Data Analysis and Statistics for Geography, Environmental Science, and Engineering

v

Contents

Preface..............................................................................................................................................xv

Acknowledgments ........................................................................................................................... xix

Author .............................................................................................................................................xxi

PART I Introduction to Probability, Statistics, Time Series,

and Spatial Analysis

Chapter 1 Introduction ..................................................................................................................3

1.1 Brief History of Statistical and Probabilistic Analysis .....................................3

1.2 Computers ..........................................................................................................4

1.3 Applications .......................................................................................................4

1.4 Types of Variables .............................................................................................4

1.4.1 Discrete.................................................................................................5

1.4.2 Continuous............................................................................................5

1.4.3 Discretization .......................................................................................5

1.4.4 Independent vs. Dependent Variables ..................................................6

1.5 Probability Theory and Random Variables .......................................................6

1.6 Methodology ......................................................................................................6

1.7 Descriptive Statistics .........................................................................................7

1.8 Inferential Statistics ........................................................................................... 7

1.9 Predictors, Models, and Regression ..................................................................7

1.10 Time Series ........................................................................................................ 8

1.11 Spatial Data Analysis ........................................................................................8

1.12 Matrices and Multiple Dimensions ...................................................................8

1.13 Other Approaches: Process-Based Models .......................................................9

1.14 Baby Steps: Calculations and Graphs ................................................................9

1.14.1 Mean, Variance, and Standard Deviation of a Sample ........................9

1.14.2 Simple Graphs as Text: Stem-and-Leaf Plots ..................................... 10

1.14.3 Histograms ......................................................................................... 11

1.15 Exercises .......................................................................................................... 11

1.16 Computer Session: Introduction to R .............................................................. 11

1.16.1 Working Directory ............................................................................ 11

1.16.2 Installing R ........................................................................................ 11

1.16.3 Personalize the R GUI Shortcut ........................................................ 11

1.16.4 Running R..........................................................................................13

1.16.5 Basic R Skills .................................................................................... 13

1.16.6 R Console .......................................................................................... 15

1.16.7 Scripts ................................................................................................15

1.16.8 Graphics Device ................................................................................ 16

1.16.9 Downloading Data Files .................................................................... 17

1.16.10 Read a Simple Text Data File ............................................................ 17

1.16.11 Simple Statistics ................................................................................19

1.16.12 Simple Graphs as Text: Stem-and-Leaf Plots ....................................20

vi Contents

1.16.13 Simple Graphs to a Graphics Window ..............................................20

1.16.14 Addressing Entries of an Array ......................................................... 20

1.16.15 Example: Salinity ..............................................................................22

1.16.16 CSV Text Files ...................................................................................23

1.16.17 Store Your Data Files and Objects ....................................................24

1.16.18 Command History and Long Sequences of Commands ...................25

1.16.19 Editing Data in Objects .....................................................................25

1.16.20 Cleanup and Close R Session ............................................................26

1.16.21 Computer Exercises ........................................................................... 26

Supplementary Reading .............................................................................................27

Chapter 2 Probability Theory .....................................................................................................29

2.1 Events and Probabilities ..................................................................................29

2.2 Algebra of Events ............................................................................................29

2.3 Combinations ................................................................................................... 31

2.4 Probability Trees .............................................................................................32

2.5 Conditional Probability ................................................................................... 33

2.6 Testing Water Quality: False Negative and False Positive ..............................34

2.7 Bayes’ Theorem ...............................................................................................35

2.8 Generalization of Bayes’ Rule to Many Events ...............................................36

2.9 Bio-Sensing .....................................................................................................36

2.10 Decision Making .............................................................................................37

2.11 Exercises ..........................................................................................................39

2.12 Computer Session: Introduction to Rcmdr, Programming,

andMultiplePlots ............................................................................................40

