Preface ........................................................ xь
1 Complex ecological data sets
1.0 Numerical analysis of ecological data ................... 1
1.1 Spatial structure, spatial dependence, spatial
correlation ............................................. 8
1 - Origin of spatial structures ....................... 10
2 - Tests of significance in the presence of spatial
correlation ............................................ 17
3 - Classical sampling and spatial structure ........... 21
1.2 Statistical testing by permutation ..................... 22
1 - Classical tests of significance .................... 22
2 - Permutation tests .................................. 25
3 - Numerical example .................................. 28
4 - Remarks on permutation tests ....................... 29
1.3 Computer programs and packages ......................... 32
1.4 Ecological descriptors ................................. 33
1 - Mathematical types of descriptors .................. 34
2 - Intensive, extensive, additive, and non-additive
descriptors ........................................ 37
1.5 Coding ................................................. 39
1 - Linear transformation .............................. 40
2 - Nonlinear transformations .......................... 41
3 - Combining descriptors .............................. 43
4 - Ranging and standardization ........................ 43
5 - Implicit transformation in association
coefficients ....................................... 45
6 - Normalization ...................................... 45
7 - Dummy variable coding .............................. 52
1.6 Missing dЈta ........................................... 54
1 - Delete rows or columns ............................. 55
2 - Accommodate algorithms to missing data ............. 55
3 - Estimate missing values ............................ 55
1.7 Software ............................................... 57
2 Matrix algebra: a summary
2.0 Matrix algebra ......................................... 59
2.1 The ecological data matrix ............................. 60
2.2 Association matrices ................................... 63
2.3 Special matrices ....................................... 64
2.4 Vectors and scaling .................................... 69
2.0 Matrix addition and multiplication ..................... 71
2.6 Determinant............................................. 76
2.7 Rank of a matrix ....................................... 80
2.8 Matrix inversion ....................................... 82
2.9 Eigenvalues and eigenvectors ........................... 89
1 - Computation ........................................ 90
2 - Numerical examples ................................. 92
2.10 Some properties of eigenvalues and eigenvectors ........ 99
2.11 Singular value decomposition .......................... 103
2.12 Software .............................................. 107
3 Dimensional analysis in ecology
3.0 Dimensional analysis .................................. 109
3.1 Dimensions ............................................ 110
3.2 Fundamental jrinciples and the Pi theorem ............. 115
3.3 The complete set of dimensionless products ............ 130
3.4 Scale factors and models .............................. 138
4 Multidimensional quantitative data
4.0 Multidimensional statistics ........................... 143
4.1 Multidimensional variables and dispersion matrix ...... 144
4.2 Correlation matrix .................................... 151
4.3 Multinormal distribution .............................. 157
4.4 Principal axes ........................................ 165
4.5 Multiple and partial correlations ..................... 171
1 - Multiple linear correlation ....................... 173
2 - Partial correlation ............................... 175
3 - Tests of statistical significance ................. 180
4 - Causal modelling using correlations ............... 182
4.6 Tests of normality and multinormality ................. 187
4.7 Software .............................................. 194
5 Multidimensional semiquantitative data
5.0 Nonparametric statistics .............................. 195
5.1 Quantitative, semiquantitative, and qualitative
multivariates ......................................... 197
5.2 One-dimensional nonparametric statistics .............. 201
5.3 Rank correlations ..................................... 205
1 - Spearman r ........................................ 205
2 - Kendall r ......................................... 209
5.4 Coefficient of concordance ............................ 213
1 - Computing Kendall W ............................... 214
2 - Testing the significance of W ..................... 216
3 - Contributions of individual variables to
Kendall's concordance ............................. 217
6 Multidimensional qualitative data
6.0 General principles .................................... 219
6.l Information and entropy ............................... 220
6.2 Two-way contingency tables ............................ 228
6.3 Multiway contingency tables ........................... 235
6.4 Contingency tables: correspondence .................... 243
6.5 Species diversity ..................................... 247
1 - Diversity ......................................... 250
2 - Evenness, equitability ............................ 255
3 - Species diversity through space ................... 258
6.6 Software .............................................. 264
7 Ecological resemblance
7.0 The basis for clustering and ordination ............... 265
7.1 Q and R analyses ...................................... 266
7.2 Association coefficients .............................. 269
1 - Similarity, distance, and dependence
coefficients ...................................... 270
2 - The double-zero problem ........................... 271
7.3 Q mode: similarity coefficients ....................... 273
1 - Symmetrical binary coefficients ................... 273
2 - Asymmetrical binary coefficients .................. 275
3 - Symmetrical quantitative coefficients ............. 278
4 - Asymmetrical quantitative coefficients ............ 284
5 - Probabilistic coefficients ........................ 288
1A Q mode: distance coefficients ......................... 295
1 - Metric distances .................................. 299
2 - Semimetrics ....................................... 310
7.5 R mode: coefficients of dependence .................... 313
1 - Descriptors other than species abundances ......... 313
2 - Species abundances: biological associations ....... 316
