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Statistical Models in Environmental and Life Sciences University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 2006 Statistical models in environmental and life sciences Lakshminarayan Rajaram University of South Florida Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the American Studies Commons Scholar Commons Citation Rajaram, Lakshminarayan, "Statistical models in environmental and life sciences" (2006). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/2668 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Statistical Models in Environmental and Life Sciences by Lakshminarayan Rajaram A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: Chris P. Tsokos, Ph.D. Marcus Mcwaters, Ph.D. Kandethody Ramachandran, Ph.D. Geoffrey Okogbaa, Ph.D. Date of Approval: April 6, 2006 Keywords: trivariate normal, simulation, covariance structure, bioavailability, extreme value distribution © Copyright 2006 , Lakshminarayan Rajaram Dedication I dedicate this dissertation to my wife, Ha, to my son, Ganesh, and to my brother, Swamy. Especially to Swamy, who spent his entire life nurturing all of us, our parents and siblings, with his unflinching devotion and love. His loving words of encouragement from the depths of his magnanimous heart and that ever-present radiant smile have solely contributed to my commitment to complete the doctoral degree. Acknowledgments I take this opportunity to acknowledge a number of people who have helped me to succeed in accomplishing my goal. Firstly, I would like to express my profound appreciation to Dr. Chris P. Tsokos for suggesting the research problems, for his continual support and encouragement all through my graduate studies. I also want to thank the other members of my committee-Dr. Marcus McWaters, Dr. K. Ramachandran, and Dr. Okogbaa- for their guidance and support. Finally, I like to express a special note of thanks to Dr. A.N.V. Rao for all his encouragement. Table of Contents List of Tables v List of Figure vii Abstract xi Chapter One Review of Literature and Statistical Modeling Using Extreme Value Distributions 1.0 Introduction and the Focus of Chapter One 1 1.1 Literature Review of Extreme Value Theory 3 1.2 Probability Framework for Extreme Value Theory 6 1.3 Generalized Extreme Value (GEV) Distribution 11 1.3.1 Probability Density Function and Maximum Likelihood 11 1.3.2 Quantile 13 1.3.3 Model Diagnostics 14 1.3.4 GEV Modeling of Non-stationary Processes 16 1.4 Gumbel Distribution 18 1.4.1 Probability Density Function and Maximum Likelihood 18 1.4.2 Quantile 20 1.4.3 Model Diagnostics 20 1.4.4 Statistical Modeling of Annual Maximum Discharge 21 1.5 Frechet Distribution 24 1.5.1 Probability Density Function and Maximum Likelihood 24 1.5.2 Quantile 25 1.5.3 Model Diagnostics 26 1.6 Weibull Distribution 27 1.6.1 Probability Density Function and Maximum Likelihood 27 1.6.2 Quantile 28 1.6.3 Model Diagnostics 29 1.6.4 Statistical Modeling of Annual Maximum Storm Surge 30 1.7 Generalized Pareto Distribution 32 1.7.1 Probability Density Function and Maximum Likelihood 33 1.7.2 Quantile 34 1.7.3 Model Diagnostics and Methods of Selecting a Threshold Value 36 1.7.4 Statistical Modeling of Daily Precipitation Data 37 1.8 Aims of the Present Research 41 i Chapter Two Statistical Modeling of Annual Monthly Maximum Rainfall Using the Generalized Extreme Value Distribution 2.0 Introduction 44 2.1 Focus of Chapter Two 45 2.1.1 Data 45 2.1.2 Methodology 48 2.1.3 Models 49 2.2 Results and Discussions 50 2.3 Similarity Profiles by Each of the Three Climatic Zones 57 2.4 Conclusion 64 Chapter Three Mixed Statistical Model for the Pharmacokinetic Parameter, Maximum Drug Concentration (Cmax): Small Samples 3.0 Introduction 65 3.1 Outline of Each Section of Chapter Three 66 3.2 Pharmacokinetics 66 3.2.1 Bioavailability and Assessment of Bioavailability 67 3.2.2 Introduction to Warfarin 68 3.2.3 Maximum Drug Concentration (Cmax) Data 70 3.3 Focus of the Chapter Three 71 3.4 Statistical Methods for Small Samples 72 3.4.1 Generalized Extreme Value and Gumbel Distributions 72 3.4.2 Weibull Distribution 72 3.4.3 Pareto Distribution 74 3.5 Mixture of Two Extreme Value Distributions 75 3.5.1 Introduction 75 3.5.2 Mixture of Two Gumbel Distributions 77 3.5.3 Mixture of Two Pareto Distributions 80 3.5.4 Mixture of Two Weibull Distributions 83 3.6 Results of the Modeling on Sample Sizes 50 and 100 88 3.6.1 Mixture of Two