Zelditch M.L. (и др.) Geometric Morphometrics for Biologists: a primer
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Index 4/6/2004 17: 31 page 438
438 INDEX
Characters (contd)
using PCA to find, 372–6
Circle construction of principal axes, 61–2
Clock model, Gould’s for analysis of
heterochrony, 339
Clustering, 307
Coding phylogenetic characters, 378–9
Comparing means, see ANOVA, MANOVA
and specific test statistics
Comparing patterns of (co)variance, 308–14
Complex numbers, 137–8
Compression/dilation, 137–42
Computer-based statistical methods, 189–208
basic statistical concepts, 190–1
constructing confidence intervals, 191–2
hypothesis testing, 193–4
number of repetitions, 205–6
power, 205
resampling-based methods
bootstrap, 195–9
jackknife, 200–1
Monte Carlo simulations, 201–2
permutation tests, 199–200
Configuration of landmarks, 75
Configuration matrix, 76–9
configuration space, 77
position or location, 77–8
size, 78–9
Configuration spaces, 77, 390–1
Consensus configuration, see Reference
configuration
Consistency of relative position of landmarks,
27–8
Continuous functions and models of
deformations, 132
Continuous variables, 190
Contraction, 413
Contrast, 47
CoordGen program, 69–70
Coplanarity of landmarks, 30–1
Correlation, 413
Correlation coefficient, 235
Correspondence of landmarks, 25
Covariance, 217, 413
Criteria for selecting landmarks, see
Landmarks, criteria for choosing
Curves without landmarks, 395–6
CVA, see Canonical variates analysis
CVAGen program, 184–6, 224
Deformation, 131–2
decomposition of, 134–150, 391–2
see also Thin-plate spline, uniform
transformations, non-uniform
transformations, partial warps
Degrees of freedom, 88, 97–8, 214, 215
Depth of field, 43
Digitization of images, 48–9
Discontinuous functions and models of
deformations, 132
Discontinuous variables, 210
Discrete variables, 190
Discriminant function, see Canonical variates
analysis
Disparity, 293–5
confidence intervals, 298–302
hypotheses explaining temporal
patterns, 294
measurement, 297–8
metrics, 296–7
multi-group analysis, 316–17
ontogenetic and interclade comparisons,
300–2
partial, 302
relationship to variation, 293
software, 314–17
DisparityBox, 314–17
SpaceAngle, 317
structure, analysis of
nearest-neighbor analysis, 303–8
patterns of (co)variance, 308–14
DisparityBox program, 314–17
Distance, 81, see also Euclidean distance, full
Procrustes distance, Mahalanobis distance,
partial Procrustes distance
Dot (inner) product, between two vectors,
251–2
Dummy codes, using in regression, 240
Edge registration, see Baseline registration
Eigenanalysis, 146
Eigenvalues, see Eigenvectors
Eigenvectors, 163–4, 167
eigenvectors of the bending energy matrix
(partial warps), 146, 149
relationship to singular axes, 262, 263
see also Principal components, partial warps
Electronically available resources, see Web
resources
Elements of matrix, 77
Error, 193, 230, 232
Error function, 232
Euclidean approximation to Kendall’s shape
space, 87, 95–6
Euclidean distance, 97, 297
Euclidean space, 86, 87, 95, 96, 149, 392
Explicit uniform components, 136, 137
Explicit uniform deformations, see Explicit
uniform components
Explicit uniform term, see Explicit uniform
components
F-test, 197, 212–14, 219
Fiber, 81–3
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INDEX 439
Flury hierarchy, 308
Focus, 43
Full Procrustes distance, 85, 115, 390
Full Procrustes superimposition, 94, 96,
99, 115
Gaussian distribution, 191, 305
Generalized least squares Procrustes
superimposition, 113–19, 298
Geodesic distance, see Partial Procrustes
