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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.

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slide1

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

presentation purpose
PRESENTATION PURPOSE

Introduce Techniques That Can

Improve Fab Process Control Significantly:

  • Reduce Variation
  • Improve Yield
  • Increase Other Efficiencies.
outline
OUTLINE

Statistical Validity

Temporal Structure & True Time Series Analysis

Special Cause Variation

Intervention Analysis

Intervention Example From Semi

Conclusions

slide4

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.

statistical validity 1
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.)

slide6

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.
statistical validity 3
STATISTICAL VALIDITY 3
  • Most Manufacturing Data Are Serially Dependent,
  • Not Drawn Independently:
statistical validity 4
STATISTICAL VALIDITY 4

What If a Lottery Operated With Auto-Dependent

(Magnetized)

Data?

16

13

9

15

8

4

7

1

statistical validity 413
STATISTICAL VALIDITY 4

Numbers

Would Be Drawn

In Patterns,

(Even With

Tumbling).

4

16

13

15

8

9

1

slide14

STATISTICAL VALIDITY 5

  • Many Confirming Studies:
  • 80+ Percent of Industrial Processes Have Temporal Structure.

See:

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

statistical validity 6
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.
slide16

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.

statistical validity 8
STATISTICAL VALIDITY 8
  • Dependence Cannot Be Swept Away:
    • Cannot Fix With Random Sorts
    • Cannot Avoid byReducing Sampling Rate
    • Lose Validity With Preconceived Models.
the opportunity
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
temporal structure
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.
true time series analysis 1
TRUE TIME SERIES ANALYSIS 1
  • Many Time Series Methods;
  • Only True Time Series Analysis Satisfies iidn.
true time series analysis 2
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.
true time series analysis 3
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

true time series analysis 4
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
slide24

DETECTION FOLLOWS MODEL

  • Control Chart Detection Techniques Only After Valid Model Estimated.
  • Special Causes Revealed in iidn Residuals.
slide25

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.
special cause variation
SPECIAL CAUSE VARIATION
  • Special Cause Variation Takes Many Forms:

Pulses

Level Shifts

Seasonal Pulses

Seasonal Pulse Changes

Trends

Trend Shifts

Here, Called Interventions

slide27

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.
slide28

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
slide29

INTERVENTION ANALYSIS3

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

at

slide30

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.
slide31

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
slide32

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

slide33

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

slide34

INTERVENTION ANALYSIS8

  • Simulation of I as a Dummy

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

slide35

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.
slide36

INTERVENTION ANALYSIS10

  • Final Model May Include Conventional

Time Series Terms (AR, MA).

  • Final Error Term Must Not Violate iidn.
slide37

INTERVENTION EXAMPLE1

COF of CMP Process Slurry.

Data With Permission from Ara Philipossian,

Dept. of Chemical Engineering, U. of Arizona

slide38

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

slide39

INTERVENTION EXAMPLE3

Autocorrelation Function of COF

Initial Insufficient Model Residuals.

Residuals Contain Information.

slide40

INTERVENTION EXAMPLE4

  • I.e., Intervention Structure Masks Underlying Temporal Structure.
  • Masking the Temporal Structure Distorted its Parameter Estimates.
slide41

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

slide42

INTERVENTION EXAMPLE7

COF

Modeled With Interventions Removed.

slide43

INTERVENTION EXAMPLE6

Autocorrelation Function of COF

Final Model Residuals.

Residuals Are Random.

slide44

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.
slide45

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,
slide46

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

slide47

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.

slide48

SUMMARY

  • Process Control On Verge Of Revolution.
  • APC Designs With Robust Software Architecture Is Infrastructure Enabler.
  • Automated Time Series Modeling Is Analytics Enabler.
slide49

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.