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Optimal Design of Experiments for Functional Responses

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Optimal Design of Experiments for Functional Responses Optimal Design of Experiments for Functional Responses by Moein Saleh A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved November 2015 by the Graduate Supervisory Committee: Rong Pan, Chair Douglas C. Montgomery George Runger Ming-Hung Kao ARIZONA STATE UNIVERSITY December 2015 ABSTRACT Functional or dynamic responses are prevalent in experiments in the fields of engi- neering, medicine, and the sciences, but proposals for optimal designs are still sparse for this type of response. Experiments with dynamic responses result in multiple re- sponses taken over a spectrum variable, so the design matrix for a dynamic response have more complicated structures. In the literature, the optimal design problem for some functional responses has been solved using genetic algorithm (GA) and ap- proximate design methods. The goal of this dissertation is to develop fast computer algorithms for calculating exact D-optimal designs. First, we demonstrated how the traditional exchange methods could be improved to generate a computationally efficient algorithm for finding G-optimal designs. The proposed two-stage algorithm, which is called the cCEA, uses a clustering-based ap- proach to restrict the set of possible candidates for PEA, and then improves the G-efficiency using CEA. The second major contribution of this dissertation is the development of fast algorithms for constructing D-optimal designs that determine the optimal sequence of stimuli in fMRI studies. The update formula for the determinant of the information matrix was improved by exploiting the sparseness of the information matrix, leading to faster computation times. The proposed algorithm outperforms genetic algorithm with respect to computational efficiency and D-efficiency. The third contribution is a study of optimal experimental designs for more general functional response models. First, the B-spline system is proposed to be used as the non-parametric smoother of response function and an algorithm is developed to determine D-optimal sampling points of a spectrum variable. Second, we proposed a two-step algorithm for finding the optimal design for both sampling points and experimental settings. In the first step, the matrix of experimental settings is held i fixed while the algorithm optimizes the determinant of the information matrix for a mixed effects model to find the optimal sampling times. In the second step, the optimal sampling times obtained from the first step is held fixed while the algorithm iterates on the information matrix to find the optimal experimental settings. The designs constructed by this approach yield superior performance over other designs found in literature. ii To my parents - Mrs. Fatemeh Shariati and Mr. Jaafar Saleh, who missed us dearly in their empty nest these passed years, and to all my teachers. iii ACKNOWLEDGEMENTS First, I would like to extend my deepest gratitude to my advisor, Dr. Rong Pan, for his guidance and support during my PhD. His vision, knowledge, and encouragement always inspired me and helped me stay focused and pursue my objectives. I am for- ever grateful for his generosity. I would like to extend a very special thanks to Dr. Ming-Hung Kao for providing me with the opportunity to work on a research under his supervision. Without him, the research would not have been as successful or as enjoyable. He was the constant source of support throughout and provided me with immeasurable encouragement and guidence. Dr. Douglas C. Montgomery is also highly deserving of my gratitude for his extensive support during my PhD. The time and feedback he generously shared with me during the research made it possible to reach the conclusion of the study. I would like to thank Dr. George Runger for partaking his time and effort as my committee member to help me fulfill the degree requirements. I am heartily thankful to Dr. Bingrong Jiang, Dr. Glade Erikson, Jon Ruterman, Vincent He, and Anand Narasimhan who provided me with the opportunity to learn and grow in the Discover Financial Services and also continue to work on this research work. This journey would not have been the same without the immeasurable help and sup- port of my close friends: Tina Pourshams, Mickey Mancenido, Petek Yontay, Nima Tajbakhsh, Hossein Vashani. Thank you all for being there every step of the way. Last, but not the least, I want to thank my loving family: my mother, Fatemeh Shariati, my father, Jaafar Salehm and my big brother, Mehdi Saleh, for their un- conditional love, support, and care. Without them, I would never have been able to complete this, or any other endeavor in life. iv TABLE OF CONTENTS Page LIST OF TABLES . vii LIST OF FIGURES . ix CHAPTER 1 INTRODUCTION . 