distance
Geometric shape, 11, 72–3
GLS, see Generalized least squares Procrustes
superimposition
Goodall’s F-test, 117, 122, 130, 221,
225, 236
Grand mean, 213, 298–9
Great circle distance, see Partial Procrustes
distance
Grouping variable in MANOVA, 171
Growth:
decoupling from development, 337–9
describing changes in shape related to, see
Ontogenetic trajectory
profiles, 332
Heterochrony, 321, 323, 324
formalisms for, 339–41
Homology of landmarks, 25–7
Hotelling-Lawley trace, 219
Hotelling’s T
2
-test, 58, 65, 112, 218,
224, 240
Hyperplanes, 309
largest possible angle between, 309–12
Hypersphere, 80
Image acquisition, 39
camera
aperture, 39–40
image capture, 44–5
lens, 40–4
digitization, 48–9
image formation, 40
improving image, 46–7
magnification, 42
saving image, 45–6
system checks, 44
Implicit uniform deformations, 136,
142–3
Implicit uniform terms, see Implicit uniform
deformations
Induced correlation, 111–12
Induced covariance, 110, 112, 119
Influential observations, 119
Isometry, 240, 331, 337, 338
Isotropic change, 62
Jackknife test, 200–1, 417
JPEG format for image files, 45
Kendall’s shape space:
Three-dimensional landmark configurations,
in, 390, 395–6, 399, 402
Two-dimensional landmark configurations,
in, 86–95
“Kluge–Kerfoot” phenomenon, 296
Lagrange multiplier, 165, 176
Landmark configuration, 14–15, 76
Landmarks, 6, 7, 23–50
Bookstein’s typology of, 31
digitizing, 48–9
criteria for choosing, 24–31
adequate coverage of the form, 28–9
consistency of relative position, 27–8
coplanarity of, 30–1
homology, 25–7
repeatability of, 30
applied to specific cases
external body form, piranha, 34–7
scapula, squirrel, 32–4
skull, rodent, 37–39
configuration of, 14–5, 75
curves without, 395–6
definition of, 75
differential weighting of, 399
image acquisition, manipulation and
enhancement, 39–47
superimposing configurations of, 387–9
three-dimensional configuration of, 386–7
software, for digitizing, 48
Latent variables, 262
Least squares, 119, 232–3
Linear regression, 230, 262
Linear combinations, 5, 155–6, 162, 164,
170, 264
Linearity and regression, 236–40
Mahalanobis distance, 177–8
MakeFan program, 406
MANCOVA, 58, 217, 336
MANOVA, 58, 65, 209–28, 210–23
analytic techniques
categorical with continuous variable,
217–18
complex trait with categorical variable,
218–22
simple trait with more than two groups,
214–15
simple trait with two groups, 210–14
two or more categorical variables, 215–17
software, 223–8
see also Computer-based statistical methods,
F-test, Goodall’s F-test
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440 INDEX
Means, comparing statistically, see ANOVA,
MANOVA, computer-based statistical
methods, Student’s t-test, F-test, Goodall’s
F-test
Standard error of, 194, 200
Mean square, 215
Measurement theory, 2–10
Mglinstaller, 20
Monte Carlo simulations, 201–2, 305
Morphological disparity, see Disparity
Multiple regression, 262–3, 264–5
Multiple triangles, 65–8
Multivariate analysis of covariance,
see MANCOVA
Multivariate analysis of variance,
see MANOVA
Multivariate regression, see Regression
Nearest-neighbor analysis, 303–8
Negative allometry, 330
Neoteny, 338, 340
Non-affine transformations, see Non-uniform
transformations
Non-uniform deformation, see Non-uniform
transformations
Non-uniform transformations, 143–51, 391–2
algebraic introduction, 146–50
intuitive introduction, 143–6
see also Bending energy, partial warps,
uniform transformations
Normal distribution, 191, 194–5, 211, 212,
215, 219, 234
Normalized vector, 251, 256, see also
Orthonormal basis
Ontogenetic scaling, 336–7
