Developments in Environmental Modelling, 20
Numerical Ecology
SECOND ENGLISH EDITION
Developments in Environmental Modelling
1.ENERGY AND ECOLOGICAL MODELLING edited by W.J. Mitsch, R.W. Bossermann and J.M. Klopatek, 1981
2.WATER MANAGEMENT MODELS IN PRACTICE: A CASE STUDY OF THE ASWAN HIGH DAM by D. Whittington and G. Guariso, 1983
3.NUMERICAL ECOLOGY by L. Legendre and P. Legendre, 1983
4A. APPLICATION OF ECOLOGICAL MODELLING IN ENVIRONMENTAL MANAGEMENT PART A edited by S.E. Jørgensen, 1983
4B. APPLICATION OF ECOLOGICAL MODELLING IN ENVIRONMENTAL MANAGEMENT PART B edited by S.E. Jørgensen and W.J. Mitsch, 1983
5.ANALYSIS OF ECOLOGICAL SYSTEMS: STATE-OF-THE-ART IN ECOLOGICAL MODELLING edited by W.K. Lauenroth, G.V. Skogerboe and M. Flug, 1983
6.MODELLING THE FATE AND EFFECT OF TOXIC SUBSTANCES IN THE ENVIRONMENT edited by S.E. Jørgensen, 1984
7.MATHEMATICAL MODELS IN BIOLOGICAL WASTE WATER TREATMENT edited by S.E. Jørgensen and M.J. Gromiec, 1985
8.FRESHWATER ECOSYSTEMS: MODELLING AND SIMULATION by M. Stra˘skraba and A.H. Gnauck, 1985
9.FUNDAMENTALS OF ECOLOGICAL MODELLING by S.E. Jørgensen, 1986
10.AGRICULTURAL NONPOINT SOURCE POLLUTION: MODEL SELECTION AND APPLICATION edited by A. Giorgini and F. Zingales, 1986
11.MATHEMATICAL MODELLING OF ENVIRONMENTAL AND ECOLOGICAL SYSTEMS edited by J.B. Shukla, T.G. Hallam and V. Capasso, 1987
12.WETLAND MODELLING edited by W.J. Mitsch, M. Stra˘skraba and S.E. Jørgensen, 1988
13.ADVANCES IN ENVIRONMENTAL MODELLING edited by A. Marani, 1988
14.MATHEMATICAL SUBMODELS IN WATER QUALITY SYSTEMS edited by S.E. Jørgensen and M.J. Gromiec, 1989
15.ENVIRONMENTAL MODELS: EMISSIONS AND CONSEQUENCES edited by J. Fenhann, H. Larsen, G.A. Mackenzie and B. Rasmussen, 1990
16.MODELLING IN ECOTOXICOLOGY edited by S.E. Jørgensen, 1990
17.MODELLING IN ENVIRONMENTAL CHEMISTRY edited by S.E. Jørgensen, 1991
18.INTRODUCTION TO ENVIRONMENTAL MANAGEMENT edited by P.E. Hansen and S.E. Jørgensen, 1991
19.FUNDAMENTALS OF ECOLOGICAL MODELLING by S.E. Jørgensen, 1994
Contents
Preface xi
1. Complex ecological data sets
1.0 |
Numerical analysis of ecological data |
1 |
1.1 |
Autocorrelation and spatial structure |
8 |
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1 – Types of spatial structures, 11; 2 – Tests of statistical significance in the presence |
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of autocorrelation, 12; 3 – Classical sampling and spatial structure, 16 |
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1.2 |
Statistical testing by permutation 17 |
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1 – Classical tests of significance, 17; 2 – Permutation tests, 20; 3 – Numerical |
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example, 22; 4 – Remarks on permutation tests, 24 |
1.3Computers 26
1.4 Ecological descriptors 27
1 – Mathematical types of descriptor, 28; 2 – Intensive, extensive, additive, and nonadditive descriptors, 31
1.5 Coding 33
1 – Linear transformation, 34; 2 – Nonlinear transformations, 35; 3 – Combining descriptors, 37; 4 – Ranging and standardization, 37; 5 – Implicit transformation in association coefficients, 39; 6 – Normalization, 39; 7 – Dummy variable (binary) coding, 46
1.6 Missing data 47
1 – Deleting rows or columns, 48; 2 – Accommodating algorithms to missing data, 48; 3 – Estimating missing values, 48
2. Matrix algebra: a summary
2.0 |
Matrix algebra |
51 |
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2.1 |
The ecological data matrix 52 |
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2.2 |
Association matrices |
55 |
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2.3 |
Special matrices |
56 |
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2.4 |
Vectors and scaling |
61 |
vi |
Contents |
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2.5 |
Matrix addition and multiplication 63 |
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2.6 |
Determinant 68 |
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2.7 |
The rank of a matrix 72 |
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2.8 |
Matrix inversion 73 |
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2.9 |
Eigenvalues and eigenvectors |
80 |
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1 – Computation, 81; 2 – Numerical examples, 83 |
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2.10 |
Some properties of eigenvalues and eigenvectors 90 |
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2.11 |
Singular value decomposition |
94 |
3. Dimensional analysis in ecology
3.0 Dimensional analysis 97
3.1Dimensions 98
3.2 |
Fundamental principles and the Pi theorem |
103 |
3.3 |
The complete set of dimensionless products |
118 |
3.4 |
Scale factors and models 126 |
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4. Multidimensional quantitative data
4.0 |
Multidimensional statistics |
131 |
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4.1 |
Multidimensional variables and dispersion matrix 132 |
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4.2 |
Correlation matrix |
139 |
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4.3 |
Multinormal distribution |
144 |
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4.4 |
Principal axes 152 |
