c) S σ

35 Competition for minimal L L 1.25 III σ ls 2 abcs 2 statistic S 2 S R

36 Competition for minimal L L 1.5 ARL log S 2 a + b log(s 2 + c) S 2 S S 2 log S 2 a + b log(s 2 + c) S 2 S R σ

37 Competition for minimal L L 1.5 II ARL difference log S 2 a + b log(s 2 + c) S 2 S R log S 2 a + b log(s 2 + c) S 2 S S σ

38 Competition for minimal L L 1.5 III σ ls 2 abcs 2 statistic S 2 S R

39 Conclusions 1 None of the statistics provides ARL performance that is symmetric in σ. 2 The log S 2 seems to be the worst approach, even beaten by R. 3 The newer a + b log(s 2 + c) is considerably better than log S 2. But, these efforts do not really pay off. 4 There is no reason to deploy log based approaches at all. This is supported also by one-sided results (both EWMA and CUSUM). 5 For application, one should prefer S 2 and S. The latter is the most popular quantity at AMTC.

40 Conclusions 1 None of the statistics provides ARL performance that is symmetric in σ. 2 The log S 2 seems to be the worst approach, even beaten by R. 3 The newer a + b log(s 2 + c) is considerably better than log S 2. But, these efforts do not really pay off. 4 There is no reason to deploy log based approaches at all. This is supported also by one-sided results (both EWMA and CUSUM). 5 For application, one should prefer S 2 and S. The latter is the most popular quantity at AMTC.

41 Conclusions 1 None of the statistics provides ARL performance that is symmetric in σ. 2 The log S 2 seems to be the worst approach, even beaten by R. 3 The newer a + b log(s 2 + c) is considerably better than log S 2. But, these efforts do not really pay off. 4 There is no reason to deploy log based approaches at all. This is supported also by one-sided results (both EWMA and CUSUM). 5 For application, one should prefer S 2 and S. The latter is the most popular quantity at AMTC.

42 Conclusions 1 None of the statistics provides ARL performance that is symmetric in σ. 2 The log S 2 seems to be the worst approach, even beaten by R. 3 The newer a + b log(s 2 + c) is considerably better than log S 2. But, these efforts do not really pay off. 4 There is no reason to deploy log based approaches at all. This is supported also by one-sided results (both EWMA and CUSUM). 5 For application, one should prefer S 2 and S. The latter is the most popular quantity at AMTC.

43 Conclusions 1 None of the statistics provides ARL performance that is symmetric in σ. 2 The log S 2 seems to be the worst approach, even beaten by R. 3 The newer a + b log(s 2 + c) is considerably better than log S 2. But, these efforts do not really pay off. 4 There is no reason to deploy log based approaches at all. This is supported also by one-sided results (both EWMA and CUSUM). 5 For application, one should prefer S 2 and S. The latter is the most popular quantity at AMTC.

44 Backup

45 Some upper variance charts Slightly modified and shortened update of Table 5 in Chang & Gan (1995) the EWMA schemes are also one-sided and equipped with a lower reflecting barrier, see Knoth (2005) for more details. CUSUM-S 2 CUSUM-ln S 2 EWMA-S 2 EWMA-ln S 2 k h = kh ln = 0.28 σ h h = hh ln = c = c ln =

46 Numerical handling of the sample range R Bland, Gilber, Kapadia & Owen (1966): P(R/σ r) = n φ(x) ( Φ(x + r) Φ(x) ) n 1 dx.

47 ARL integral equations and it s solution cu L(z) = 1 + L(x) 1 c l λ f ( ) x (1 λ) z dx, z [c l, c u ]. λ 1 log S 2 : Gauss-Legendre Nyström, 2 others: collocation with piece-wise Chebyshev polynomials, 3 validated with Monte Carlo with 10 8 replicates.

48 The λ hunt for minimal out-of-control ARL E 1 (L) Mittag et al. (1998) true local min e λ λ = , c = , Ê (L) = ± 0.091, Ê 1(L) = ± , 10 9 rep. P (L = 1) 0.4!