Lab Manual for Models of Ecological Data Statistical Computation for Environmental Sciences in R

About this book

The environmental sciences are undergoing a revolution in the use of models and data. Facing ecological data sets of unprecedented size and complexity, environmental scientists are struggling to understand and exploit powerful new statistical tools for making sense of ecological processes. In "Models for Ecological Data", James Clark introduces ecologists to these modern methods in modeling and computation. Assuming only basic courses in calculus and statistics, the text introduces readers to basic maximum likelihood and then works up to more advanced topics in Bayesian modeling and computation. Clark covers both classical statistical approaches and powerful new computational tools and describes how complexity can motivate a shift from classical to Bayesian methods.

Through this lab manual, the book introduces readers to the practical work of data modeling and computation in the language R.

Based on a successful course at Duke University and National Science Foundation-funded institutes on hierarchical modeling, "Models for Ecological Data" will enable ecologists and other environmental scientists to develop useful models that make sense of ecological data. It features: consistent treatment from classical to modern Bayes; underlying distribution theory to algorithm development; many examples and applications; does not assume statistical background; extensive supporting appendixes; and, accompanying lab manual in R.

Contents

Preface ix Part I. Introduction 1 Chapter 1: Models in Context 3 1.1 Complexity and Obscurity in Nature and in Models 3 1.2 Making the Connections: Data, Inference, and Decision 5 1.3 Two Elements of Models: Known and Unknown 13 1.4 Learning with Models: Hypotheses and Quantification 19 1.5 Estimation versus Forward Simulation 23 1.6 Statistical Pragmatism 24 Chapter 2: Model Elements: Application to Population Growth 27 2.1 A Model and Data Example 27 2.2 Model State and Time 30 2.3 Stochasticity for the Unknown 42 2.4 Additional Background on Process Models 44 Part II. Elements of Inference 45 Chapter 3: Point Estimation: Maximum Likelihood and the Method of Moments 3.1 Introduction 47 3.2 Likelihood 47 3.3 A Binomial Model 53 3.4 Combining the Binomial and Exponential 54 3.5 Maximum Likelihood Estimates for the Normal Distribution 56 3.6 Population Growth 57 3.7 Application: Fecundity 60 3.8 Survival Analysis Using Maximum Likelihood 62 3.9 Design Matrixes 68 3.10 Numerical Methods for MLE 71 3.11 Moment Matching 71 3.12 Common Sampling Distributions and Dispersion 74 3.13 Assumptions and Next Steps 76 Chapter 4: Elements of the Bayesian Approach 77 4.1 The Bayesian Approach 78 4.2 The Normal Distribution 84 4.3 Subjective Probability and the Role of the Prior 91 Chapter 5: Confidence Envelopes and Prediction Intervals 93 5.1 Classical Interval Estimation 95 5.2 Bayesian Credible Intervals 115 5.3 Likelihood Profile for Multiple Parameters 120 5.4 Confidence Intervals for Several Parameters: Linear Regression 122 5.5 Which Confidence Envelope to Use 130 5.6 Predictive Intervals 133 5.7 Uncertainty and Variability 141 5.8 When Is It Bayesian? 142 Chapter 6: Model Assessment and Selection 143 6.1 Using Statistics to Evaluate Models 143 6.2 The Role of Hypothesis Tests 144 6.3 Nested Models 144 6.4 Additional Considerations for Classical Model Selection 151 6.5 Bayesian Model Assessment 154 6.6 Additional Thoughts on Bayesian Model Assessment 159 Part III. Larger Models 161 Chapter 7: Computational Bayes: Introduction to Tools Simulation 163 7.1 Simulation to Obtain the Posterior 163 7.2 Some Basic Simulation Techniques 164 7.3 Markov Chain Monte Carlo Simulation 173 7.4 Application: Bayesian Analysis for Regression 189 7.5 Using MCMC 202 7.6 Computation for Bayesian Model Selection 205 7.7 Priors on the Response 209 7.8 The Basics Are Now Behind Us 212 Chapter 8: A Closer Look at Hierarchical Structures 213 8.1 Hierarchical Models for Context 213 8.2 Mixed and Generalized Linear Models 216 8.3 Application: Growth Responses to CO2 230 8.4 Thinking Conditionally 235 8.5 Two Applications to Trees 241 8.6 Noninformative Priors in Hierarchical Settings 249 8.7 From Simple Models to Graphs 249 Part IV. More Advance Methods 251 Chapter 9: Time 9.1 Why Is Time Important? 253 9.2 Time Series Terminology 254 9.3 Descriptive Elements of Time Series Models 255 9.4 The Frequency Domain 264 9.5 Application: Detecting Density Dependence in Population Time Series 264 9.6 Bayesian State Space Models 272 9.7 Application: Black Noddy on Heron Island 282 9.8 Nonlinear State Space Models 289 9.9 Lags 297 9.10 Regime Change 298 9.11 Constraints on Time Series Data 300 9.12 Additional Sources of Variablity 301 9.13 Alternatives to the Gibbs Sampler 302 9.14 More on Longitudinal Data Structures 302 9.15 Intervention and Treatment Effects 309 9.16 Capture-Recapture Studies 318 9.17 Structured Models as Matrixes 329 9.18 Structure as Systems of Difference Equations 336 9.19 Time Series, Population Regulation, and Stochasticity 347 Chapter 10: Space-Time 353 10.1 A Deterministic Model for a Stochastic Spatial Process 354 10.2 Classical Inference on Population Movement 359 10.3 Island Biogeography and Metapopulations 378 10.4 Estimation of Passive Dispersal 388 10.5 A Bayesian Framework 397 10.6 Models for Explicit Space 401 10.7 Point-Referenced Data 403 10.8 Block-Referenced Data and Misalignment 412 10.9 Hierarchical Treatment of Space 415 10.10 Application: A Spatio-Temporal Model of Population Spread 424 10.11 How to Handle Space 432 Chapter 11: Some Concluding Perspectives 435 11.1 Models, Data, and Decision 435 11.2 The Promise of Graphical Models, Improved Algorithms, and Faster Computers 437 11.3 Predictions and What to Do with Them 444 11.4 Some Remarks on Software 456 Appendix A Taylor Series 457 Appendix B Some Notes on Differential and Difference Equations 464 Appendix C Basic Matrix Algebra 486 Appendix D Probability Models 502 Appendix E Basic Life History Calculations 541 Appendix F Common Distributions 573 Appendix G Common Conjugate Likelihood-Prior Pairs 583 References 585 Index 615

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Biography

James S. Clark is H. L. Blomquist Professor in the Nicholas School for the Environment, Professor of Statistics and Decisions Sciences, and Professor of Biology at Duke University. An award-winning researcher on the ecological impact of climate change, he is the author of more than one hundred papers for leading journals, including "Science, Nature, and Ecology". In 2005, he was elected to the American Academy of Arts and Sciences.