7.6 Choice of a coefficient ............................... 320
7.7 Transformations for community composition data ........ 327
1 - Transformation formulas ........................... 328
2 - Numerical example ................................. 332
3 - Beals smoothing ................................... 334
7.8 Software ............................................. 334
8 Cluster analysis
8.0 A search for discontinuities .......................... 337
8.1 Definitions ........................................... 338
8.2 The basic model: single linkage clustering ............ 341
8.3 Cophenetic matrix and ultrametric property ............ 346
1 - Cophenetic matrix ................................. 346
2 - Ultrametric property .............................. 347
8.4 The panoply of methods ................................ 347
1 - Sequential versus simultaneous algorithms ......... 347
2 - Agglomeration versus division ..................... 348
3 - Monothetic versus polythetic methods .............. 348
4 - Hierarchical versus non-hierarchical methods ...... 348
5 - Constrained clustering methods .................... 349
6 - Probabilistic versus non-probabilistic methods .... 349
8.5 Hierarchical agglomerative clustering ................. 350
1 - Single linkage agglomerative clustering ........... 350
2 - Complete linkage agglomerative clustering ......... 350
3 - Intermediate linkage clustering ................... 351
4 - Unweighted arithmetic average clustering (UPGMA) .. 352
5 - Weighted arithmetic average clustering (WPGMA) .... 355
6 - Unweighted centroid clustering (UPGMC) ............ 357
7 - Weighted centroid clustering (WPGMC) .............. 360
8 - Ward's minimum variance method .................... 360
9 - General agglomerative clustering model ............ 367
10 - Flexible clustering .............................. 370
11 - Information analysis ............................. 372
8.6 Reversals ............................................. 376
8.7 Hierarchical divisive clustering ...................... 377
1 - Monothetic methods ................................ 377
2 - Polythetic methods ................................ 379
3 - Division in ordination space ...................... 380
4 - Twinspan .......................................... 381
8.8 Partitioning by Д"-means .............................. 383
8.9 Species clustering: biological associations ........... 389
1 - Non-hierarchical complete linkage clustering ...... 392
2 - Concordance analysis .............................. 395
3 - Indicator species ................................. 397
8.10 Seriation ............................................. 403
8.11 Multivariate regression trees (MRT) ................... 406
8.12 Clustering statistics ................................. 411
1 - Connectedness and isolation ....................... 411
2 - Cophenetic correlation and related measures ....... 412
8.13 Cluster validation .................................... 415
8.14 Cluster representation and choice of a method ......... 418
8.15 Software .............................................. 423
9 Ordination in reduced space
9.0 Projecting data sets in a few dimensions .............. 425
9.1 Principal component analysis (PCA) .................... 429
1 - Computing the eigenvectors of a dispersion
matrix ............................................ 431
2 - Computing and representing the principal
components ........................................ 432
3 - Contributions of the descriptors .................. 434
4 - PCA biplots ....................................... 443
5 - Principal components of a correlation matrix ...... 445
6 - The meaningful components ......................... 448
7 — Misuses of principal component analysis ........... 450
8 - Ecological applications ........................... 452
9 - Algorithms ........................................ 456
10 - Metric ordination of community composition data .. 462
9.2 Correspondence analysis (CA) .......................... 464
1 - Computation ....................................... 466
2 - Numerical example ................................. 471
3 - Interpretation .................................... 476
4 - Site xspecies data tables ......................... 477
5 - Arch effect and detrended correspondence
analysis .......................................... 482
6 - Ecological applications ........................... 487
7 - Algorithms ........................................ 490
9.3 Principal coordinate analysis (PCoA) .................. 492
1 - Computation ....................................... 493
2 - Numerical example ................................. 494
3 - Rationale of the method ........................... 497
4 - Negative eigenvalues .............................. 500
5 - Ecological applications ........................... 506
6 - Algorithm ......................................... 511
9.4 Nonmetric multidimensional scaling (nMDS) ............. 512
9.5 Software .............................................. 519
10 Interpretation of ecological structures
10.0 Ecological structures ................................. 521
10.1 Clustering and ordination ............................. 522
10.2 The mathematics of ecological interpretation .......... 526
1 - Explaining ecological structures .................. 530
2 - Forecasting ecological structures ................. 532
3 - Ecological prediction ............................. 534
10.3 Regression ............................................ 536
1 - Simple linear regression: model I ................. 539
2 - Simple linear regression: model II ................ 543
3 - Multiple linear regression ........................ 555
4 - Polynomial regression ............................. 568
5 - Partial linear regression and variation
partitioning ...................................... 570
6 - Nonlinear regression .............................. 583
7 - Logistic regression ............................... 584
8 - Splines and lowess smoothing ...................... 589
10.4 Path analysis ......................................... 592
10.5 Matrix comparisons .................................... 597
1 - Two association matrices: Mantel test ............. 598
2 - More than two association matrices ................ 604
3 - Anosim test ....................................... 608
4 - Procrustes test ................................... 611
10.6 The fourth-corner problem ............................. 613
1 - Comparing two qualitative variables ............... 614
2 - Test of statistical significance .................. 616