1 2 Functional Data Analysis Models . 8 2.1 Review of FDA Approaches . 10 2.1.1 Smoothing Techniques for Analyzing Functional Data . 11 2.1.2 Mixed Effects Model . 17 2.1.3 Varying Coefficient Model . 24 2.2 Model Comparison . 29 2.3 Conclusion . 35 3 Optimal Experimental Designs for Linear Models . 37 3.1 Alphabetic Optimality Criteria . 39 3.2 Exchange Algorithms for Finding D, A and I Optimality . 40 3.3 An Algorithm for Finding Exact G-Optimal Designs . 45 3.3.1 The Clustering-Based Coordinate Exchange Algorithm (cCEA) 47 3.4 Results . 55 3.5 Conclusion . 61 4 Optimal Experimental Designs for fMRI Experiments . 64 4.1 Experimental Design of fMRI . 66 4.2 Linear Models for Estimation and Detection Problems . 69 4.3 Design Matrix Construction for fMRI Problem . 71 4.4 Proposed Algorithms . 76 4.4.1 The Estimation Problem . 76 v CHAPTER Page 4.4.2 The Detection Problem . 79 4.4.3 Multi-Objective Optimization . 81 4.5 Comparing CEA and GA for fMRI Experimental Design Problems . 82 4.5.1 Performance of the Estimation Algorithm in fMRI Experi- ments................................................... 82 4.5.2 Performance of the Detection Algorithm in fMRI Experiments 85 4.6 Conclusion . 85 5 Optimal Experimental Designs for Dynamic Responses . 89 5.1 Introduction . 91 5.2 Regression Models for Dynamic Responses . 94 5.2.1 B-Spline Model . 95 5.2.2 Mixed Effects Model . 98 5.3 Optimal Sampling Times for Dynamic Response . 101 5.3.1 D-Optimal Design of Sampling Time . 102 5.3.2 Algorithm for Finding D-Optimal Sampling Times . 106 5.3.3 Robust Sampling Plans . 108 5.4 Optimal Design of Experiments with Dynamic Responses . 109 5.4.1 The Two-Step Approach . 110 5.4.2 Two Engineering Examples . 113 5.5 Conclusion . 115 6 Summary and Conclusion . 118 REFERENCES . 123 vi LIST OF TABLES Table Page 3.1 G-Optimal Designs for Experiments with Two Factors . 56 3.2 G-Optimal Designs for Experiments with Three Factors. 57 3.3 G-Optimal Designs for Experiments with Four Factors . 58 3.4 G-Optimal Designs for Experiments with Five Factors . 59 3.5 Compare the G-Efficiencies of Optimal Designs Generated by Different Algorithms. 62 3.6 Compare the Computation Times (in Seconds) of the cCEA with Ro- driguez et apl.'s Algorithm with One Run. 63 4.1 Comparing the Determinants of the Designs Obtained by Genetic Algo- rithm and Coordinate Exchange Algorithm for the Estimation Problem with τISI = 2 and τTR =2.......................................... 83 4.2 Comparing the Determinants of the Designs Obtained by Genetic Algo- rithm and Coordinate Exchange Algorithm for the Estimation Problem with τISI = 3 and τTR =2.......................................... 84 4.3 Comparing the Determinants of the Designs Obtained by Genetic Algo- rithm and Coordinate Exchange Algorithm for the Detection Problem with τISI = 2 and τTR =2.......................................... 85 4.4 Comparing the Determinants of the Designs Obtained by Genetic Algo- rithm and Coordinate Exchange Algorithm for the Detection Problem with τISI = 2 and τTR =2.......................................... 86 5.1 Optimal Sampling Times for an Order-4 B-Spline Basis System . 111 5.2 Optimal Experimental Designs of an Order-4 B-Spline System with 3 Experimental Factors and 55 Experimental Units . 112 vii Table Page 5.3 Determinants of the Information Matrices of Experimental Designs De- rived from Three Approaches. 113 5.4 Determinants of the Information Matrix from the Optimal Design, Standard Design, and Engineer Suggested Design. 114 5.5 Two Design Alternatives for the Engineer-Mapping Example . 116 viii LIST OF FIGURES Figure Page 2.1 A Set of Five Fourier Bases Defined on the Range [0,1] with Period Equals to 0.5 . 13 2.2 A Set of Six B-Spline Bases Defined on the Range [0,1] with the Knots Located at τ = f0; 0:3; 0:6; 0:9; 1g.................................... 14 2.3 Order Four B-Spline Bases System with 7,8,9 and 10 Bases Func- tions Which Are Derived from the Knot Points That Are Located at f0,0.3,0.6,0.9,1g, f0,0.3,0.6,0.6,0.9,1g, f0,0.3,0.6,0.6,0.6,0.9,1g and f0,0.3,0.6,0.6,0.6,0.6,0.9,1g, Respectively from Left to Right. 16 2.4 The Accumulated HRF of Three HRFs That Correspond to a Design Sequence D = f1101:::g with τISI = 3 and τTR = 2. The Accumulated HRF, Drawn by a Blue Solid Curve, Is Equal to the Summation of HRFs, Drawn by Black Dotted Curves. 19 2.5 108 Profile Curves Derived from the Experiments Conducted for De- signing the Electrical Alternator. The Data Is Available for x1 = 1375, x2 = 1500, x3 = 1750, x4 = 2000, x5 = 2500, x6 = 3500 and x7 = 5000 Which Are Shown by Circles. These Points Are Connected by Straight Lines. 30 2.6 B-Spline Bases Used for the Mixed Effects and the Varying Coefficient Models.
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