EWMA Control Charts Monitoring Normal Variance When No Standard is Given

Transcription

1 EWMA Control Charts Monitoring Normal Variance When No Standard is Given Sven Knoth Joint Research Conference (QPRC & SRC), Seattle, June 25, 2014

2 Outline 1 Introduction 2 EWMA charts for (mean and) variance 3 Effects of Parameter Estimation 4 RL numerics with parameter uncertainty 5 Wrap up 6 (Mean Monitoring) S 2 EWMA & prerun uncertainty 2 / 41

3 Introduction 1 Control charts are core tools of Statistical Process Control (SPC). 2 Aim: Detect as soon as possible changes in a sequentially observed process with low number of false alarms. 3 Run Length (RL): # of observations until signal. 4 Chart performance is evaluated by looking at: Average Run Length (ARL) & more complicated expectations (worst case, conditional...). Standard deviation. Quantiles. Whole distribution, that is, CDF etc. 5 Charts are designed to meet certain ARL and, less frequently, RL quantile patterns. S 2 EWMA & prerun uncertainty 3 / 41

4 Basic assumptions and notation simplified Observations X i are independent and normally distributed. Consider the change point model: X i N (0, 1) for i < τ, and for i τ (change point τ) { N (δ, 1) shift in mean X i N (0, 2 ) scale change. Denote L the stopping time of the corresponding control chart. S 2 EWMA & prerun uncertainty 4 / 41

5 EWMA chart as one example of the SPC classics or Z 0 = z 0 = µ 0 = 0, Z i = (1 λ)z i 1 + λx i with λ (0, 1], i = 1, 2,... { } λ L fix = min i 1 : Z i > c x 2 λ { λ ( ) } L vacl = min i 1 : Z i > c x 1 (1 λ) 2 λ 2i. S. W. Roberts (1959) Control-charts-tests based on geometric moving averages. Technometrics 1, J. M. Lucas and M. S. Saccucci (1990) Exponentially weighted moving average control schemes: Properties and enhancements. Technometrics 32, S 2 EWMA & prerun uncertainty 5 / 41

6 Two examples from a Mask Shop (some type of a semiconductor company such as the AMTC) 1 CD (critical dimension) uniformity: Measure a certain number ( ) of, e. g., lines of, e. g., nominal size 250 nm on a single plate, calculate sample mean CD and standard deviation S CD, chart both. S 2 EWMA & prerun uncertainty 6 / 41

7 Two examples from a Mask Shop (some type of a semiconductor company such as the AMTC) 1 CD (critical dimension) uniformity: Measure a certain number ( ) of, e. g., lines of, e. g., nominal size 250 nm on a single plate, calculate sample mean CD and standard deviation S CD, chart both. 2 Gauge repeatability CD-SEM (scanning electron microscope): Repeat a few times (e. g., 5) the measurement of a given line, calculate standard deviation S R, chart it. S 2 EWMA & prerun uncertainty 6 / 41

8 Data example Short-term repeatability monitoring of CD SEM Metrology/MET121/SPC_HT usl 04 Mar 08:00 18 Mar 08:00 250_Space_3s_Y_ShortTerm ucl CL 18 Mar 09 01:58 21 Jan 04 Feb 18 Feb 04 Mar 18 Mar time span: 21 Jan Mar 09 creation time: 03/18/09 16:42 WinSPC timestamp S 2 EWMA & prerun uncertainty 7 / 41 (0 points in WinSPC excluded)

9 EWMA chart for variance Now: subsamples of size n (all time favorite is n = 5), X ij, j = 1,..., n, Si 2 = 1 n ( Xij n 1 X ) 2 n i, Xi = 1 X ij. n j=1 Z 0 = z 0 = σ 2 0 = 1, Z i = (1 λ)z i 1 + λs 2 i with λ (0, 1], L upper = min {i 1 : Z i > c u }, L two = min {i 1 : Z i > c u or Z i < c l }. j=1 A. W. Wortham and L. J. Ringer (1971) Control via exponential smoothing. The Logistics Review 7, J. F. MacGregor and T. J. Harris (1993) The exponentially weighted moving variance. Journal of Quality Technology 25, S 2 EWMA & prerun uncertainty 8 / 41

10 Performance measurement The base collection E τ (...) expectation of... for given change point τ, zero-state ARL { E1 (L), τ = 1 ( early change) L =. E (L), τ = (no change) S 2 EWMA & prerun uncertainty 9 / 41

11 Performance measurement Median RL (and other quantiles) zero-state RL median (MRL): L 0.5 = inf { n 1 : P(L n) 0.5 } so that P(L L 0.5 1) < 0.5, P(L L 0.5 ) 0.5. denote with L α general quantiles etc. S 2 EWMA & prerun uncertainty 10 / 41

12 EWMA chart for variance trouble Calculation of the ARL and other performance measures for EWMA in case of χ 2 distribution with the usual tool box (Markov chain approximation, Nyström) does not provide accurate numbers. For EWMA applied to S 2, problems were cured in S. Knoth (2005) Accurate ARL computation for EWMA-S 2 control charts. Statistics and Computing 15, S. Knoth (2007) Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance. Sequential Analysis 26, S 2 EWMA & prerun uncertainty 11 / 41

