Introduction to Probability and Statistics for Ecosystem Managers

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.

“Ultimately, the ecosystem manager who works through this volume and works through the many examples provided will emerge with a valuable toolbox of analysis techniques.”  (The Quarterly Review of Biology, 1 December 2015)

• List of figures xiiiList of tables xviiPreface xixAcknowledgments xxiList of abbreviations xxiii1 Introduction 11.1 The textbook’s purpose 11.1.1 The textbook’s focus on ecosystem management 21.1.2 Reader level, prerequisites, and typical reader jobs 31.2 The textbook’s pedagogical approach 41.2.1 General points 41.2.2 Use of this textbook for self-study 41.2.3 Learning resources 51.3 Chapter summaries 71.4 Installing and running R Commander 91.4.1 Running R 91.4.2 Starting an R Commander session 91.4.3 Terminating an R Commander session 101.5 Introductory R Commander session 101.6 Teaching probability through simulation 131.6.1 The frequentist statistical inference paradigm 141.7 Summary 152 Probability and simulation 172.1 Introduction 172.2 Basic probability 172.2.1 Definitions 172.2.2 Independence 202.3 Random variables 222.3.1 Definitions 222.3.2 Simulating random variables 262.3.3 A random variable’s expected value (mean) and variance 262.3.4 Details of the normal (Gaussian) distribution 282.3.5 Distribution approximations 302.4 Joint distributions 312.4.1 Definition 312.4.2 Mixed variables 322.4.3 Marginal distribution 322.4.4 Conditional distributions 332.4.5 Independent random variables 342.5 Influence diagrams 342.5.1 Definitions 342.5.2 Example of a Bayesian network in ecosystem management 362.5.3 Modeling causal relationships with an influence diagram 382.6 Advantages of influence diagrams in ecosystem management 402.7 Two ecosystem management Bayesian networks 412.7.1 Waterbody eutrophication 412.7.2 Wildlife population viability 412.8 Influence diagram sensitivity analysis 412.9 Drawbacks to influence diagrams 423 Application of probability: Models of political decision making in ecosystem management 433.1 Introduction 433.2 Influence diagram models of decision making 433.2.1 Ecosystem status perception nodes 443.2.2 Image nodes 443.2.3 Economic, militaristic, and institutional goal nodes 453.2.4 Audience effect nodes 453.2.5 Resource nodes 463.2.6 Action and target nodes 463.2.7 Overall goal attainment node 473.2.8 How a group influence diagram reaches a decision 473.2.9 An advantage of this decision-making architecture 473.2.10 Evaluation dimensions 473.3 Rhino poachers: A simplified model 503.4 Policymakers: A simplified model 573.5 Conclusions 594 Statistical inference I: Basic ideas and parameter estimation 614.1 Definitions of some fundamental terms 614.2 Estimating the PDF and CDF 624.2.1 Histograms 624.2.2 Ogive 644.3 Measures of central tendency and dispersion 644.4 Sample quantiles 654.4.1 Sample quartiles 654.4.2 Sample deciles and percentiles 654.5 Distribution of a statistic 654.5.1 Basic setup in statistics 654.5.2 Sampling distributions 664.5.3 Normal quantile–quantile plot 664.6 The central limit theorem 684.7 Parameter estimation 684.7.1 Bias, variance, and efficiency 694.8 Interval estimates 704.8.1 A confidence interval for μ when σ2 is known 704.9 Basic regression analysis 714.9.1 Definitions and fundamental characteristics 714.9.2 The regression model 724.9.3 Correlation 744.9.4 Sampling distributions 754.9.5 Prediction and estimation 764.9.6 Misuse of regression models 764.10 General methods of parameter estimation 794.10.1 Maximum likelihood 794.10.2 Minimum Hellinger distance 804.10.3 Consistency analysis 805 Statistical inference II: Hypothesis tests 835.1 Introduction 835.2 Hypothesis tests: General definitions and properties 835.2.1 Definitions and procedure 835.2.2 Confidence intervals and hypothesis tests 855.2.3 Types of mistakes 855.2.4 One way to set the test’s level 865.2.5 The z -test for hypotheses about μ 895.2.6 p-Values 915.3 Power 925.3.1 Power curves 935.4 t-Tests and a test for equal variances 955.4.1 The t -test 955.4.2 Two-sample t -tests 955.4.3 Tests for paired data 965.4.4 Testing for equal variances 985.5 Hypothesis tests on the regression model 985.5.1 