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

- 1 -

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 [email protected]

Coauthor:

David P. Reilly

FounderAutomatic Forecasting Systems

215 675 [email protected]

September 11, 2002

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Introduce Techniques That CanImprove Fab Process Control

Significantly:

• Reduce Variation• Improve Yield• Increase Other Efficiencies.

PRESENTATION PURPOSE

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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

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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

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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.)

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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.

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Most Manufacturing Data Are Serially Dependent,Not Drawn Independently:


Lag

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

Aut

ocor

rela

tion.8

.3

-.3

-.8

Confidence Limits

Coefficient

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15

1

4

13 16

7

89

What If a Lottery Operated With

Auto-Dependent(Magnetized)

Data?

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15

1

4

13 16

7

89

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15

7

8

13

9

16

1

4

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15

17

8

13

9

164

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15

1 8

13

9

164

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15

1

13

9

164

8

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

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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.

e.g.:

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

But,

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

at

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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,… ,

etc.

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,… ,

etc.

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,… ,

etc.

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

COF

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

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

0.000

0.200

0.400

0.600

0.800

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.

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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

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

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

0.000

0.200

0.400

0.600

0.800

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.




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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.

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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.

Documents

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