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- 1. Spanos EE290H F05 CUSUM, MA and EWMA Control Charts Increasing the sensitivity and getting ready for automated control: The Cumulative Sum chart, the Moving Average and the Exponentially Weighted Moving Average Charts. Lecture 14: CUSUM and EWMA 1
- 2. Spanos EE290H F05 Shewhart Charts cannot detect small shifts The charts discussed so far are variations of the Shewhart chart: each new point depends only on one subgroup. Shewhart charts are sensitive to large process shifts. The probability of detecting small shifts fast is rather small: Fig 6-13 pp 195 Montgomery. Lecture 14: CUSUM and EWMA 2
- 3. Spanos EE290H F05 Cumulative-Sum Chart If each point on the chart is the cumulative history (integral) of the process, systematic shifts are easily detected. Large, abrupt shifts are not detected as fast as in a Shewhart chart. CUSUM charts are built on the principle of Maximum Likelihood Estimation (MLE). Lecture 14: CUSUM and EWMA 3
- 4. Spanos EE290H F05 Maximum Likelihood Estimation The "correct" choice of probability density function (pdf) moments maximizes the collective likelihood of the observations. If x is distributed with a pdf(x,θ) with unknown θ, then θ can be estimated by solving the problem: m max θ Π pdf( xi,θ ) i=1 This concept is good for estimation as well as for comparison. Lecture 14: CUSUM and EWMA 4
- 5. Spanos EE290H F05 Maximum Likelihood Estimation Example To estimate the mean value of a normal distribution, collect the observations x1,x2, ... ,xm and solve the non-linear programming problem: ⎧ ⎫ ⎧ 2 ⎫ 2 ˆ ˆ 1 ⎛ x −μ ⎞ 1 ⎛ x −μ ⎞ ⎞ ⎛ 1 ⎪m ⎪ m − ⎜ i − ⎜ i ⎟ ⎪ ⎟ ⎪ 1 ⎪ 2⎝ σ ⎠ 2 ⎝ σ ⎠ ⎟⎪ ⎜ max μ ⎨Π e e ⎬ or min μ ⎨iΣ1 log⎜ ˆ ˆ ⎟⎬ i =1 σ 2π = σ 2π ⎪ ⎪ ⎪ ⎝ ⎠⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ Lecture 14: CUSUM and EWMA 5
- 6. Spanos EE290H F05 MLE Control Schemes If a process can have a "good" or a "bad" state (with the control variable distributed with a pdf fG or fB respectively). This statistic will be small when the process is "good" and large when "bad": m f B(x i) Σ log f G(xi) i=1 m m log(Π pi ) = ∑ log( pi ) i =1 Lecture 14: CUSUM and EWMA i =1 6
- 7. Spanos EE290H F05 MLE Control Schemes (cont.) Note that this counts from the beginning of the process. We choose the best k points as "calibration" and we get: m k f B(x i) f B(x i) S m = Σ log - min k < m Σ log >L f G(x i) f G(x i) i=1 i=1 or f B(x m) S m = max (S m-1+log , 0) > L f G(x m) This way, the statistic Sm keeps a cumulative score of all the "bad" points. Notice that we need to know what the "bad" process is! Lecture 14: CUSUM and EWMA 7
- 8. Spanos EE290H F05 The Cumulative Sum chart If θ is a mean value of a normal distribution, is simplified to: m S m = Σ (x i - µ 0) i=1 where μ0 is the target mean of the process. This can be monitored with V-shaped or tabular “limits”. Advantages The CUSUM chart is very effective for small shifts and when the subgroup size n=1. Disadvantages The CUSUM is relatively slow to respond to large shifts. Also, special patterns are hard to see and analyze. Lecture 14: CUSUM and EWMA 8
- 9. Spanos EE290H F05 Shewhart small shift Example 15 10 5 0 -5 -10 -15 UCL=13.4 µ0=-0.1 20 40 60 80 100 120 140 0 CUSUM small shift 0 LCL=-13.6 160 20 40 60 80 100 120 140 160 120 100 80 60 40 20 0 -20 -40 -60 Lecture 14: CUSUM and EWMA 9
- 10. Spanos EE290H F05 The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ Need to set L(0) (i.e. the run length when the process is in control), and L(δ) (i.e. the run-length for a specific deviation). Figure 7-3 Montgomery pp 227 Lecture 14: CUSUM and EWMA 10
- 11. Spanos EE290H F05 The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ δ is the amount of shift (normalized to σ) that we wish to detect with type I error α and type II error β. Α is a scaling factor: it is the horizontal distance between successive points in terms of unit distance on the vertical axis. 2 ln 1- β d= 2 α δ δ= Δ θ = tan -1 δ σx 2A d Lecture 14: CUSUM and EWMA θ 11
