ADVANCED INTERVENTION ANALYSIS of Tool Data for Improved Process Control

1. ADVANCED INTERVENTION ANALYSIS of Tool Data for Improved Process Control Presenter: Rob Firmin, Ph.D. Managing Director Foliage Software Systems 408 321 8444 rfirmin@foliage.com Coauthor: David P. Reilly Founder Automatic Forecasting Systems 215 675 0652 dave@autobox.com September 11, 2002

2. PRESENTATION PURPOSE Introduce Techniques That Can Improve Fab Process Control Significantly: • Reduce Variation • Improve Yield • Increase Other Efficiencies.

3. OUTLINE Statistical Validity Temporal Structure & True Time Series Analysis Special Cause Variation Intervention Analysis Intervention Example From Semi Conclusions

4. APC Effect on Process Control • APC Infrastructure Will Have Profound Effects. • More Data, Compatible Formats. • Equally Important: APC Benefits Open Door to More Advanced Statistical Methods Advanced Methods Address Problems With Enhanced Validity.

5. STATISTICAL VALIDITY 1 • Statistical Analysis Requires iidn to Be Valid. • Iidn: Independent, Identically Distributed and Normal Observations. P(A|B) = P(A) and P(B|A) = P(B) (Applies to Each Value and to Each Combination of Values.)

6. STATISTICAL VALIDITY 2 • Statistical Analysis Requires iidn to Be Valid. • Iidn: Independent, Identically Distributed and Normal Observations. P(A|B) = P(A) and P(B|A) = P(B) (Applies to Each Value and to Each Combination of Values.) • Conventional Techniques Applied to Most Time Series Data Are Not Valid.

7. STATISTICAL VALIDITY 3 • Most Manufacturing Data Are Serially Dependent, • Not Drawn Independently:

8. STATISTICAL VALIDITY 4 What If a Lottery Operated With Auto-Dependent (Magnetized) Data? 16 13 9 15 8 4 7 1

9. STATISTICAL VALIDITY 4 16 13 9 15 8 4 7 1

10. STATISTICAL VALIDITY 4 16 13 15 9 8 4 7 1

11. STATISTICAL VALIDITY 4 16 13 4 15 9 7 1 8

12. STATISTICAL VALIDITY 4 4 16 13 15 9 1 8

13. STATISTICAL VALIDITY 4 Numbers Would Be Drawn In Patterns, (Even With Tumbling). 4 16 13 15 8 9 1

14. STATISTICAL VALIDITY 5 • Many Confirming Studies: • 80+ Percent of Industrial Processes Have Temporal Structure. See: Alwan, L. C., H. V. Roberts (1995)

15. STATISTICAL VALIDITY 6 • Consequences of Non-iidn: • Probability Statements Are Invalid: • Mean May ≠ Expected Value, • Hypothesis Tests May Be Invalid. • Models Are Incorrect: • Failures of Necessity and Sufficiency. • Forecasting Is Invalid.

16. STATISTICAL VALIDITY 7 Consequences of Non-iidn: • Conventional Control Charts Lead to Erroneous Conclusions & Under- & Over- Control. • E.G., x and R control charts: Operator Shift Changes  Higher Within Group Variance Positive Autocorrelation  Lower Within Group Variance.

17. STATISTICAL VALIDITY 8 • Dependence Cannot Be Swept Away: • Cannot Fix With Random Sorts • Cannot Avoid byReducing Sampling Rate • Lose Validity With Preconceived Models.

18. THE OPPORTUNITY • Valid Time Series Models Separate the Process from its Noise. • 1 - R2 of a Valid Model = Natural Variation • R2 = Potential Control Improvement • = ∑ (yi – y)2/ ∑ (yi – y)2 • = Model Variation/Total Variation

19. TEMPORAL STRUCTURE • Temporal Structure: Form of Any Specific Time Series Dependence. • Temporal Structure Estimated as: • Autoregressive (AR) • Moving Average (MA) • Integrated (Differenced) AR & MA = ARIMA • Interventions Are Extensions.