Prediction and estimation confidence intervals 1035.5.2 Multiple regression 1045.5.3 Original scale prediction in regression 1065.6 Brief introduction to vectors and matrices 1065.6.1 Basic definitions 1065.6.2 Inverse of a matrix 1085.6.3 Random vectors and random matrices 1085.7 Matrix form of multiple regression 1095.7.1 Generalized least squares 1115.8 Hypothesis testing with the delete-d jackknife 1115.8.1 Background 1115.8.2 A one-sample delete-d jackknife test 1115.8.3 Testing classifier error rates 1145.8.4 Important points about this test 1155.8.5 Parameter confidence intervals 1156 Introduction to spatial statistics 1176.1 Overview 1176.1.1 Types of spatial processes 1186.2 Spatial statistics and GIS 1186.2.1 Types of spatial data 1186.3 QGIS 1216.3.1 Capabilities 1226.3.2 Installing QGIS 1226.3.3 Documentation and tutorials 1226.3.4 Installing plugins 1236.3.5 How to convert a text file to a shapefile 1236.4 Continuous spatial processes 1256.4.1 Definitions 1256.4.2 Graphical tools for exploring continuous spatial data 1276.4.3 Third- and fourth-order cumulant minimization 1326.4.4 Best linear unbiased predictor 1326.4.5 Kriging variance 1346.4.6 Model-fitting diagnostics 1366.4.7 Kriging within a window 1376.5 Spatial point processes 1386.5.1 Definitions 1386.5.2 Marked spatial point processes 1496.5.3 Conclusions 1506.6 Continuously valued multivariate processes 1516.6.1 Fitting multivariate covariance functions 1516.6.2 Cokriging: The MWRCK procedure 1557 Introduction to spatio-temporal statistics 1597.1 Introduction 1597.2 Representing time in a GIS 1597.2.1 The QGIS Time Manager plugin 1607.2.2 A Clifford algebra-based spatio-temporal data structure 1637.2.3 A raster- and event-based spatio-temporal data model 1637.2.4 Application of ESTDM to a land cover study 1667.3 Spatio-temporal prediction: MCSTK 1667.3.1 Algorithms 1667.3.2 Covariogram model and its estimator 1697.4 Multivariate processes 1747.4.1 Definitions 1757.4.2 Transformations 1757.4.3 Covariograms and cross-covariograms 1807.4.4 Parameter estimation 1817.4.5 Prediction algorithms 1827.4.6 Cross-validation 1837.4.7 Summary 1907.5 Spatio-temporal point processes 1907.6 Marked spatio-temporal point processes 1957.6.1 A mark semivariogram estimator 1968 Application of statistical inference: Estimating the parameters of an individual-based model 1998.1 Overview 1998.2 A simple IBM and its estimation 2008.2.1 Simple IBM 2008.2.2 Parameter estimation 2018.3 Fitting IBMs with MSHD 2048.3.1 Ergodicity 2068.3.2 Observable random variables from IBM output 2078.4 Further properties of parameter estimators 2078.4.1 Consistency 2078.4.2 Robustness 2088.5 Parameter confidence intervals for a nonergodic model 2098.6 Rhino-supporting ecosystem influence diagram 2098.6.1 Spatial effects on poaching 2108.6.2 IBM variables 2138.6.3 Initial conditions and hypothesis values of parameters 2148.6.4 Mapping functions 2158.6.5 Realism of ecosystem influence diagram output 2178.7 Estimation of rhino IBM parameters 2198.7.1 Parameter confidence intervals 2209 Guiding an influence diagram’s learning 2239.1 Introduction 2239.2 Online learning of Bayesian network parameters 2249.2.1 Basic algorithm using simulation 2249.2.2 Updating influence diagrams 2259.3 Learning an influence diagram’s structure 2299.3.1 Minimum description length score function 2299.3.2 Description length of an edge 2299.3.3 Random generation of DAGs 2309.3.4 Algorithm to detect and delete cycles 2309.3.5 Mutate functions 2319.3.6 MDLEP algorithm 2329.3.7 Using MDLEP to learn influence diagram structure 2329.4 Feedback-based learning for group decision-making diagrams 2339.4.1 Definitions and algorithm 2339.5 Summary and conclusions 23410 Fitting and testing a political–ecological simulator 23510.1 Introduction 23510.1.1 Background on rhino poaching 23610.1.2 Scenarios wherein rhino poaching is reduced 23710.2 EMT simulator construction 23710.2.1 Modeled groups 23710.2.2 Rhino-supporting ecosystem influence diagram 24810.3 Consistency analysis estimates of simulator parameters 24810.4 MPEMP computation 25110.4.1 Setup 25110.4.2 Solution 25310.5 Conclusions 254Appendix 257Simpson’s rule in two dimensions 257References 263Index 275