Differential Smoothing in BEWMA Chart.pptx

BEWMA literature review

1. Differential Smoothing in BEWMA Chart Author: Ick Huh (2014)

2. Background and Introduction Bivariate EWMA (BEWMA) Chart with Differential Smoothing Performance Comparison BEWMA and Its Variants Standard BEWMA Chart Proposed BEWMA Chart Double BEWMA (dBEWMA) Chart Multiple Univariate Charts for Independent Quality Indicators Numerical Results Sensitivity Analysis Discussion Outline ‹#›

3. Background and Introduction ‹#›

4. Background ## Analogue of Univariate EWMA to Multivariate EWMA univariate multivariate ‹#›

5. Mathematical form of MEWMA where W = 0, R = diag(r 1 , r 2 , … , r p ), 0 < r j <= 1, j = 1, 2, … , p. And I is the p x p identity matrix. ‹#›

6. Situations where varying of smoothing may be justified Situations where varying of smoothing may be justified departures in process mean may be different across quality variables some variables may evolve over time at a much different pace than other variables the level of correlation between variables could vary substantially In addition, i f 2 variables are highly correlated, the same degree of smoothing might be suffice. It has also been suggested that when variables are independent, separate univariate charts might be justified. If there is no a priori reason to weight past observations differently, then r_1 = r_2 = … = r_n ‹#›

7. Objective of the study T o investigate whether gains in average run length of BEWMA are achieved by using varying degrees of smoothing on the quality variables, especially for bivariate case. ‹#›

8. BEWMA Chart with Differential Smoothing ‹#›

9. Formal definition of BEWMA Chart Suppose 2 variables (X 1 , X 2 ) and X t = (X 1t , X 2t )’ represent the observations at time t and X t has bivariate normal distribution. When the process operates on-target, Assume that mean shifts to , covariance matrix remains the same. By taking , referring to equation of (2), then: ‹#›

10. Out-of-control signal T he chart signals when where W = (0, 0)’ and H(> 0) is chosen to achieve a specified in-control ARL and ∑ w is covariance matrix of W t . ‹#›

11. Out-of-control signal(2) ‹#›

12. ARL computation using Markov Chain Consider: A tilted ellipse in W 1t and W 2t centered at (0,0) is formed based on equation above. The ellipse’s interior forms the in-control chart region. Further, we need to determine: The rectangle [-UCL1,UCL1] x [-UCL2,UCL2] that circumscribes the ellipse Divide [-UCL1,UCL1] to 2*m1+1 intervals. Do similar things to [-UCL2,UCL2] The transient states of the proposed bivariate Markov chain consist of all subrectangles whose centers fall inside the ellipse (i.e. inside the control region) ‹#›

13. Transition Probability ‹#›

14. Transition Probability (2) ‹#›

15. R Code demo ‹#›

16. Performance Comparison ‹#›

17. Assumptions where ARL = 200 is chosen for all competing charts. Further, the ARL opt (i.e. the ARL 1 value from the optimal chart – the one who gives the smallest ARL 1 value) is searched by varying the smoothing parameters. ‹#›

18. Illustration of ‹#›

19. Standard and Proposed BEWMA chart The standard BEWMA chart use a single smoothing r. Thus, the ARL opt is obtained from varying the smoothing parameter r for a given non-centrality parameter and correlation coefficient . The differential smoothing BEWMA chart has smoothing parameter r 1 and r 2 . Given a correlation coefficient and mean shift , the smoothing parameters r 1 and r 2 are varied to obtain the ARL opt . ‹#›

20. Double BEWMA chart Similar to standard MEWMA chart, the ARL opt is obtained from varying the smoothing parameter r for a given non-centrality parameter and correlation coefficient . ‹#›

21. Multiple univariate charts for independent quality indicators When the quality indicators are independent, the benefit of a single multivariate chart might be diminished. It has been suggested that using individual univariate chart is more efficient which also enables us to instantly diagnose which quality indicators trigger the out-of-control process. Given mean shift , the smoothing parameters r 1 and r 2 are varied to obtain the ARL opt . ‹#›

22. Correlation = 52 points from elips figure 1 (a) Mean shifts Numerical results (1) Differentially smoothed BEWMA with constant 𝛿 and ρ ‹#›

23. Numerical results ( 1 ) Differentially smoothed BEWMA with constant 𝛿 and ρ Findings: Symmetry behaviour on ARL opt ARL opt range is 24.68 to 26.69 Reduction efficiency in ARL opt (note: ARLstd = 26.68) = 7.5% ARLopt exceed standard when mean is same (point 1) Differentially smoothed BEWMA perform better in many directions max min ‹#›

24. Findings: Small mean shift ( 𝛿 0.25 - 0.50), ARL opt dBEWMA > standard BEWMA Differentially BEWMA as good as standard BEWMA Standard BEWMA is superior in most direction for small mean shift. Numerical results (2) Differentially smoothed BEWMA with different 𝛿 and ρ ‹#›