20. TRUE TIME SERIES ANALYSIS 1 • Many Time Series Methods; • Only True Time Series Analysis Satisfies iidn.

21. TRUE TIME SERIES ANALYSIS 2 • Many Time Series Methods; • Only True Time Series Analysis Satisfies iidn. • Proper Identification, Estimation and Diagnostics • Result in iidn Residuals.

22. TRUE TIME SERIES ANALYSIS 3 • Manual Step 1: • Identify Appropriate Subset of Models • Render Series Stationary, Homogeneous & Normal. • e.g.: Ñ1lnYt = lnYt – lnYt-1 Ñ1: first difference

23. TRUE TIME SERIES ANALYSIS 4 • Manual Step 1: • Identify Appropriate Subset of Models • Render Series Stationary, Homogeneous & Normal. • Ñ1lnYt = lnYt – lnYt-1 • Manual Step 2: • Estimate Model • e.g.: Ñ1lnYt = f Ñ1lnYt - q at-1 +at • Manual Step 3: • Diagnose Model

24. DETECTION FOLLOWS MODEL • Control Chart Detection Techniques Only After Valid Model Estimated. • Special Causes Revealed in iidn Residuals.

25. ADJUSTMENT NEEDS NO CAUSE • Feed-Forward/ Feed-Back Schemes: Based on Valid Time Series Models. • Feed-Forward/ Feed-Back Works With or Without Knowledge of Cause. • Most Temporal Structure Not Traced to Cause.

26. SPECIAL CAUSE VARIATION • Special Cause Variation Takes Many Forms: Pulses Level Shifts Seasonal Pulses Seasonal Pulse Changes Trends Trend Shifts Here, Called Interventions

27. INTERVENTION ANALYSIS1 • Conventional Time Series Blends Interventions into Model, Biasing Parameter Estimates. • Intervention Variables Can Be Estimated Separately. • Intervention Variables Free the Underlying Temporal Structure to Be Modeled Accurately.

28. INTERVENTION ANALYSIS2 • AFS Autobox Technique • Start With Simple Model, e.g., : • Yt = B0 + B1Yt-1 + at , • B0: Intercept • B1Yt-1: AR(1) Term • But, • at May Not Be Random: • Omitted Data Variables or Interventions

29. INTERVENTION ANALYSIS3 • Expand at to Include Unknown Variables: • at = Random Component V + Interventions I • Yt = B0 + B1Yt-1 + B2It + Vt at

30. INTERVENTION ANALYSIS4 • Iterate All Possible Intervention Periods With Dummy = 1 for Timing of Intervention Effect. • Compare Error Variance for All Models, Including Base Model. • Minimum Mean Squared Error Wins.

31. INTERVENTION ANALYSIS5 • Simulation of I as a Dummy E.g., to Look for a Pulse P : P model 1 = 1,0,0,0,0,0,0,… P model 2 = 0,1,0,0,0,0,0,… , etc. • Yt = B0 + B1Yt-1 + B2Pt + Vt

32. INTERVENTION ANALYSIS6 • Simulation of I as a Dummy To Look for a Level Shift L : L model 1 = 0,1,1,1,1,1,1,… L model 2 = 0,0,1,1,1,1,1,… , etc. Yt = B0 + B1Yt-1 + B2Pt + B3Lt + Vt

33. INTERVENTION ANALYSIS7 • Simulation of I as a Dummy To Look for a Seasonal Pulse S : S model 1 = 1,0,0,1,0,0,1,0,… S model 2 = 0,1,0,0,1,0,0,1,… , etc. Yt = B0 + B1Yt-1 + B2Pt + B3Lt + B4St + Vt

34. INTERVENTION ANALYSIS8 • Simulation of I as a Dummy The Same Process Is Applied to Trend, Trend Shifts and Other Patterns.