3 - Permutational models .............................. 618
4 - Other types of comparisons among variables ........ 621
10.7 Software .............................................. 622
11 Canonical analysis
11.0 Principles of canonical analysis ...................... 625
11.1 Redundancy analysis (RDA) ............................. 629
1 - Simple RDA, 630
2 - Statistics in simple RDA .......................... 632
3 - The algebra of simple RDA ......................... 635
4 - Numerical examples, simple RDA .................... 642
5 - RDA and CCA of community composition data ......... 646
6 - Partial RDA ....................................... 649
7 - Statistics in partial RDA ......................... 651
8 — Tests of significance in partial RDA .............. 651
9 — Numerical example, partial RDA .................... 653
10 - Some applications of partial RDA ................. 654
11 - Variation partitioning by RDA .................... 658
11.2 Canonical correspondence analysis (CCA) ............... 661
1 - The algebra of canonical correspondence analysis .. 662
2 - Numerical example ................................. 667
11.3 Linear discriminant analysis (LDA) .................... 673
1 - The algebra of discriminant analysis .............. 676
2 — Statistics in linear discriminant analysis ........ 682
3 - Numerical example ................................. 683
11.4 Canonical correlation analysis (CCorA) ................ 690
1 - The algebra of canonical correlation analysis ..... 691
2 - Statistics in canonical correlation analysis ...... 694
3 - Applications of CCorA ............................. 694
11.5 Co-inertia (CoIA) and Procrustes (Proc) analyses ...... 696
1 - The algebra of co-inertia analysis (CoIA) ......... 697
2 - Symmetric Procrustes analysis (Proc) .............. 703
3 - Canonical correlation, Procrustes, or co-inertia
analysis? ......................................... 705
11.6 Canonical analysis of community composition data ...... 706
11.7 Software .............................................. 709
12 Ecological data series
12.0 Ecological series ..................................... 711
12.1 Characteristics of data series and research
objectives ............................................ 714
12.2 Trend extraction and numerical niters ................. 722
12.3 Periodic variability: correlogram ..................... 727
1 - Autocovariance and autocorrelation ................ 728
2 - Cross-covariance and cross-correlation ............ 735
12.4 Periodic variability: periodogram ..................... 739
1 - Periodogram of Whittaker and Robinson ............. 739
2 - Contingency periodogram of , Legendre et al. ...... 744
3 - Periodogram of Schuster ........................... 747
4 - Periodogram of Dutilleul .......................... 751
5 - Harmonic regression ............................... 753
12.5 Periodic variability: spectral and wavelet analyses ... 754
1 - Series of a single variable ....................... 754
2 - Multidimensional series ........................... 759
3 - Maximum entropy spectral analysis ................. 763
4 - Wavelet analysis .................................. 766
12.6 Detection of discontinuities in multivariate series ... 768
1 - Ordinations in reduced space ...................... 768
2 - Segmenting data series ............................ 769
3 - Webster's method .................................. 770
4 - Time-constrained clustering by MRT ................ 773
5 - Chronological clustering .......................... 773
12.7 Box-Jenkins models .................................... 780
12.8 Software .............................................. 782
13 Spatial analysis
13.0 Spatial patterns ...................................... 785
13.1 Structure functions ................................... 792
1 - Spatial correlograms .............................. 793
2 - Interpretation of all-directional
correlograms ...................................... 800
3 - Variogram ......................................... 807
4 - Multivariate variogram ............................ 813
5 - Spatial covariance, semi-variance, correlation,
cross-correlation ................................. 816
6 - Multivariate Mantel correlogram ................... 819
13.2 Maps .................................................. 821
1 - Trend-surface analysis............................. 822
2 - Interpolated maps ................................. 829
3 - Measures of fit ................................... 833
13.3 Patches and boundaries ................................ 834
1 - Connection networks ............................... 834
2 - Space-constrained clustering ...................... 839
3 - Ecological boundaries ............................. 844
4 - Dispersal ......................................... 847
13.4 Unconstrained and constrained ordination maps ......... 849
13.5 Spatial modelling through canonical analysis .......... 852
13.6 Software .............................................. 857
14 Multiscale analysis: spatial eigenfunctions
14.0 Introduction to multiscale analysis ................... 859
14.1 Distance-based Moran's eigenvector maps (dbMEM) ....... 861
1 - Algorithm, 862
2 - Numerical examples ................................ 864
3 - Ecological applications ........................... 869
4 - Interpretation of the fractions ................... 877
14.2 Moran's eigenvector maps (MEM), general form .......... 881
1 - Algorithm described through an example ............ 881
2 - Different types of MEM eigenfunctions ............. 884
14.3 Asymmetric eigenvector maps (AEM) ..................... 888
1 - Algorithm described through an example ............ 888
2 - Ecological applications ........................... 892
14.4 Multiscale ordination (MSO) ........................... 894
14.5 Other eigenfunction-based methods of spatial
analysis .............................................. 900
1 — Space-time interaction ............................ 900
2 - Multiscale codependence analysis .................. 901
3 — Estimating and controlling for spatial structure
in modelling ...................................... 902
14.6 Multiscale analysis of beta diversity ................. 903
14.7 Software .............................................. 904
References .................................................... 907
References to cited works .................................. 907
References to R packages ................................... 963
Subject index ................................................. 969
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