Pareto Distributions 88 3.6.2 Mixture of Two Weibull Distributions 89 3.7 Summary and Conclusions 95 Chapter Four Statistical Model for the Pharmacokinetic Parameter, Maximum Drug Concentration (Cmax): Large Samples 4.0 Introduction 97 4.1 Outline of Each Section of Chapter Four 97 4.2 Generalized Extreme Value (GEV) Distribution 98 4.2.1 Derivations of Normal Equations 100 4.2.2 Validity of the GEV Model 101 4.3 Gumbel Distribution (Type I Extreme Value Distribution) 102 4.3.1 Derivations of Normal Equations 102 4.3.2 Validity of the Gumbel Model 103 ii 4.4 Weibull Distribution (Type III Extreme Value Distribution) 104 4.4.1 Derivations of Normal Equations 104 4.4.2 Validity of the Weibull Model 105 4.5 Generalized Pareto Distribution 105 4.6 Results of the Modeling on Large Samples 107 4.6.1 Assessment of Generalized Extreme Value Model 107 4.6.2 Assessment of Gumbel Model 110 4.6.3 Assessment of Weibull Model 111 4.6.4 Assessment of Generalized Pareto Model 111 4.7 Return Level Profiles Based on the Quartiles 113 4.8 Summary and Conclusions 114 Chapter Five Statistical Modeling of a Pharmacokinetic System 5.0 Introduction 116 5.1 Focus of the Chapter Five 119 5.2 A Pharmacokinetic Model for the Antibiotic Drug, Coumermycin A1 120 5.3 Simulation of Rate Constants Using Trivariate Normal Distributions 124 5.4 Discussion of the Results 128 5.4.1 Effects of Varying the Values of the Standard Deviation and the Correlation Structure for Studying the Behaviors of x1(t), x2(t) and x3(t) 129 5.4.2 Summary of the Effects of the Standard Deviation and the Correlation Structure on the Behaviors of x1(t), x2(t) and x3(t) 130 5.5 Stochastic Behavior of the Drug Concentration in Each Compartment 136 5.5.1 Effects of Varying the Values of the Standard Deviation and the Correlation Structure on the Deterministic (xi(t)) and Stochastic (E[xi(t)]) behaviors of the Individual Compartments 136 5.5.2 Summary of the Effects of the Standard deviation and the Correlation Structure on the Deterministic and the Stochastic Characterizations 137 5.6 Simulation of Rate Constants Using Trivariate Exponential Distribution 142 5.6.1 Introduction 142 5.6.2 Discussion of Results 144 5.7 Conclusion 148 Chapter Six Statistical Modeling of a Pharmacokinetic System Using a System of Delay Random Differential Equations 6.0 Introduction 149 6.1 Focus of Chapter Six 151 6.2 Pharmacokinetic Model 154 6.3 Discussion of the Results 156 6.3.1 Overall Behavior of the Drug Concentration in all Three Compartments 156 6.3.2 Comparison of Deterministic Behavior Between Delay and Non-delay Models 158 iii 6.3.3 Comparison of Deterministic and Stochastic behaviors of the Delay Models 163 6.4 Conclusion 168 Chapter Seven Future Research Studies 7.0 Possible Extensions of the Present Research Studies 169 References 171 Bibliography 179 About the author End Page iv List of Tables Table 1.6.1 Estimates of shape and scale parameters with their standard errors from 3 from 3 and 2-parameter Weibull fits on the data set containing 111 annual maximum storm surge. 31 Table 1.7.1 Estimates of scale and shape parameters with standard errors from the generalized Pareto model on the Fort Collins precipitation data. 38 Table 1.7.2 95% Confidence interval estimation using profile likelihood for the GPD shape parameter and 100-year return level. 38 Table 2.1 Tabulation of the location, climate zone (central, north, or south), years of data, latitude and longitude for all forty four locations. 46 Table 2.2 Summary of the stationary or non-stationary form of the maximum rainfall data for 56 years of data for the forty four locations. 52 Table 2.3 Return levels of the extreme rainfall with profile deviance of quantiles. 54 Table 3.1 Maximum likelihood estimators of the parameters with standard errors from the mixture of two Gumbel models fit to the mixture of two Gumbel data sets simulated using n = 1000,μ1 = 200, σ 1= σ 2 = 40, μ2 = 200*d , p = 0.40 . 78 Table 3.2 Maximum likelihood estimators of the parameters with standard errors from the mixture of two Gumbel models fit to the mixture of two Gumbel data sets simulated using n = 500,μ1 = 200, σ 1= σ 2 = 40, μ2 = 200*d , p = 0.40 . 79 Table 3.3 Maximum likelihood estimators of the parameters with standard errors from the mixture of two Gumbel models fit to the mixture of two Gumbel data sets simulated using n = 500,μ1 = 100, σ 1=σ 2 = 40, μ2 = 200*d , p = 0.40 .
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