13 EWMA chart for variance exemplary setup Choose λ = 0.1. Set in-control ARL to E (L) = 500. c u = and c l = , c u = , respectively. Look at ARL vs. σ profiles and specific CDF shapes. S 2 EWMA & prerun uncertainty 12 / 41

14 Exemplary EWMA chart figures ARL vs. σ RL CDF for σ {0.8, 1, 1.2} ARL chart type upper two P (L l) chart type/stdev upper two σ = 1 σ = 1.2 σ = σ l S 2 EWMA & prerun uncertainty 13 / 41

15 Now, the in-control parameters are unknown. Thus, instead of utilizing N (µ 0, σ0 2 ) or N (0, 1) one has to rely on pre-run (phase I) estimates like x and s 2. L. A. Jones, C. W. Champ, and S. E. Rigdon (2001) The performance of exponentially weighted moving average charts with estimated parameters. Technometrics 43, L. A. Jones (2002) The Statistical Design of EWMA Control Charts with Estimated Parameters. Journal of Quality Technology 34, W. A. Jensen, L. A. Jones-Farmer, C. W. Champ, and W. H. Woodall (2006) Effects of Parameter Estimation on Control Chart Properties: A Literature Review. Journal of Quality Technology 38, S 2 EWMA & prerun uncertainty 14 / 41

16 Effects of Parameter Estimation on EWMA... could be evaluated by considering certain (typical and extreme ones) conditional and marginal performance patterns (ARL, CDF, RL quantiles etc.). S 2 EWMA & prerun uncertainty 15 / 41

17 Framework for variance monitoring We collect in phase I m samples X ij each of size n and make use of the estimator ˆσ 0 2 = 1 m si 2. m i=1 S 2 EWMA & prerun uncertainty 16 / 41

18 Exemplary EWMA chart phase I with samples. Marginal performance (average over ˆσ 0 2 ), n = 5, in-control. upper two-sided P (L l) # phase I samples P (L l) # phase I samples l l S 2 EWMA & prerun uncertainty 17 / 41

19 Monte Carlo study P (L 348), m {10, 20,..., 200, } for upper chart Monte Carlo Inf Monte Carlo minus numerical 1e 04 5e 05 0e+00 5e 05 1e 04 Inf numerical numerical 10 8 replicates S 2 EWMA & prerun uncertainty 18 / 41

20 Monte Carlo study P (L 349), m {10, 20,..., 200, } for two-sided chart Monte Carlo Inf Monte Carlo minus numerical 1e 04 5e 05 0e+00 5e 05 1e 04 Inf numerical numerical 10 8 replicates S 2 EWMA & prerun uncertainty 19 / 41

21 Exemplary EWMA chart phase I with samples. Marginal performance (average over ˆσ 2 0), n = 5, in-control upper only. P (L l) # phase I samples l S 2 EWMA & prerun uncertainty 20 / 41

22 Exemplary EWMA chart phase I with samples. Marginal performance (average over ˆσ 2 0), n = 5, in-control upper only. P (L l) # phase I samples l in 10 5 multiples S 2 EWMA & prerun uncertainty 21 / 41

23 Monte Carlo study P (L 10 5 ), m {10, 20,..., 200, } for upper chart Monte Carlo In Monte Carlo minus numerical 2e 04 1e 04 0e+00 1e 04 2e In numerical numerical 10 7 replicates S 2 EWMA & prerun uncertainty 22 / 41

24 Usage with phase I estimates Changes in startup and threshold Z 0 = z 0 = 1, Z i = (1 λ)z i 1 + λs 2 i, L upper = min {i N : Z i > c u }. Z 0 = z 0 = ˆσ 2 0, Z i = (1 λ) Z i 1 + λs 2 i, L upper = min {i N : Z } i > ˆσ 0c 2 u. S 2 EWMA & prerun uncertainty 23 / 41

25 ARL calculation Let L(λ, c u, z 0, ) be the ARL for the classical EWMA chart. Then L(λ, s 2 c u, s 2 z 0, ) provides the EWMA ARL conditioned on phase I estimate ˆσ 2 0 = s2 for scale change. = for marginal ARL: L total (λ, c u, z 0, ) = 0 fˆσ 2 0 (s 2 )L(λ, s 2 c u, s 2 z 0, ) ds 2. The integral is truncated and solved with numerical quadrature. L(λ, s 2 c u, s 2 z 0, ) from S. Knoth (2005) Accurate ARL computation for EWMA-S 2 control charts. Statistics and Computing 15, S 2 EWMA & prerun uncertainty 24 / 41

