1 Introduction................................................................................................ 1
1.1 Why Numerical Ecology?................................................................... 1
1.2 Why R?............................................................................................... 2
1.3 Readership and Structure of the Book................................................ 2
1.4 How to Use This Book........................................................................ 3
1.5 The Data Sets...................................................................................... 4
1.5.1 The Doubs Fish Data.............................................................. 4
1.5.2 The Oribatid Mite Data........................................................... 5
1.6 A Quick Reminder about Help Sources.............................................. 7
1.7 Now It Is Time.................................................................................... 7
2 Exploratory Data Analysis........................................................................ 9
2.1 Objectives........................................................................................... 9
2.2 Data Exploration................................................................................. 9
2.2.1 Data Extraction....................................................................... 9
2.2.2 Species Data: First Contact..................................................... 10
2.2.3 Species Data: A Closer Look.................................................. 12
2.2.4 Species Data Transformation.................................................. 18
2.2.5 Environmental Data................................................................ 24
2.3 Conclusion.......................................................................................... 30
3 Association Measures and Matrices......................................................... 31
3.1 Objectives........................................................................................... 31
3.2 The Main Categories of Association Measures
(Short Overview)................................................................................ 31
3.2.1 Q Mode and R Mode............................................................... 32
3.2.2 Symmetrical or Asymmetrical Coefficients
in Q Mode: The Double-Zero Problem................................... 32
viii Contents
3.2.3 Association Measures for Qualitative or
Quantitative Data.................................................................. 33
3.2.4 To Summarize…................................................................... 33
3.3 Q Mode: Computing Distance Matrices Among Objects................. 34
3.3.1 Q Mode: Quantitative Species Data..................................... 34
3.3.2 Q Mode: Binary (Presence–Absence) Species Data............ 36
3.3.3 Q Mode: Quantitative Data
(Excluding Species Abundances)......................................... 41
3.3.4 Q Mode: Binary Data (Excluding Species
Presence–Absence Data)...................................................... 43
3.3.5 Q Mode: Mixed Types, Including Categorical
(Qualitative Multiclass) Variables........................................ 44
3.4 R Mode: Computing Dependence Matrices Among Variables......... 46
3.4.1 R Mode: Species Abundance Data....................................... 46
3.4.2 R Mode: Species Presence–Absence Data........................... 47
3.4.3 R Mode: Quantitative and Ordinal Data
(Other than Species Abundances)........................................ 47
3.4.4 R Mode: Binary Data (Other than Species
Abundance Data).................................................................. 49
3.5 Pre-transformations for Species Data............................................... 50
3.6 Conclusion........................................................................................ 50
4 Cluster Analysis......................................................................................... 53
4.1 Objectives......................................................................................... 53
4.2 Clustering Overview......................................................................... 53
4.3 Hierarchical Clustering Based on Links........................................... 56
4.3.1 Single Linkage Agglomerative Clustering........................... 56
4.3.2 Complete Linkage Agglomerative Clustering...................... 57
4.4 Average Agglomerative Clustering................................................... 59
4.5 Ward’s Minimum Variance Clustering............................................. 61
4.6 Flexible Clustering............................................................................ 63
4.7 Interpreting and Comparing Hierarchical Clustering Results.......... 63
4.7.1 Introduction.......................................................................... 63
4.7.2 Cophenetic Correlation........................................................ 63
4.7.3 Looking for Interpretable Clusters....................................... 65
4.8 Non-hierarchical Clustering.............................................................. 79
4.8.1 k-Means Partitioning............................................................ 80
4.8.2 Partitioning Around Medoids............................................... 84
4.9 Comparison with Environmental Data............................................. 87
4.9.1 Comparing a Typology with External Data
(ANOVA Approach)............................................................. 88
4.9.2 Comparing Two Typologies
(Contingency Table Approach)............................................ 91
Contents ix
4.10 Species Assemblages........................................................................ 91
4.10.1 Simple Statistics on Group Contents.................................. 91
4.10.2 Kendall’s W Coefficient of Concordance........................... 92
4.10.3 Species Assemblages in Presence–Absence Data.............. 95
4.10.4 IndVal: Species Indicator Values........................................ 97
4.11 Multivariate Regression Trees: Constrained Clustering................... 99
4.11.1 Introduction........................................................................ 99
4.11.2 Computation (Principle)..................................................... 99
4.11.3 Application Using Packages mvpart
and MVPARTwrap............................................................. 102
4.11.4 Combining MRT and IndVal.............................................. 107
4.11.5 MRT as a “Chronological” Clustering Method.................. 108
4.12 A Very Different Approach: Fuzzy Clustering................................. 110
4.12.1 Fuzzy c-Means Clustering Using cluster’s
Function fanny()................................................................. 110
4.13 Conclusion........................................................................................ 114
5 Unconstrained Ordination........................................................................ 115
5.1 Objectives......................................................................................... 115
