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- 0 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com ADVANCED INTERVENTION ANALYSIS of Tool Data for Improved Process Control Presenter: Rob Firmin,

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Presentation on theme: "- 0 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com ADVANCED INTERVENTION ANALYSIS of Tool Data for Improved Process Control Presenter: Rob Firmin,"— Presentation transcript:

1 - 0 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com ADVANCED INTERVENTION ANALYSIS of Tool Data for Improved Process Control Presenter: Rob Firmin, Ph.D. Managing Director Foliage Software Systems Coauthor: David P. Reilly Founder Automatic Forecasting Systems September 11, 2002

2 - 1 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com Introduce Techniques That Can Improve Fab Process Control Significantly: Reduce Variation Improve Yield Increase Other Efficiencies. PRESENTATION PURPOSE

3 - 2 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com 1.Statistical Validity 2.Temporal Structure & True Time Series Analysis 3.Special Cause Variation 4.Intervention Analysis 5.Intervention Example From Semi 6.Conclusions OUTLINE

4 - 3 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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. APC Effect on Process Control

5 - 4 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 - 5 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 - 6 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com Most Manufacturing Data Are Serially Dependent, Not Drawn Independently: STATISTICAL VALIDITY 3

8 - 7 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY What If a Lottery Operated With Auto-Dependent (Magnetized) Data?

9 - 8 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY

10 - 9 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY

11 AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY

12 AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY

13 AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY Numbers Would Be Drawn In Patterns, (Even With Tumbling).

14 AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY 5 Many Confirming Studies: 80+ Percent of Industrial Processes Have Temporal Structure. See: Alwan, L. C., H. V. Roberts (1995)

15 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com STATISTICAL VALIDITY 8 Dependence Cannot Be Swept Away: Cannot Fix With Random Sorts Cannot Avoid by Reducing Sampling Rate Lose Validity With Preconceived Models.

18 AEC/ APC XIV Symposium 2002 foliage.comautobox.com THE OPPORTUNITY Valid Time Series Models Separate the Process from its Noise. 1 - R 2 of a Valid Model = Natural Variation R 2 = Potential Control Improvement = (y i – y) 2 / (y i – y) 2 = Model Variation/Total Variation

19 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com TRUE TIME SERIES ANALYSIS 1 Many Time Series Methods; Only True Time Series Analysis Satisfies iidn.

21 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com TRUE TIME SERIES ANALYSIS 3 Manual Step 1: Identify Appropriate Subset of Models Render Series Stationary, Homogeneous & Normal. e.g.: 1 lnY t = lnY t – lnY t-1 1 : first difference

23 AEC/ APC XIV Symposium 2002 foliage.comautobox.com TRUE TIME SERIES ANALYSIS 4 Manual Step 1: Identify Appropriate Subset of Models Render Series Stationary, Homogeneous & Normal. 1 lnY t = lnY t – lnY t-1 Manual Step 2: Estimate Model e.g.: 1 lnY t = 1 lnY t - a t-1 + a t Manual Step 3: Diagnose Model

24 AEC/ APC XIV Symposium 2002 foliage.comautobox.com DETECTION FOLLOWS MODEL Control Chart Detection Techniques Only After Valid Model Estimated. Special Causes Revealed in iidn Residuals.

25 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com SPECIAL CAUSE VARIATION Special Cause Variation Takes Many Forms: Pulses Level Shifts Seasonal Pulses Seasonal Pulse Changes Trends Trend Shifts Here, Called Interventions

27 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 1 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 2 AFS Autobox Technique Start With Simple Model, e.g., : Y t = B 0 + B 1 Y t-1 + a t, B 0 : Intercept B 1 Y t-1 : AR(1) Term But, a t May Not Be Random: Omitted Data Variables or Interventions

29 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 3 Expand a t to Include Unknown Variables: a t = Random Component V + Interventions I Y t = B 0 + B 1 Y t-1 + B 2 I t + V t atat

30 AEC/ APC XIV Symposium 2002 foliage.comautobox.com 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. INTERVENTION ANALYSIS 4

31 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 5 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. Y t = B 0 + B 1 Y t-1 + B 2 P t + V t

32 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 6 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. Y t = B 0 + B 1 Y t-1 + B 2 P t + B 3 L t + V t

33 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 7 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. Y t = B 0 + B 1 Y t-1 + B 2 P t + B 3 L t + B 4 S t + V t

34 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 8 Simulation of I as a Dummy The Same Process Is Applied to Trend, Trend Shifts and Other Patterns.