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4.5 |
Multiple and partial correlations |
158 |
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1 – Multiple linear |
correlation, 158; |
2 – Partial |
correlation, 161; 3 – Tests of |
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statistical significance, 164; |
4 – Interpretation of |
correlation coefficients, 166; |
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5 – Causal modelling using correlations, 169 |
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4.6 |
Multinormal conditional distribution |
173 |
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4.7 |
Tests of normality and multinormality |
178 |
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5. Multidimensional semiquantitative data
5.0 |
Nonparametric statistics 185 |
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5.1 |
Quantitative, semiquantitative, and qualitative multivariates 186 |
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5.2 |
One-dimensional nonparametric statistics 191 |
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5.3 |
Multidimensional ranking tests |
194 |
Contents |
vii |
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6. Multidimensional qualitative data
6.0 |
General principles |
207 |
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6.1 |
Information and entropy 208 |
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6.2 |
Two-way contingency tables |
216 |
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6.3 |
Multiway contingency tables |
222 |
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6.4 |
Contingency tables: correspondence 230 |
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6.5 |
Species diversity |
235 |
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1 – Diversity, 239; 2 – Evenness, equitability, 243
7. Ecological resemblance
7.0 |
The basis for clustering and ordination |
247 |
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7.1 |
Q and R analyses |
248 |
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7.2 |
Association coefficients |
251 |
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7.3 |
Q mode: similarity coefficients |
253 |
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1 |
– Symmetrical binary coefficients, 254; 2 – Asymmetrical binary coefficients, 256; |
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3 |
– Symmetrical |
quantitative |
coefficients, 258; 4 – Asymmetrical |
quantitative |
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coefficients, 264; 5 – Probabilistic coefficients, 268 |
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7.4 |
Q mode: distance coefficients |
274 |
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1 |
– Metric distances, 276; 2 – Semimetrics, 286 |
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7.5 |
R mode: coefficients of dependence |
288 |
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1 |
– Descriptors |
other |
than |
species |
abundances, 289; 2 – Species |
abundances: |
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biological associations, 291 |
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7.6 |
Choice of a coefficient |
295 |
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7.7 |
Computer programs and packages |
302 |
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8. Cluster analysis
8.0 A search for discontinuities 303
8.1Definitions 305
8.2 |
The basic model: single linkage clustering |
308 |
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8.3 |
Cophenetic matrix and ultrametric property |
312 |
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1 – Cophenetic matrix, 312; 2 – Ultrametric property, 313 |
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8.4 |
The panoply of methods |
314 |
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1 – Sequential |
versus simultaneous |
algorithms, 314; 2 – Agglomeration versus |
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division, 314; |
3 – Monothetic |
versus |
polythetic |
methods, 314; 4 – Hierarchical |
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versus non-hierarchical methods, 315; 5 – Probabilistic versus non-probabilistic |
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methods, 315 |
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viii |
Contents |
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8.5 Hierarchical agglomerative clustering |
316 |
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1 – Single linkage agglomerative |
clustering, 316; 2 – Complete |
linkage |
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agglomerative |
clustering, 316; 3 – Intermediate |
linkage clustering, 318; |
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4 – Unweighted |
arithmetic average |
clustering (UPGMA), 319; 5 – Weighted |
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arithmetic average clustering (WPGMA), 321; 6 – Unweighted centroid clustering |
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(UPGMC), 322; |
7 – Weighted centroid |
clustering |
(WPGMC), 324; |
8 – Ward’s |
minimum variance method, 329; 9 – General agglomerative clustering model, 333; 10 – Flexible clustering, 335; 11 – Information analysis, 336