25. Numerical results (2) Differentially smoothed BEWMA with different 𝛿 and ρ Findings: Large mean shift ( 𝛿 1.00 - 2.00), ARL opt dBEWMA < standard BEWMA Differentially BEWMA only in a few directions better than the standard BEWMA dBEWMA is superior in most direction for large mean shift. ‹#›

26. The salient point is that r 1 and r 2 differ the most when the mean directions are near the reference point P1, P2, P3, and P4. On the other hand, r 1 ≈ r 2 for mean directions where 𝛿 1 ≈ 𝛿 2 The performance of the proposed chart is similar to the standard BEWMA chart. Numerical results (3) Differentially smoothed BEWMA with different ρ by r opt = (r 1 ,r 2 )’ ‹#›

27. The combined chart performs better in some directions and much worse in others. When 𝛿 is large, the standard BEWMA is superior in most direction. The differentially smoothed BEWMA shows superior performance in all directions. Numerical results (4) The Combined Univariate EWMA with different 𝛿 ‹#›

28. Sensitivity Analysis ‹#›

29. Differentially Smoothed BEWMA Chart Issues There is an interesting issue on the topic of robustness in the differentially smoothed BEWMA chart on the misspecification of the correlation coefficient ( ρ). The Chart calibration is done using the value of (ρ) for the correlation coefficient, but the true correlation sometimes is ρ+ε or ρ-ε ρ is treated as a process parameter requiring estimation from phase I process data. ‹#›

30. Table 2 present the result of a sensitivity study using 6 correlation ( ρ) misspecification cases: For example, the true correlation may be 0.20+0.05=0.25, but the chart is calibrated using 0.20. Markov chain is used to compute the ARL based on the misspecified and true correlation coefficients using the control limit from the misspecified correlation. ‹#›

31. To assess the effect of the misspecification we used the absolute relative deviation percentage (ADRP). ARL(M) and ARL(T) are the ARL from the misspecified and true correlations. For in-control situation, (table 2) the effect has little impact on ARL performance. But in (table 3) the difference in ARL increases with increasing correlation for out of control scenarios. Also note that the smaller the jump in mean, the smaller the difference in ARL ‹#›

32. Discussion ‹#›

33. Differential Smoothing / Equal Smoothing? Adopting the optimal ARL as a criterion, the results reported that noticeable gains in optimal average run length are achieved using differential smoothing when there is a substantial difference in size among individual variable mean jumps. Example Suppose there are two quality variables X 1 and X 2 , with specific mean jump where (0.20 , 1.99). The optimal weight for Univariate EWMA chart for X 1 to dete ct σ 1 = 0.20 with ARL =200 is r 1 =0.02 and the optimal weight for X 2 to detect σ 2 = 1.99 is r 2 =0.43. The difference between r 1 and r 2 means that when this process is monitored with a BEWMA chart for (X 1 ,X 2 ), the optimal weights will differ. ‹#›

34. Detection Power loss When a chart has a certain mean vector departure but the mean vector departure is there is a question of detection power loss. An example case is where a differentially smoothed BEWMA chart is designed to detect a mean vector shift where a component does not change at all. Hence, the smoothing parameter for the variable with unchanged mean will be very small. Correlation ( ρ) between two variables would play an important role in chart performance when the chart is used in other directions, when there is no correlation the chart performs poorly in every direction. When correlation increases, the chart tends to perform better in many directions. ‹#›

35. Markov Chain Method The Markov chain method developed for this paper for avg ARL calculation is accurate and efficient for the p = 2 (BEWMA chart) case. The conditional argument to calculate transition probabilities extends to any number of quality variables ( p) . Limitation However, the number of transient states grows quickly with p . If p is large, simulations may be preferred for computing ARLs. ‹#›

36. Weighing, the numerical challenge. Finding the optimal value for weight, is the problem in this proposed method. The standard chart only uses one level of smoothing, thus users needed to optimize over single variable (r) only. When differential smoothing is applied, optimization is required on several variables, such as r 1 , r 2 ,...r p . It will take some time to determine the optimal values. ‹#›

37. Thank you! ‹#›

38. Appendix ‹#›

39. Chart Inertia The problem noted by Lowry et al. in 1992, in the context of the standard MEWMA chart. The problem arises when after a period of normal operation, the process goes off target at a sampling period where the last value of the chart statistic is far from the chart control limit. With small smoothing parameters, the chart can take a long time to reach the control limit, so it’s inefficient at detecting the change in process mean. Findings reveal that increasing the smoothing amounts r 1 and r 2 in small amounts without causing a substantial increase in the off-target ARL. ‹#›

40. Alternative Analytical Method In 1995, Rigdon provided an alternative analytical method for obtaining the ARL of the standard MEWMA control chart by solving single and double integral equations. Numerical methods are based on equal smoothing parameters and they are directionally invariant but the differential smoothing scheme is not. The writers are interested to investigate the adaptation required to make the methods work under the differentially smoothed scheme. ‹#›