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Analyzing Supersaturated Designs Using Biased Estimation QPRC 2003 By Adnan Bashir and James Simpson May 23,2003 FAMU-FSU College of Engineering, Department of Industrial Engineering

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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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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Motivation example Composites Production INPUTS (Factors) Resin Flow Rate ( x 1 ) Type of Resin (x 2 ) Gate Location ( x 3 ) Fiber Weave ( x 4 ) Mold Complexity ( x 5 ) Fiber Weight ( x 6 ) Curing Type ( x 7 ) Pressure ( x 8 ) OUTPUTS (Responses) Fiber Permeability Product Quality Tensile Strength Noise Process Raw Materials FAMU-FSU College of Engineering, Department of Industrial Engineering

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Motivation example (continued) Response y = Tensile strength Each experiment costs $500, requires 8 hours, budget $3,000 (6 experiments) FAMU-FSU College of Engineering, Department of Industrial Engineering 12345678Y 111111111 2 11 3 111 4 11 1 51 1 1 1 611 1 1 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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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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) FAMU-FSU College of Engineering, Department of Industrial Engineering

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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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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Ridge Regression Motivation Consider a centered, scaled matrix, X* Consider adding k > 0 to each diagonal of X*'X*, say k = 0.1 FAMU-FSU College of Engineering, Department of Industrial Engineering Ordinary Least SquaresRidge Regression

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Ridge regression estimates where k is referred to as a shrinkage parameter Thus, FAMU-FSU College of Engineering, Department of Industrial Engineering Geometric interpretation of ridge regression

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Ridge Regression, (continued) Shrinkage parameter FAMU-FSU College of Engineering, Department of Industrial Engineering 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) FAMU-FSU College of Engineering, Department of Industrial Engineering

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Read X, Y Select the best 1-factor model By OLS (k=0) 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 Proposed Analysis Method FAMU-FSU College of Engineering, Department of Industrial Engineering Contd.

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Proposed Analysis Method Is the stopping rule satisfied? Adding 1 factor at a time to the best 3-factor model, from the remaining variables to get the best 4-factor model Is the stopping rule satisfied? 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 FAMU-FSU College of Engineering, Department of Industrial Engineering Yes No

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Selecting the Best Model FAMU-FSU College of Engineering, Department of Industrial Engineering Where diff: user defined tolerance CpCp

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Method Comparison-Monte Carlo Simulation & Design of Experiments Factors considered in the simulation study III Fractional Factorial Design Matrix FAMU-FSU College of Engineering, Department of Industrial Engineering

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Analysis Method Comparison The performance measures, Type I and Type II errors FAMU-FSU College of Engineering, Department of Industrial Engineering

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Case Studies with Corresponding Models FAMU-FSU College of Engineering, Department of Industrial Engineering

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Method Comparison Results, Type I errors FAMU-FSU College of Engineering, Department of Industrial Engineering FactorsType I errors (%) No. of No. of Sig.Collin-ErrorAverage RunsFactors earityVariance Proposed Stepwise 12203H3.0014.1012.92 18403L3.008.9316.43 18407H3.008.7014.56 12207L3.009.0612.52 12407L0.502.8810.42 18203L0.500.0013.26 18207H0.500.0013.27 12403H0.500.0017.28 15305M1.756.5610.67 15305M1.756.8011.76 15305M1.755.2011.28

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Method Comparison Results, Type II errors FAMU-FSU College of Engineering, Department of Industrial Engineering FactorsType II errors (%) No. of No. of Sig.Collin-ErrorAverage RunsFactors earityVariance Proposed Stepwise 12203H3.008.6763.40 18403L3.000.00 18407H3.0037.2049.86 12207L3.0026.1930.53 12407L0.5017.5014.86 18203L0.500.00 18207H0.502.983.36 12403H0.500.00 15305M1.753.203.53 15305M1.754.004.80 15305M1.752.003.60

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Factors Contributing to Method Performance Type II Errors Stepwise Method FAMU-FSU College of Engineering, Department of Industrial Engineering var

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Factors Contributing to Method Performance Type II Errors Proposed Method FAMU-FSU College of Engineering, Department of Industrial Engineering var

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Summary Results FAMU-FSU College of Engineering, Department of Industrial Engineering 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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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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 FAMU-FSU College of Engineering, Department of Industrial Engineering

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