- 12. Spanos EE290H F05 ARL vs. Deviation for V-Mask CUSUM Lecture 14: CUSUM and EWMA 12
- 13. Spanos EE290H F05 CUSUM chart of furnace Temperature difference Detect 2Co, σ =1.5Co, (i.e. δ=1.33), α=.0027 β=0.05, A=1 => θ = 18.43o, d = 6.6 50 3 40 2 30 II 20 III 0 I 10 1 -1 0 -2 -10 -3 0 20 Lecture 14: CUSUM and EWMA 40 60 80 100 0 20 40 60 80 100 13
- 14. Spanos EE290H F05 Tabular CUSUM A tabular form is easier to implement in a CAM system Ci+ = max [ 0, xi - ( μo + k ) + Ci-1+ ] Ci- = max [ 0, ( μo - k ) - xi + h Ci-1- ] d θ C0+ = C0- = 0 k = (δ/2)/σ h = dσxtan(θ) Lecture 14: CUSUM and EWMA 14
- 15. Spanos EE290H F05 Tabular CUSUM Example Lecture 14: CUSUM and EWMA 15
- 16. Spanos EE290H F05 Various Tabular CUSUM Representations Lecture 14: CUSUM and EWMA 16
- 17. Spanos EE290H F05 CUSUM Enhancements To speed up CUSUM response one can use "modified" V masks: 15 10 5 0 -5 0 20 40 60 80 100 Other solutions include the application of Fast Initial Response (FIR) CUSUM, or the use of combined CUSUMShewhart charts. Lecture 14: CUSUM and EWMA 17
- 18. Spanos EE290H F05 General MLE Control Schemes Since the MLE principle is so general, control schemes can be built to detect: • single or multivariate deviation in means • deviation in variances • deviation in covariances An important point to remember is that MLE schemes need, implicitly or explicitly, a definition of the "bad" process. The calculation of the ARL is complex but possible. Lecture 14: CUSUM and EWMA 18
- 19. Spanos EE290H F05 Control Charts Based on Weighted Averages Small shifts can be detected more easily when multiple samples are combined. Consider the average over a "moving window" that contains w subgroups of size n: Mt = x t + x t-1 + ... +x t-w+1 w σ2 V(M t) = n w The 3-sigma control limits for Mt are: UCL = x + 3 σ nw LCL = x - 3 σ nw Limits are wider during start-up and stabilize after the first w groups have been collected. Lecture 14: CUSUM and EWMA 19
- 20. Spanos EE290H F05 Example - Moving average chart w = 10 15 10 5 0 -5 -10 0 20 40 60 80 100 120 140 160 sample Lecture 14: CUSUM and EWMA 20
- 21. Spanos EE290H F05 The Exponentially Weighted Moving Average If the CUSUM chart is the sum of the entire process history, maybe a weighed sum of the recent history would be more meaningful: zt = λxt + (1 - λ)zt -1 0 < λ < 1 z0 = x It can be shown that the weights decrease geometrically and that they sum up to unity. t-1 z t = λ Σ ( 1 - λ )j x t - j + ( 1 - λ )t z 0 j=0 UCL = x + 3 σ LCL = x - 3 σ Lecture 14: CUSUM and EWMA λ (2-λ)n λ (2-λ)n 21
- 22. Spanos EE290H F05 Two example Weighting Envelopes relative importance 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 10 EWMA 0.6 EWMA 0.1 Lecture 14: CUSUM and EWMA 20 30 40 50 -> age of sample 22
- 23. Spanos EE290H F05 EWMA Comparisons 15 λ=0.6 10 5 0 -5 -10 0 20 40 60 0 20 40 60 80 100 120 140 160 80 100 sample 120 140 160 15 10 λ=0.1 5 0 -5 -10 Lecture 14: CUSUM and EWMA 23
- 24. Spanos EE290H F05 Another View of the EWMA • The EWMA value zt is a forecast of the sample at the t+1 period. • Because of this, EWMA belongs to a general category of filters that are known as “time series” filters. xt = f (xt - 1, xt - 2, xt - 3 ,... ) xt- xt = at Usually: p xt = Σ φixt - i + i=1 q Σ θjat - j j=1 • The proper formulation of these filters can be used for forecasting and feedback / feed-forward control! • Also, for quality control purposes, these filters can be used to translate a non-IIND signal to an IIND residual... Lecture 14: CUSUM and EWMA 24
- 25. Spanos EE290H F05 Summary so far… While simple control charts are great tools for visualizing the process,it is possible to look at them from another perspective: Control charts are useful “summaries” of the process statistics. Charts can be designed to increase sensitivity without sacrificing type I error. It is this type of advanced charts that can form the foundation of the automation control of the (near) future. Next stop before we get there: multivariate and modelbased SPC! Lecture 14: CUSUM and EWMA 25

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