- 0 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com ADVANCED INTERVENTION ANALYSIS of Tool Data...

AEC/ APC XIV Symposium 2002

foliage.com autobox.com

ADVANCED INTERVENTION ANALYSIS

of Tool Data for Improved Process Control

Presenter:

Rob Firmin, Ph.D.

Managing DirectorFoliage Software Systems

408 321 8444rfirmin@foliage.com

Coauthor:

David P. Reilly

FounderAutomatic Forecasting Systems

215 675 0652dave@autobox.com

September 11, 2002

Introduce Techniques That CanImprove Fab Process Control

Significantly:

• Reduce Variation• Improve Yield• Increase Other Efficiencies.

PRESENTATION PURPOSE

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

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

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.)

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.

Most Manufacturing Data Are Serially Dependent,Not Drawn Independently:

tion.8

Confidence Limits

Coefficient

What If a Lottery Operated With

Auto-Dependent(Magnetized)

- 10 -

- 11 -

- 12 -

- 13 -

NumbersWould Be Drawn

In Patterns,(Even With

Tumbling).

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Many Confirming Studies:

80+ Percent of Industrial Processes Have Temporal Structure.

See: Alwan, L. C., H. V. Roberts (1995)

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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.

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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 GroupVariancePositive Autocorrelation Lower

WithinGroup Variance.

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Dependence Cannot Be Swept Away:

Cannot Fix With Random Sorts Cannot Avoid by Reducing Sampling Rate Lose Validity With Preconceived Models.

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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 =

ARIMAInterventions Are Extensions.

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TRUE TIME SERIES ANALYSIS 1

Many Time Series Methods;Only True Time Series Analysis Satisfies iidn.

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Many Time Series Methods;Only True Time Series Analysis Satisfies iidn.

Proper Identification, Estimation and DiagnosticsResult in iidn Residuals.

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Manual Step 1:Identify Appropriate Subset of ModelsRender Series Stationary, Homogeneous & Normal.

1lnYt = lnYt – lnYt-1

1: first difference

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Manual Step 1:Identify Appropriate Subset of ModelsRender Series Stationary, Homogeneous & Normal.

1lnYt = lnYt – lnYt-1

Manual Step 2:Estimate Model

e.g.: 1lnYt = 1lnYt - at-1 + at

Manual Step 3:Diagnose Model

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DETECTION FOLLOWS MODEL

Control Chart Detection Techniques Only After Valid Model Estimated.

Special Causes Revealed in iidn Residuals.

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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.

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SPECIAL CAUSE VARIATION

Special Cause Variation Takes Many Forms:

PulsesLevel ShiftsSeasonal PulsesSeasonal Pulse ChangesTrendsTrend Shifts

Here, Called Interventions

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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.

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AFS Autobox Technique

Start With Simple Model, e.g., :

Yt = B0 + B1Yt-1 + at ,

B0: Intercept

B1Yt-1: AR(1) Term

at May Not Be Random:

Omitted Data Variables or Interventions

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Expand at to Include Unknown Variables:

at = Random Component V + Interventions I

Yt = B0 + B1Yt-1 + B2It + Vt

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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.

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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,… ,

Yt = B0 + B1Yt-1 + B2Pt + Vt

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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,… ,

Yt = B0 + B1Yt-1 + B2Pt + B3Lt + Vt

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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,… ,

Yt = B0 + B1Yt-1 + B2Pt + B3Lt + B4St + Vt

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The Same Process Is Applied to Trend, Trend Shifts and Other Patterns.

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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.

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Final Model May Include Conventional Time Series Terms (AR, MA).

Final Error Term Must Not Violate iidn.

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COF of CMP Process Slurry.

Data With Permission from Ara Philipossian,Dept. of Chemical Engineering, U. of Arizona

INTERVENTION EXAMPLE1

0 100 200 300 400 500 600 700 800

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Yt = 0.058164 + (1- 0.841B1) at/(1- 0.997B1)

Initial Model:

Autobox Recognized That the AR and MA Terms Approximately Cancel:

Yt = 0.20834 + at

N = 720 Seconds

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Autocorrelation Function of COFInitial Insufficient Model Residuals.

Residuals Contain Information.

Residuals ACFCOF Insufficient Model

-0.800

-0.600

-0.400

-0.200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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I.e., Intervention Structure Masks Underlying Temporal Structure.

Masking the Temporal Structure Distorted its Parameter Estimates.

- 41 -

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

Final Model:

Obs 187 Obs 196

Obs 212 Obs 474 Obs 492

Intervention Process

Non-white Noise Process

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COFModeled With Interventions Removed.

COFBenchmarked Without Interventions

0 100 200 300 400 500 600 700 800

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Autocorrelation Function of COFFinal Model Residuals.

Residuals Are Random.

Residuals ACFCOF Final Model

-0.800

-0.600

-0.400

-0.200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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INTERVENTION ANALYSIS

ACCOMPLISHMENTSa) 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.

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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,

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Improves Yield

IMPLICATIONS

Time Series Models are Complicated.

- 47 -

Improves Yield

Increases Other Efficiencies.

IMPLICATIONS

Time Series Models are Complicated.

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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.

- 0 - AEC/ APC XIV Symposium 2002 foliage.comautobox.com ADVANCED INTERVENTION ANALYSIS of Tool Data...

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