2.12.1 R Commander....................................................................................40

2.12.2 Package Installation and Loading .....................................................40

2.12.3 R GUI SDI Option: Best for R Commander ...................................... 43

2.12.4 How to Import a Text Data File Using Rcmdr .................................. 43

2.12.5 Simple Graphs on a Text Window ..................................................... 45

2.12.6 Simple Graphs on a Graphics Window: Histograms .........................46

2.12.7 More than One Variable: Reading Files and Plot Variables ..............47

2.12.7.1 Using the R Console ...........................................................48

2.12.7.2 Using the R Commander .................................................... 51

2.12.8 Programming Loops .......................................................................... 53

2.12.9 Application: Bayes’ Theorem ............................................................ 54

2.12.10 Application: Decision Making ..........................................................55

2.12.11 More on Graphics Windows ..............................................................55

2.12.12 Editing Data in Objects .....................................................................56

2.12.13 Clean Up and Exit .............................................................................56

2.12.14 Additional GUIs to Use R .................................................................57

2.12.15 Modifying the R Commander ........................................................... 57

2.12.16 Other Packages to Be Used in the Book ............................................57

2.12.17 Computer Exercises ...........................................................................58

Supplementary Reading .............................................................................................58

Chapter 3 Random Variables, Distributions, Moments, and Statistics ....................................... 59

3.1 Random Variables ...........................................................................................59

3.2 Distributions ....................................................................................................59

viiContents

3.2.1 Probability Mass and Density Functions (pmf and pdf) ....................59

3.2.2 Cumulative Functions (cmf and cdf) ..................................................62

3.2.3 Histograms .........................................................................................62

3.3 Moments .......................................................................................................... 63

3.3.1 First Moment or Mean ........................................................................63

3.3.2 Second Central Moment or Variance .................................................64

3.3.3 Population and Sample ....................................................................... 66

3.3.4 Other Statistics and Ways of Characterizing a Sample ......................67

3.4 Some Important RV and Distributions ............................................................ 68

3.5 Application Examples: Species Diversity ........................................................72

3.6 Central Limit Theorem ....................................................................................72

3.7 Random Number Generation ...........................................................................73

3.8 Exercises .......................................................................................................... 74

3.9 Computer Session: Probability and Descriptive Statistics ..............................75

3.9.1 Descriptive Statistics: Categorical Data, Table, and Pie Chart .......... 75

3.9.2 Using a Previously Generated Object or a Dataset ............................78

3.9.3 Summary of Descriptive Statistics and Histogram ............................ 78

3.9.4 Density Approximation ...................................................................... 81

3.9.5 Theoretical Distribution: Example Binomial Distribution ................82

3.9.6 Application Example: Species Diversity ............................................ 86

3.9.7 Random Number Generation .............................................................86

3.9.8 Comparing Sample and Theoretical Distributions:

ExampleBinomial ..............................................................................89

3.9.9 Programming Application: Central Limit Theorem ..........................90

3.9.10 Sampling: Function Sample ...............................................................92

3.9.11 Cleanup and Close R Session ............................................................. 92

3.9.12 Computer Exercises ............................................................................93

Supplementary Reading .............................................................................................93

Chapter 4 Exploratory Analysis and Introduction to Inferential Statistics .................................95

4.1 Exploratory Data Analysis (EDA) ...................................................................95

4.1.1 Index Plot ...........................................................................................95

4.1.2 Boxplot ...............................................................................................95

4.1.3 Empirical Cumulative Distribution Function (ecdf) .......................... 96

4.1.4 Quantile–Quantile (q – q) Plots ..........................................................98

4.1.5 Combining Plots for Exploratory Data Analysis (EDA) .................... 98

4.2 Relationships: Covariance and Correlation .....................................................98

4.2.1 Serial Data: Time Series and Autocorrelation ................................. 101

4.3 Statistical Inference ....................................................................................... 102

4.3.1 Hypothesis Testing ........................................................................... 103

4.3.2 p-Value .............................................................................................. 105

4.3.3 Power ................................................................................................ 105

4.3.4 Condence Intervals......................................................................... 107

4.4 Statistical Methods ........................................................................................ 109

4.5 Parametric Methods ...................................................................................... 110

4.5.1 Z Test or Standard Normal ............................................................... 110

4.5.2 The t-Test .......................................................................................... 110

4.5.3 The F Test ......................................................................................... 111

4.5.4 Correlation ........................................................................................ 112