26 Convolution for CDF and quantiles Using p n (z) = P(L > n Z 0 = z): p n,total (z; λ, c u, ) = 0 fˆσ 2 0 (s 2 )p n (s 2 z; λ, s 2 c u, ) ds 2 To increase computing speed the geometric tail behavior of p n (s 2 i z; λ, s2 i c u, ) at each quadrature node s 2 i is individually (for large n) exploited. Unfortunately, it is lacking for p n,total (z; λ, c u, ). p n(s 2 z; λ, s 2 c u, ) from S. Knoth (2007) Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance. Sequential Analysis 26, S 2 EWMA & prerun uncertainty 25 / 41

27 Should we use literature on ensuring a certain conditional ARL performance with a given probability? Albers & Kallenberg (2004) Are estimated control charts in control? Statistics: A Journal of Theoretical and Applied Statistics 38, Capizzi & Masarotto (2010) Combined Shewhart-EWMA control charts with estimated parameters. Journal of Statistical Computation and Simulation 80, Gandy & Kvaløy (2013) Guaranteed conditional performance of control charts via bootstrap methods. Scandinavian Journal of Statistics 40, S 2 EWMA & prerun uncertainty 26 / 41

28 Or should we make use of an already existing paper about variance EWMA? Maravelakis & Castagliola (2009) An EWMA chart for monitoring the process standard deviation when parameters are estimated. Computational Statistics & Data Analysis 53, S 2 EWMA & prerun uncertainty 27 / 41

29 Or should we make use of an already existing paper about variance EWMA? Maravelakis & Castagliola (2009) An EWMA chart for monitoring the process standard deviation when parameters are estimated. Computational Statistics & Data Analysis 53, ln S 2 EWMA with reflecting barrier ln(σ0). 2 Calibrated to marginal E (L) = Table 2, n = 5, = 1.2, λ = 0.01 m K UCL E 1(L) ? S 2 EWMA & prerun uncertainty 27 / 41

30 We propose to readjust the EWMA thresholds to ensure the originally intended in-control behavior in terms of a properly chosen RL quantile. Example (n = 5): Given λ = 0.1 and known in-control value σ0 2: Choose c u = and (c l = , c u = ) so that P (L 1000) = S 2 EWMA & prerun uncertainty 28 / 41

31 Readjustment figures modified (c l, c u) corrected critical value upper two known estimated phase I samples (m) S 2 EWMA & prerun uncertainty 29 / 41

32 Readjustment figures initial QRL 0 restoring (c l, c u) and impact to in-control cdf upper two-sided P (L l) # phase I samples P (L l) # phase I samples l l (marginal) P (L 1000) = 0.25 S 2 EWMA & prerun uncertainty 30 / 41

33 Readjustment figures initial QRL 0 restoring (c l, c u) and impact to ooc behavior upper (σ = 1.2) two-sided (σ {0.8, 1.2}) P (L l) # phase I samples P (L l) # phase I samples σ = 1.2 σ = l l (marginal) P (L 1000) = 0.25 S 2 EWMA & prerun uncertainty 31 / 41

34 Readjustment figures initial QRL 0 restoring c u and impact to ooc behavior II margin ooc ARL for σ = phase I samples (m) (marginal) P (L 1000) = 0.25 S 2 EWMA & prerun uncertainty 32 / 41

35 Readjustment figures initial QRL 0 restoring (c l, c u) and impact to ooc behavior III margin ooc ARL for σ = 1.2 and σ = σ = 1.2 σ = 0.8 known estimated phase I samples (m) (marginal) P (L 1000) = 0.25 S 2 EWMA & prerun uncertainty 33 / 41

36 Wrap up Collection of routines presented to calculate various marginal RL characteristics (for averaging over phase I estimates). Impact of parameter uncertainty illustrated. Propose to use QRL instead of ARL calibration if parameter uncertainty is present. Recommend to use phase I size of 50 and more (30 seems to be a more pragmatic compromise). Algorithms are implemented in the package spc version It will be extended to simultaneous monitoring of mean and variance. S 2 EWMA & prerun uncertainty 34 / 41

37 Backup mean monitoring S 2 EWMA & prerun uncertainty 35 / 41

38 Exemplary EWMA chart phase I with 50 and 100 obs. Marginal performance (average over ˆµ 0 and ˆσ 2 0 ). in-control out of control (δ = 1) P(L l) limits/phase I size fix vacl n = n = 50 n = 100 P(L l) limits/phase I size fix vacl n = n = 50 n = l l S 2 EWMA & prerun uncertainty 36 / 41

39 Readjustment figures modified c x corrected critical value known parameters/mode both sigma mu none QRL ARL phase I size S 2 EWMA & prerun uncertainty 37 / 41

40 Readjustment figures initial ARL 0 restoring c x and impact to cdf P(L l) n= l S 2 EWMA & prerun uncertainty 38 / 41

41 Readjustment figures initial QRL 0 restoring c x and impact to cdf P(L l) n= l S 2 EWMA & prerun uncertainty 39 / 41

42 Readjustment figures initial QRL 0 restoring c x and impact to out-of-control behavior P(L l) n= l S 2 EWMA & prerun uncertainty 40 / 41

43 Readjustment figures initial QRL 0 restoring c x and impact to out-of-control behavior II margin ooc ARL for δ = phase I size S 2 EWMA & prerun uncertainty 41 / 41