8.6Reversals 341
8.7 |
Hierarchical divisive clustering |
343 |
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1 – Monothetic methods, 343; 2 – Polythetic |
methods, 345; |
3 – Division in |
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ordination space, 346; 4 – TWINSPAN, 347 |
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8.8 |
Partitioning by K-means 349 |
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8.9 |
Species clustering: biological associations |
355 |
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1 – Non-hierarchical |
complete |
linkage clustering, 358; |
2 – Probabilistic |
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clustering, 361; 3 – Indicator species, 368 |
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8.10 |
Seriation 371 |
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8.11 |
Clustering statistics |
374 |
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1 – Connectedness and isolation, 374; 2 – Cophenetic correlation and related |
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measures, 375 |
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8.12 |
Cluster validation |
378 |
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8.13 |
Cluster representation and choice of a method 381 |
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9. Ordination in reduced space
9.0 |
Projecting data sets in a few dimensions |
387 |
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9.1 |
Principal component analysis (PCA) |
391 |
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1 |
– Computing the eigenvectors, 392; 2 – Computing and representing the principal |
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components, 394; |
3 – Contributions of |
descriptors, 395; |
4 – Biplots, 403; |
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5 |
– Principal |
components |
of |
a |
correlation |
matrix, 406; 6 – The meaningful |
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components, 409; |
7 – Misuses |
of |
principal |
components, 411; |
8 – Ecological |
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applications, 415; 9 – Algorithms, 418 |
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9.2 |
Principal coordinate analysis (PCoA) |
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424 |
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1 |
– Computation, 425; |
2 – Numerical |
example, 427; 3 – Rationale of the |
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method, 429; |
4 – Negative |
eigenvalues, 432; |
5 – Ecological applications, 438; |
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6 |
– Algorithms, 443 |
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9.3 |
Nonmetric multidimensional scaling (MDS) |
444 |
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9.4 |
Correspondence analysis (CA) |
451 |
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1 |
– Computation, 452; |
2 – Numerical |
example, 457; 3 – Interpretation, 461; |
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4 |
– Site |
species |
data |
tables, 462; |
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5 – Arch effect, 465; |
6 – Ecological |
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applications, 472; 7 – Algorithms, 473 |
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9.5 |
Factor analysis |
476 |
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Contents |
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ix |
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10. Interpretation of ecological structures |
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10.0 |
Ecological structures |
481 |
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10.1 |
Clustering and ordination |
482 |
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10.2 |
The mathematics of ecological interpretation |
486 |
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10.3 |
Regression |
497 |
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1 |
– Simple |
linear |
regression: |
model I, 500; |
2 – Simple linear |
regression: |
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model II, 504; 3 – Multiple linear regression, 517; 4 – Polynomial regression, 526; |
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5 |
– Partial |
linear |
regression, 528; 6 – Nonlinear regression, 536; |
7 – Logistic |
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regression, 538; 8 – Splines and LOWESS smoothing, 542 |
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10.4 |
Path analysis |
546 |
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10.5 |
Matrix comparisons |
551 |
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1 |
– Mantel test, |
552; |
2 – More |
than two matrices, 557; 3 – ANOSIM test, 560; |
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– Procrustes analysis, 563 |
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10.6 |
The 4th-corner problem |
565 |
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1 – Comparing two qualitative variables, 566; 2 – Test of statistical significance, 567; 3 – Permutational models, 569; 4 – Other types of comparison among variables, 571
