ADVANCED INTERVENTION ANALYSIS

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_at_foliage.com
Coauthor David P. Reilly Founder Automatic
Forecasting Systems 215 675 0652 dave_at_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

1. Statistical Validity
2. Temporal Structure True Time Series Analysis
3. Special Cause Variation
4. Intervention Analysis
5. Intervention Example From Semi
6. 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(AB) P(A) and P(BA) 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(AB) P(A) and P(BA) 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 by Reducing 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 a t-1
  a t
• 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., AFSs Autobox
  Fully Automatic, Including Intervention
  Analysis).
• Intervention Analysis Increases Model
  ValidityImproves Fab Process Control,

46
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
  ValidityImproves Fab Process Control,

Improves Yield
47
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
  ValidityImproves 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.