5.2 Ordination Overview........................................................................ 115
5.2.1 Multidimensional Space..................................................... 115
5.2.2 Ordination in Reduced Space............................................. 116
5.3 Principal Component Analysis......................................................... 117
5.3.1 Overview............................................................................ 117
5.3.2 PCA on the Environmental Variables of the Doubs
Data Set Using rda()........................................................... 118
5.3.3 PCA on Transformed Species Data.................................... 128
5.3.4 Domain of Application of PCA.......................................... 130
5.3.5 PCA Using Function PCA()............................................... 131
5.4 Correspondence Analysis................................................................. 132
5.4.1 Introduction........................................................................ 132
5.4.2 CA Using Function cca() of Package vegan....................... 133
5.4.3 CA Using Function CA()................................................... 138
5.4.4 Arch Effect and Detrended Correspondence Analysis....... 139
5.4.5 Multiple Correspondence Analysis.................................... 140
5.5 Principal Coordinate Analysis.......................................................... 140
5.5.1 Introduction........................................................................ 140
5.5.2 Application to the Doubs Data Set
Using cmdscale and vegan................................................. 141
5.5.3 Application to the Doubs Data Set Using pcoa()............... 143
5.6 Nonmetric Multidimensional Scaling............................................... 145
5.6.1 Introduction........................................................................ 145
5.6.2 Application to the Fish Data............................................... 146
5.7 Handwritten Ordination Function..................................................... 149
x Contents
6 Canonical Ordination................................................................................ 153
6.1 Objectives......................................................................................... 153
6.2 Canonical Ordination Overview....................................................... 154
6.3 Redundancy Analysis....................................................................... 154
6.3.1 Introduction........................................................................ 154
6.3.2 RDA of the Doubs River Data............................................ 156
6.3.3 A Hand-Written RDA Function......................................... 195
6.4 Canonical Correspondence Analysis................................................ 198
6.4.1 Introduction........................................................................ 198
6.4.2 CCA of the Doubs Data..................................................... 199
6.5 Linear Discriminant Analysis........................................................... 207
6.5.1 Introduction........................................................................ 207
6.5.2 Discriminant Analysis Using lda()..................................... 208
6.6 Other Asymmetrical Analyses.......................................................... 210
6.7 Symmetrical Analysis of Two (or More) Data Sets.......................... 211
6.8 Canonical Correlation Analysis........................................................ 211
6.8.1 Introduction........................................................................ 211
6.8.2 Canonical Correlation Analysis using CCorA................... 212
6.9 Co-inertia Analysis........................................................................... 214
6.9.1 Introduction........................................................................ 214
6.9.2 Co-inertia Analysis Using ade4.......................................... 215
6.10 Multiple Factor Analysis.................................................................. 218
6.10.1 Introduction........................................................................ 218
6.10.2 Multiple Factor Analysis Using FactoMineR.................... 219
6.11 Conclusion........................................................................................ 224
7 Spatial Analysis of Ecological Data.......................................................... 227
7.1 Objectives......................................................................................... 227
7.2 Spatial Structures and Spatial Analysis: A Short Overview............. 228
7.2.1 Introduction........................................................................ 228
7.2.2 Induced Spatial Dependence and Spatial Autocorrelation. 229
7.2.3 Spatial Scale....................................................................... 230
7.2.4 Spatial Heterogeneity......................................................... 231
7.2.5 Spatial Correlation or Autocorrelation Functions
and Spatial Correlograms................................................... 232
7.2.6 Testing for the Presence of Spatial Correlation:
Conditions.......................................................................... 236
7.2.7 Modelling Spatial Structures.............................................. 238
7.3 Multivariate Trend-Surface Analysis................................................ 238
7.3.1 Introduction........................................................................ 238
7.3.2 Trend-Surface Analysis in Practice.................................... 239
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling............ 243
7.4.1 Introduction........................................................................ 243
Contents xi
7.4.2 Classical Distance-Based MEM, Formerly Called
Principal Coordinates of Neighbour Matrices.................... 244
7.4.3 MEM in a Wider Context: Weights Other
than Geographic Distances................................................. 263
7.4.4 MEM with Positive or Negative Spatial Correlation:
Which Ones Should Be Used?........................................... 278
7.4.5 Asymmetric Eigenvector Maps :
When Directionality Matters.............................................. 279
7.5 Another Way to Look at Spatial Structures:
Multiscale Ordination....................................................................... 285
7.5.1 Principle............................................................................. 285
7.5.2 Application to the Mite Data: Exploratory Approach........ 286
7.5.3 Application to the Detrended Mite
and Environmental Data..................................................... 289
7.6 Conclusion........................................................................................ 292
Bibliographical References............................................................................. 293
Index.................................................................................................................. 301
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