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  • Bridging the Gap Between Space-Filling and Optimal Designs

    Design for Computer Experiments

    by

    Kathryn Kennedy

    A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree

    Doctor of Philosophy

    Approved July 2013 by the Graduate Supervisory Committee:

    Douglas C. Montgomery, Co-Chair

    Rachel T. Johnson, Co-Chair John W. Fowler

    Connie M. Borror

    ARIZONA STATE UNIVERSITY

    August 2013

  • i

    ABSTRACT

    This dissertation explores different methodologies for combining two popular

    design paradigms in the field of computer experiments. Space-filling designs are

    commonly used in order to ensure that there is good coverage of the design space, but

    they may not result in good properties when it comes to model fitting. Optimal designs

    traditionally perform very well in terms of model fitting, particularly when a polynomial

    is intended, but can result in problematic replication in the case of insignificant factors.

    By bringing these two design types together, positive properties of each can be retained

    while mitigating potential weaknesses.

    Hybrid space-filling designs, generated as Latin hypercubes augmented with I-

    optimal points, are compared to designs of each contributing component. A second

    design type called a bridge design is also evaluated, which further integrates the

    disparate design types. Bridge designs are the result of a Latin hypercube undergoing

    coordinate exchange to reach constrained D-optimality, ensuring that there is zero

    replication of factors in any one-dimensional projection. Lastly, bridge designs were

    augmented with I-optimal points with two goals in mind. Augmentation with candidate

    points generated assuming the same underlying analysis model serves to reduce the

    prediction variance without greatly compromising the space-filling property of the

    design, while augmentation with candidate points generated assuming a different

    underlying analysis model can greatly reduce the impact of model misspecification

    during the design phase.

    Each of these composite designs are compared to pure space-filling and optimal

    designs. They typically out-perform pure space-filling designs in terms of prediction

    variance and alphabetic efficiency, while maintaining comparability with pure optimal

    designs at small sample size. This justifies them as excellent candidates for initial

    experimentation.

  • ii

    DEDICATION

    I would like to dedicate this dissertation to my parents, whose unwavering support and

    tenacious confidence was invaluable over my years in the program. In particular, I’d like

    to thank my father for always leading by example, and championing my cause for time

    and fair treatment. And my mother, for her commitment to perfectionism, excellent

    literary taste, and welcomed penchant for feeding me. I would also like to thank my

    ‘little’ brother, always setting the proactive example I so wish I could emulate, and my

    friends for bearing with me as I prevaricated over the years.

  • iii

    ACKNOWLEDGMENTS

    I would like to acknowledge my committee, and thank them for their continued support

    as I worked to find time to complete the research phase of the degree requirements. Dr.

    Douglas Montgomery has been an exemplary advisor, and I am honored he agreed to

    guide me as he did my father. Dr. Rachel Johnson’s research provided an excellent

    starting point, and she has been instrumental in providing guidance as to its

    continuance. I would like to thank Dr. John Fowler, for without his assistance I could

    not have continued in the program. And last but never least, Dr. Connie Borror, who

    taught the class which originally inspired me to transition into the Industrial

    Engineering department back in my undergrad days. Many thanks as well to Dr. Bradley

    Jones, whose JMP script provided an important starting point for the latter phases of my

    research.

  • iv

    TABLE OF CONTENTS

    Page

    LIST OF TABLES ..................................................................................................................... vi

    LIST OF FIGURES ................................................................................................................... ix

    CHAPTER

    1 INTRODUCTION ................................................................................................. 1

    2 LITERATURE REVIEW ...................................................................................... 3

    Design of Computer Experiments ................................................................... 3

    Traditional Response Surface Methodology Designs ................................ 3

    Space Filling Designs ................................................................................... 5

    Analysis of Computer Experiments .............................................................. 20

    Polynomial Models .................................................................................... 21

    Gaussian Process Models .......................................................................... 23

    Augmentation of Designs for Computer Experiments ................................. 25

    3 HYBRID SPACE-FILLING DESIGNS FOR COMPUTER EXPERIMENTS .... 27

    Methodology .................................................................................................. 27

    Results ............................................................................................................ 29

    Theoretical Prediction Variance................................................................ 29

    Empirical Root Mean Squared Error ........................................................ 33

    Design Variability .......................................................................................... 40

    Theoretical Prediction Variance................................................................ 41

    Prediction Performance ............................................................................ 47

    Empirical Root Mean Squared Error ....................................................... 48

    Conclusions ................................................................................................... 50

  • v

    CHAPTER Page

    4 BRIDGE DESIGN PROPERTIES ..................................................................... 51

    Methodology .................................................................................................. 51

    Results ............................................................................................................ 53

    Second-Order ............................................................................................. 53

    Third-Order ............................................................................................... 59

    Fourth-Order .............................................................................................64

    Fifth-Order .................................................................................................69

    Theoretical Properties Summary .............................................................. 70

    Empirical Model Fitting Results ............................................................... 71

    Conclusions .................................................................................................... 79

    5 AUGMENTED BRIDGE DESIGNS .................................................................. 80

    Methodology ................................................................................................. 80

    Results ........................................................................................................... 83

    Same-Order Augmentation ...................................................................... 84

    Augmentation With Higher Order Optimal Points .................................. 92

    Discussion ................................................................................................ 102

    Conclusions .................................................................................................. 103

    6 CONCLUSIONS AND FUTURE WORK ......................................................... 105

    Future Work ................................................................................................. 106

    REFERENCES ..................................................................................................................... 109

  • vi

    LIST OF TABLES

    Table Page

    1. Minimum number of design points needed (n = p) ......................................... 28

    2. Ranges and fixed values for the experimental and fixed variables in

    Test Function 3 .........