35 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 9 Standard F Test Measures Statistical Significance of Reduction From Base Model F 1, N-k-1 [SS Sim Model – SS Base Model ]/ [SS Sim 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 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS 10 Final Model May Include Conventional Time Series Terms (AR, MA). Final Error Term Must Not Violate iidn.

37 AEC/ APC XIV Symposium 2002 foliage.comautobox.com COF of CMP Process Slurry. Data With Permission from Ara Philipossian, Dept. of Chemical Engineering, U. of Arizona INTERVENTION EXAMPLE 1

38 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 2 Y t = ( B 1 ) a t /( B 1 ) Initial Model: Autobox Recognized That the AR and MA Terms Approximately Cancel: Y t = a t N = 720 Seconds

39 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 3 Autocorrelation Function of COF Initial Insufficient Model Residuals. Residuals Contain Information.

40 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 4 I.e., Intervention Structure Masks Underlying Temporal Structure. Masking the Temporal Structure Distorted its Parameter Estimates.

41 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 5 Y t = X1 t X2 t X3 t – 0.042X4 t –0.050X5 t + ( B 3 ) a t /( B B 3 ) N = 720 R 2 = Final Model: Obs 187Obs 196 Obs 212Obs 474Obs 492 Intervention Process Non-white Noise Process

42 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 7 COF Modeled With Interventions Removed.

43 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION EXAMPLE 6 Autocorrelation Function of COF Final Model Residuals. Residuals Are Random.

44 AEC/ APC XIV Symposium 2002 foliage.comautobox.com INTERVENTION ANALYSIS ACCOMPLISHMENTS a)Undistorted Probabilistic Model b)Automatic Detection of Effect of Change in Percent Solids on Friction: Amplitude Timing c)Forecast of Friction d)Basis for Control e)All Computed Quickly.

45 AEC/ APC XIV Symposium 2002 foliage.comautobox.com IMPLICATIONS Time Series Models Are Complicated. Formerly, Extensive Manual Judgment. Can Be Automatic and Fast, (e.g., AFSs Autobox: Fully Automatic, Including Intervention Analysis). Intervention Analysis Increases Model Validity Improves Fab Process Control,

46 AEC/ APC XIV Symposium 2002 foliage.comautobox.com Improves Yield IMPLICATIONS Time Series Models are Complicated. Formerly, Extensive Manual Judgment. Can Be Automatic and Fast, (e.g., AFSs Autobox: Fully Automatic, Including Intervention Analysis). Intervention Analysis Increases Model Validity Improves Fab Process Control,

47 AEC/ APC XIV Symposium 2002 foliage.comautobox.com Improves Yield Increases Other Efficiencies. IMPLICATIONS Time Series Models are Complicated. Formerly, Extensive Manual Judgment. Can Be Automatic and Fast, (e.g., AFSs Autobox: Fully Automatic, Including Intervention Analysis). Intervention Analysis Increases Model Validity Improves Fab Process Control,

48 AEC/ APC XIV Symposium 2002 foliage.comautobox.com SUMMARY Process Control On Verge Of Revolution. APC Designs With Robust Software Architecture Is Infrastructure Enabler. Automated Time Series Modeling Is Analytics Enabler.

49 AEC/ APC XIV Symposium 2002 foliage.comautobox.com REFERENCES Alwan, Layth C Statistical Process Analysis, Irwin McGraw-Hill, New York, NY. Alwan, Layth C.; and H. V. Roberts The Pervasive Problem of Misplaced Control Limits, Applied Statistics, 44, pp Philipossian, Ara; and E. Mitchell. July/August Performing Mean Residence Time Analysis of CMP Process, Micro, pp Box, George E. P.; G. M. Jenkins; and G. C. Reinsel Times Series Analysis, Forecasting and Control, 3 rd Ed. Prentice Hall.


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