Introduction to Probability and Statistics for Eco–system Managers – Simulation and Resampling

Timothy C. Haas, Lubar School of Business Administration, University of Wisconsin at Milwaukee. Timothy Haas is involved in teaching undergraduate and graduate courses in statistical methods, pursuing decision making and environmental statistics re?search, and collaborating with faculty on application of statistics to Marketing and Eco?nomics.

Preface List of Abbreviations List of Tables List of Figures 1 Introduction 1.1 This Textbook's Purpose 1.1.1 The Textbook's Focus on Ecosystem Management 1.1.2 Reader Level, Prerequisites, and Typical Reader Jobs 1.2 This Textbook's Pedagogical Approach 1.2.1 General Points 1.2.2 Use of This Textbook for Self-Study 1.2.3 Learning Resources 1.3 Chapter Summaries 1.4 Installing and Running R Commander 1.4.1 Running R 1.4.2 Starting an R Commander Session 1.4.3 Terminating an R Commander Session 1.5 Introductory R Commander Session 1.6 Teaching Probability Through Simulation 1.6.1 The Frequentist Statistical Inference Paradigm 1.7 Summary 2 Probability and Simulation 2.1 Introduction 2.2 Basic Probability 2.2.1 Definitions 2.2.2 Independence 2.3 Random Variables 2.3.1 Definitions 2.3.2 Simulating Random Variables 2.3.3 A Random Variable's Expected Value (Mean) and Variance 2.3.4 Details of the Normal (Gaussian) Distribution 2.3.5 Distribution Approximations 2.4 Joint Distributions 2.4.1 Definition 2.4.2 Mixed Variables 2.4.3 Marginal Distribution 2.4.4 Conditional Distributions 2.4.5 Independent Random Variables 2.5 Influence Diagrams 2.5.1 Definitions 2.5.2 Example of a Bayesian Network in Ecosystem Management 2.5.3 Modeling Causal Relationships With an Influence Diagram 2.6 Advantages of Influence Diagrams in Ecosystem Management 2.7 Two Ecosystem Management Bayesian networks 2.7.1 Waterbody Eutrophication 2.7.2 Wildlife Population Viability 2.8 Influence Diagram Sensitivity Analysis 2.9 Drawbacks to Influence Diagrams 3 Application of Probability: Models of Political Decision Making in Ecosystem Management 3.1 Introduction 3.2 Influence Diagram Models of Decision Making 3.2.1 Ecosystem Status Perception Nodes 3.2.2 Image Nodes 3.2.3 Economic, Militaristic, and Institutional Goal Nodes 3.2.4 Audience Effect Nodes 3.2.5 Resource Nodes 3.2.6 Action and Target Nodes 3.2.7 Overall Goal Attainment Node 3.2.8 How a Group Influence Diagram Reaches a Decision 3.2.9 An Advantage of This Decision Making Architecture 3.2.10 Evaluation Dimensions 3.3 Rhino Poachers: A Simplified Model 3.4 Policymakers: A Simplified Model 3.5 Conclusions 4 Statistical Inference I: Basic Ideas and Parameter Estimation 4.1 Definitions of Some Fundamental Terms 4.2 Estimating the PDF and CDF 4.2.1 Histograms 4.2.2 Ogive 4.3 Measures of Central Tendency and Dispersion 4.4 Sample Quantiles 4.4.1 Sample Quartiles 4.4.2 Sample Deciles and Percentiles 4.5 Distribution of a Statistic 4.5.1 Basic Setup in Statistics 4.5.2 Sampling Distributions 4.5.3 Normal Quantile-Quantile Plot 4.6 The Central Limit Theorem 4.7 Parameter Estimation 4.7.1 Bias, Variance, and Efficiency 4.8 Interval Estimates 4.8.1 A Confidence Interval for mu When sigma2 is Known 4.9 Basic Regression Analysis 4.9.1 Definitions and Fundamental Characteristics 4.9.2 The Regression Model 4.9.3 Correlation 4.9.4 Sampling Distributions 4.9.5 Prediction and Estimation 4.9.6 Misuse of Regression Models 4.10 General Methods of Parameter Estimation 4.10.1 Maximum Likelihood 4.10.2 Minimum Hellinger Distance 4.10.3 Consistency Analysis 5 Statistical Inference II: Hypothesis Tests 5.1 Introduction 5.2 Hypothesis Tests: General Definitions and Properties 5.2.1 Definitions and Procedure 5.2.2 Confidence Intervals and Hypothesis Tests 5.2.3 Types of Mistakes 5.2.4 One Way to Set the Test's Level 5.2.5 The z-Test for Hypotheses About mu 5.2.6 P-values 5.3 Power 5.3.1 Power Curves 5.4 t-Tests and a Test for Equal Variances 5.4.1 The t-Test 5.4.2 Two-Sample