11. |
Canonical analysis |
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11.0 |
Principles of canonical analysis |
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575 |
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11.1 |
Redundancy analysis (RDA) |
579 |
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1 |
– The |
algebra |
of |
redundancy |
analysis, 580; |
2 – Numerical |
examples, 587; |
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3 |
– Algorithms, 592; |
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11.2 |
Canonical correspondence analysis (CCA) |
594 |
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1 |
– The |
algebra |
of |
canonical |
correspondence |
analysis, 594; |
2 – Numerical |
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example, 597; 3 – Algorithms, 600 |
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11.3 |
Partial RDA and CCA |
605 |
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1 |
– Applications, 605; 2 – Tests of significance, 606 |
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11.4 |
Canonical correlation analysis (CCorA) |
612 |
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11.5 |
Discriminant analysis |
616 |
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1 |
– The algebra of discriminant analysis, 620; 2 – Numerical example, 626 |
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11.6 |
Canonical analysis of species data |
633 |
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12. Ecological data series |
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12.0 |
Ecological series |
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637 |
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12.1 |
Characteristics of data series and research objectives 641 |
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12.2 |
Trend extraction and numerical filters 647 |
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12.3 |
Periodic variability: correlogram |
653 |
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1 – Autocovariance and autocorrelation, 653; 2 – Cross-covariance and crosscorrelation, 661
x |
Contents |
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12.4 Periodic variability: periodogram 665 |
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1 |
– Periodogram of Whittaker and Robinson, 665; 2 – Contingency periodogram of |
Legendre et al., 670; 3 – Periodograms of Schuster and Dutilleul, 673; |
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– Harmonic regression, 678 |
12.5 Periodic variability: spectral analysis 679
1 – Series of a single variable, 680; 2 – Multidimensional series, 683; 3 – Maximum entropy spectral analysis, 688
12.6 |
Detection of discontinuities in multivariate series 691 |
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1 |
– Ordinations in |
reduced space, 692; 2 – Segmenting data series, 693; |
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– Webster’s method, 693; 4 – Chronological clustering, 696 |
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12.7 |
Box-Jenkins models |
702 |
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12.8 |
Computer programs |
704 |
13. Spatial analysis
13.0 |
Spatial patterns |
707 |
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13.1 |
Structure functions |
712 |
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1 |
– Spatial |
correlograms, 714; |
2 – Interpretation |
of |
all-directional |
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correlograms, 721; |
3 – Variogram, 728; 4 – Spatial |
covariance, |
semi-variance, |
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correlation, cross-correlation, 733; 5 – Multivariate Mantel correlogram, 736 |
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13.2 |
Maps 738 |
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– Trend-surface analysis, 739; |
2 – Interpolated |
maps, 746; 3 – Measures of |
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fit, 751 |
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13.3 |
Patches and boundaries 751 |
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1 |
– Connection |
networks, 752; |
2 – Constrained |
clustering, 756; |
3 – Ecological |
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boundaries, 760; 4 – Dispersal, 763 |
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13.4 |
Unconstrained and constrained ordination maps |
765 |
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13.5 |
Causal modelling: partial canonical analysis |
769 |
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1 |
– Partitioning method, 771; 2 – Interpretation of the fractions, 776 |
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13.6 |
Causal modelling: partial Mantel analysis 779 |
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1 |
– Partial Mantel |
correlations, 779; |
2 – Multiple |
regression |
approach, 783; |
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3 |
– Comparison of methods, 783 |
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13.7 |
Computer programs |
785 |
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Bibliography 787
Tables 833
Subject index 839