Searching for important factors among large No. of factors in DOE

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Analyzing Supersaturated Designs Using Biased
Estimation
QPRC 2003 ByAdnan Bashir andJames Simpson
May 23,2003
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Outline

• Introduction
• Motivation example
• Research objectives

• Proposed analysis method
• Multicollinearity ridge
• Best subset model
• Simulated case studies
• Example
• Results

• Conclusion recommendations
• Future research

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Introduction

• Many studies and experiments contain a large
  number of variables
• Fewer variables are significant
• Which are those few factors? How do we find those
  factors?
• Screening experiments (Design Analysis) are
  used to find those important factors
• Several methods techniques (Design Analysis)
  are available to screen

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Motivation exampleComposites Production
Raw Materials
INPUTS (Factors) Resin Flow Rate (x1) Type of
Resin (x2) Gate Location (x3) Fiber Weave
(x4) Mold Complexity (x5) Fiber Weight
(x6) Curing Type (x7) Pressure (x8)
OUTPUTS (Responses) Fiber Permeability Product
Quality Tensile Strength
Process
Noise
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Motivation example (continued)
Response y Tensile strength
Each experiment costs 500, requires 8 hours,
budget 3,000 (6 experiments)
1 High level -1 Low level

• Supersaturated Designs number of factors m
  number of runs n
• Columns are not Orthogonal

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Research Objectives

• Propose an efficient technique to screen the
  important factors in an experiment with fewer
  number of runs
• Construct improved supersaturated designs
• Develop an accurate, reliable and efficient
  technique to analyze supersaturated designs

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Analysis of SSDs Current Methods

• Stepwise regression, most commonly used
• Lin (1993, 1995), Wang (1995), Nguyen (1996)
• All possible regressions
• Abraham, Chipman, and Vijayan (1999)
• Bayesian method
• Box and Meyer (1993)
• Investigated techniques
• Principle components, partial least squares and
  flexible regression methods (MARS CART)

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Analysis of SSDs Proposed Method

• Modified best subset via ridge regression
  (MBS-RR)
• Ridge regression for multicollinearity
• Best subset for variable selection in each model
• Criterion based selection to identify best model

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Ridge Regression Motivation
Ordinary Least Squares
Ridge Regression
Consider adding k gt 0 to each diagonal of X'X ,
say k 0.1

• Consider a centered, scaled matrix, X

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Ridge Regression

• Ridge regression estimates
• where k is referred to as a
• shrinkage parameter
• Thus,

Geometric interpretation of ridge regression
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Ridge Regression, (continued)Shrinkage parameter

• Hoerl and Kennard (1975) suggest
• where p is number of parameter
• are found from the least squares
  solution

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Shrinkage Parameter Ridge Trace
Ridge trace for nine regressors (Adapted from
Montgomery, Peck, Vining 2001)
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Proposed Analysis Method
Read X, Y
Contd.
Select the best 1-factor model By OLS (k0)
Calculate k, and find the best 2-factor model by
all possible subsets
Adding 1 factor at a time to the best 2-factor
model, from the remaining variables to get the
best 3-factor model
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Proposed Analysis Method
Is the stopping rule satisfied?
Yes
No
Adding 1 factor at a time to the best 3-factor
model, from the remaining variables to get the
best 4-factor model
Yes
Is the stopping rule satisfied?
No
Adding 1 factor at a time to the best 7-factor
model, from the remaining variables to get the
best 8-factor model
Final Model with Min. Cp
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Selecting the Best Model
Where diff user defined tolerance
Cp
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Method Comparison-Monte CarloSimulation Design
of Experiments
Factors considered in the simulation study
III Fractional Factorial Design Matrix
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Analysis Method Comparison

• The performance measures, Type I and Type II
  errors

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Case Studies with Corresponding Models
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Method Comparison Results, Type I errors
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Method Comparison Results, Type II errors
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Factors Contributing to Method PerformanceType
II Errors
Stepwise Method
var
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Factors Contributing to Method PerformanceType
II Errors
Proposed Method
var
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Summary Results
A No. of runs B No. of factors C
Multicollinearity D Error variance E No. of
Sig. factors
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Conclusions Recommendations

• SSDs Analysis Best Subset Ridge Regression
• Use ridge regression estimation
• Best subset variable selection method outperforms
  stepwise regression

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Future Research

• Analyzing SSDs
• Multiple criteria in selecting the best model
• All possible subset, 3 factor model
• Streamline program code
• Real-life case studies
• Genetic algorithm for variable selection

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Acknowledgement

• Dr. Carroll Croarkin, chair of selection
  committee for Mary G. Natrella
• Selection Committee for Mary G. Natrella
  Scholarship
• Dr. Simpson, Supervisor