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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 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.
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Most Manufacturing Data Are Serially Dependent,Not Drawn Independently:
STATISTICAL VALIDITY 3
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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STATISTICAL VALIDITY 4
15
1
4
13 16
7
89
What If a Lottery Operated With
Auto-Dependent(Magnetized)
Data?
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STATISTICAL VALIDITY 4
15
1
4
13 16
7
89
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STATISTICAL VALIDITY 4
15
7
8
13
9
16
1
4
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STATISTICAL VALIDITY 4
15
17
8
13
9
164
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STATISTICAL VALIDITY 4
15
1 8
13
9
164
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STATISTICAL VALIDITY 4
15
1
13
9
164
8
NumbersWould Be Drawn
In Patterns,(Even With
Tumbling).
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STATISTICAL VALIDITY 5
Many Confirming Studies:
80+ Percent of Industrial Processes Have Temporal Structure.
See: Alwan, L. C., H. V. Roberts (1995)
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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.
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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 GroupVariancePositive Autocorrelation Lower
WithinGroup Variance.
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STATISTICAL VALIDITY 8
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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TRUE TIME SERIES ANALYSIS 2
Many Time Series Methods;Only True Time Series Analysis Satisfies iidn.
Proper Identification, Estimation and DiagnosticsResult in iidn Residuals.
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TRUE TIME SERIES ANALYSIS 3
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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TRUE TIME SERIES ANALYSIS 4
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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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
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INTERVENTION ANALYSIS3
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.
INTERVENTION ANALYSIS4
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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
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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
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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
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INTERVENTION ANALYSIS8
Simulation of I as a Dummy
The Same Process Is Applied to Trend, Trend Shifts and Other Patterns.
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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.
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INTERVENTION ANALYSIS10
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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INTERVENTION EXAMPLE2
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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INTERVENTION EXAMPLE3
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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INTERVENTION EXAMPLE4
I.e., Intervention Structure Masks Underlying Temporal Structure.
Masking the Temporal Structure Distorted its Parameter Estimates.
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INTERVENTION EXAMPLE5
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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INTERVENTION EXAMPLE7
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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INTERVENTION EXAMPLE6
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.
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
Increases Other Efficiencies.
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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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.