35. INTERVENTION ANALYSIS9 • Standard F Test Measures Statistical Significance • of Reduction From Base Model • F1, N-k-1  [SSSim Model – SSBase Model]/ [SSSim Model /N-k-1] • k: number of parameters at each stage • SS: sum of squares • If Significant, Then Variable Is Added to Model. • Procedure Repeated for Each Intervention Type.

36. INTERVENTION ANALYSIS10 • Final Model May Include Conventional Time Series Terms (AR, MA). • Final Error Term Must Not Violate iidn.

37. INTERVENTION EXAMPLE1 COF of CMP Process Slurry. Data With Permission from Ara Philipossian, Dept. of Chemical Engineering, U. of Arizona

38. INTERVENTION EXAMPLE2 • Initial Model: Yt = 0.058164 + (1- 0.841B1) at/(1- 0.997B1) • Autobox Recognized That the AR and MA Terms Approximately Cancel: Yt = 0.20834 + at N = 720 Seconds

39. INTERVENTION EXAMPLE3 Autocorrelation Function of COF Initial Insufficient Model Residuals. Residuals Contain Information.

40. INTERVENTION EXAMPLE4 • I.e., Intervention Structure Masks Underlying Temporal Structure. • Masking the Temporal Structure Distorted its Parameter Estimates.

41. INTERVENTION EXAMPLE5 Intervention Process • Final Model: Obs 187 Obs 196 Yt = 0.19068 + 0.045X1t + 0.034X2t + 0.023X3t – 0.042X4t –0.050X5t + (1 + 0.159B3) at /(1 + 0.145B2 - 0.627B3) N = 720 R2 = 0.962 Obs 212 Obs 474 Obs 492 Non-white Noise Process

42. INTERVENTION EXAMPLE7 COF Modeled With Interventions Removed.

43. INTERVENTION EXAMPLE6 Autocorrelation Function of COF Final Model Residuals. Residuals Are Random.

44. INTERVENTION ANALYSIS ACCOMPLISHMENTS • Undistorted Probabilistic Model • Automatic Detection of Effect of Change in Percent Solids on Friction: Amplitude Timing • Forecast of Friction • Basis for Control • All Computed Quickly.

45. IMPLICATIONS • Time Series Models Are Complicated. • Formerly, Extensive Manual Judgment. • Can Be Automatic and Fast, (e.g., AFS’s Autobox: Fully Automatic, Including Intervention Analysis). • Intervention Analysis Increases Model Validity—Improves Fab Process Control,

46. IMPLICATIONS • Time Series Models are Complicated. • Formerly, Extensive Manual Judgment. • Can Be Automatic and Fast, (e.g., AFS’s Autobox: Fully Automatic, Including Intervention Analysis). • Intervention Analysis Increases Model Validity—Improves Fab Process Control, Improves Yield

47. IMPLICATIONS • Time Series Models are Complicated. • Formerly, Extensive Manual Judgment. • Can Be Automatic and Fast, (e.g., AFS’s Autobox: Fully Automatic, Including Intervention Analysis). • Intervention Analysis Increases Model Validity—Improves Fab Process Control, Improves Yield Increases Other Efficiencies.

48. SUMMARY • Process Control On Verge Of Revolution. • APC Designs With Robust Software Architecture Is Infrastructure Enabler. • Automated Time Series Modeling Is Analytics Enabler.

49. REFERENCES Alwan, Layth C. 2000. Statistical Process Analysis, Irwin McGraw-Hill, New York, NY. Alwan, Layth C.; and H. V. Roberts. 1995. “The Pervasive Problem of Misplaced Control Limits,” Applied Statistics, 44, pp. 269-278. Philipossian, Ara; and E. Mitchell. July/August 2002. “Performing Mean Residence Time Analysis of CMP Process,” Micro, pp. 85-95. Box, George E. P.; G. M. Jenkins; and G. C. Reinsel. 1994. Times Series Analysis, Forecasting and Control, 3rd Ed. Prentice Hall.