t-Tests 5.4.3 Tests for Paired Data 5.4.4 Testing for Equal Variances 5.5 Hypothesis Tests on the Regression Model 5.5.1 Prediction and Estimation Confidence Intervals 5.5.2 Multiple Regression 5.5.3 Original Scale Prediction in Regression 5.6 Brief Introduction to Vectors and Matrices 5.6.1 Basic Definitions 5.6.2 Inverse of a Matrix 5.6.3 Random Vectors and Random Matrices 5.7 Matrix Form of Multiple Regression 5.7.1 Generalized Least Squares 5.8 Hypothesis Testing with the delete-d Jackknife 5.8.1 Background 5.8.2 A One-Sample delete-d Jackknife Test 5.8.3 Testing Classifier Error Rates 5.8.4 Important Points About This Test 5.8.5 Parameter Confidence Intervals 6 Introduction to Spatial Statistics 6.1 Overview 6.1.1 Types of Spatial Processes 6.2 Spatial Statistics and GIS 6.2.1 Types of Spatial Data 6.3 QGIS 6.3.1 Capabilities 6.3.2 Installing QGIS 6.3.3 Documentation and Tutorials 6.3.4 Installing Plugins 6.3.5 How to Convert a Text File to a Shapefile 6.4 Continuous Spatial Processes 6.4.1 Definitions 6.4.2 Graphical Tools for Exploring Continuous Spatial Data 6.4.3 Third- and Fourth-Order Cumulant Minimization 6.4.4 Best Linear Unbiased Predictor 6.4.5 Kriging Variance 6.4.6 Model-Fitting Diagnostics 6.4.7 Kriging Within a Window 6.5 Spatial Point Processes 6.5.1 Definitions 6.5.2 Marked Spatial Point Processes 6.5.3 Conclusions 6.6 Continuously-Valued Multivariate Processes 6.6.1 Fitting Multivariate Covariance Functions 6.6.2 Cokriging: The MWRCK Procedure 7 Introduction to Spatio-Temporal Statistics 7.1 Introduction 7.2 Representing Time in a GIS 7.2.1 The QGIS Time Manager Plugin 7.2.2 A Clifford Algebra-based Spatio-Temporal Data Structure 7.2.3 A Raster- and Event-Based Spatio-Temporal Data Model 7.2.4 Application of ESTDM to a Land Cover Study 7.3 Spatio-Temporal Prediction: MCSTK 7.3.1 Algorithms 7.3.2 Covariogram Model and its Estimator 7.4 Multivariate Processes 7.4.1 Definitions 7.4.2 Transformations 7.4.3 Covariograms and Cross-covariograms 7.4.4 Parameter Estimation 7.4.5 Prediction Algorithms 7.4.6 Cross-validation 7.4.7 Summary 7.5 Spatio-Temporal Point Processes 7.6 Marked Spatio-Temporal Point Processes 7.6.1 A Mark Semivariogram Estimator 8 Application of Statistical Inference: Estimating the Parameters of an Individual- Based Model 8.1 Overview. 8.2 A Simple IBM and its Estimation 8.2.1 Simple IBM 8.3 Fitting IBMs with MSHD 8.3.1 Ergodicity 8.3.2 Observable Random Variables from IBM Output 8.4 Further Properties of Parameter Estimators 8.4.1 Consistency 8.4.2 Robustness 8.5 Parameter Confidence Intervals for a Nonergodic Model 8.6 Rhino-Supporting Ecosystem Influence Diagram 8.6.1 Spatial Effects on Poaching 8.6.2 IBM Variables 8.6.3 Initial Conditions and Hypothesis Values of Parameters 8.6.4 Mapping Functions 8.6.5 Realism of Ecosystem Influence Diagram Output 8.7 Estimation of Rhino IBM Parameters 8.7.1 Parameter Confidence Intervals 9 Guiding an Inuence Diagram's Learning 9.1 Introduction 9.2 Online Learning of Bayesian Network Parameters 9.2.1 Basic Algorithm Using Simulation 9.2.2 Updating Influence Diagrams 9.3 Learning an Influence Diagram's Structure 9.3.1 Minimum Description Length Score Function 9.3.2 Description Length of an Edge 9.3.3 Random Generation of DAGs 9.3.4 Algorithm to Detect and Delete Cycles 9.3.5 Mutate Functions 9.3.6 MDLEP Algorithm 9.3.7 Using MDLEP to Learn Influence Diagram Structure 9.4 Feedback-Based Learning for Group Decision Making Diagrams 9.4.1 Definitions and Algorithm 9.5 Summary and Conclusions 10 Fitting and Testing a Political-Ecological Simulator 10.1 Introduction 10.1.1 Background on Rhino Poaching 10.1.2 Scenarios Wherein Rhino Poaching is Reduced 10.2 EMT Simulator Construction 10.2.1 Modeled Groups 10.2.2 Rhino-Supporting Ecosystem Influence Diagram 10.3 Consistency Analysis Estimates of Simulator Parameters 10.4 MPEMP Computation 10.4.1 Setup 10.4.2 Solution 10.5 Conclusions Appendix A Simpson's Rule in Two Dimensions References Index