Ontogenetic allometry, 322, 324, 325, 333–4
Ontogenetic trajectory, 341–5
computing by multivariate regression,
see Regression
comparing, 336–47
see also Ontogenetic allometry
Ontogeny, in relation to phylogeny, 321–62
Ordination methods, see Principal components
analysis, canonical variates analysis
Orthogonality, 163, 164
Orthonormal basis, 163–4
Outliers, 119
Outlines, see Semilandmarks
Over-dispersion, 307
Paedomorphosis, 334, 337, 341
Parallelism, between ontogeny and phylogeny,
321, 322, 334
Parameters, 191
Partial disparity, 302
Partial least squares analysis, 261–90
comparison with CCA, 264–5
comparison with PCA, 263–4
comparison with regression, 262–3
comparing patterns of covariances across
groups, 265, 276–7
using to test hypotheses of morphological
integration, 266–76, 277–9
software, 285–9
Partial Procrustes distance, 83, 85, 87, 93–6,
99, 222
Partial Procrustes superimposition, 93, 115
Partial warp scores, 129, 143
Partial warps, 143–51, 298
algebraic introduction, 146–50
degrees of freedom, 128–9
intuitive introduction, 143–6
using as variables in conventional
multivariate analyses,
why not to use as phylogenetic characters,
369–72
see also Bending energy, non-uniform
transformations
PCA, see Principal components analysis
PCAGen program, 151, 181–4
Peramorphosis, 334, 337, 341
Permutation test, 199–200
Phylogeny:
Ontogeny and, 321–61
Inferring from morphometric data, 363–5,
367–80
Photo-editing software, 47
Pillai’s trace, 219
Pinhole camera, 40–4
Pinocchio effect, 119, 120, 132
PLS, see Partial least squares analysis
Population, 190
Positive allometry, 330
Power (statistical), 205
Power law, formalism for analysis of allometry,
330–1
Pre-shape space, 79–83
definition, 79
fibers in, 81–3
shape of, 79–80
Principal axes of a transformation, 61
variables implied by, 62–5
Principal components analysis, 15, 156–70,
328–30
algebraic description of, 161–5
comparison with PLS, 263–4
finding phylogenetic characters by, 372–6
geometric description of, 156–61
interpretation of results, 166–70
relationship to allometric coefficients,
328–30
principal component scores, 159–61
relationship to canonical variates analysis,
155, 170–1
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INDEX 441
scree plots, 168
software, 181–4
testing whether successive eigenvalues are
distinct, 169
Principal strains of a transformation, 61
Principal warps, 149, see also Partial warps
Probability distribution, 190
Procrustes distance, 82–3, 93, 94, 389–90
full, 85, 115, 390
partial, 83, 85, 87, 93–6, 99, 389
Procrustes methods, 113–21
Procrustes residuals, see Shape coordinates, GLS
Procrustes superimposition, 113–19
axes of symmetry, and, 117
degrees of freedom, 97–8
partial, 93, 115
full, 94, 96, 99, 115
Reference configuration, 83
Selecting, 96–7
Regress6 program, 255–6
Regression, 229–59, 423
assumption of linearity, 236–40
bivariate, 232–4
comparison with PLS, 262–3
comparing regressions
directions of change, 249–53
rates of response, 254
correlation coefficient, 235
MANCOVA, 240–2
multivariate, 14, 58, 235–6
regression equations
bivariate, 230, 232
multivariate, 235–6
software, 255–9, 406
standardization by, 242–4
Relative warps, 170, 423, see also Principal
components analysis
Repeatability, 30
Repeated median as a criterion for
superimposition, 119–20
Resampling-based methods, 222–3
bootstrap, 195–9
jackknife, 200–1
Monte Carlo methods, 201–2
permutation tests, 199–200
Rescaling, 52
Residual variance, 263
Resistant-fit superimposition, 119–21
Resistant-fit theta-rho analysis, see Resistant-fit
superimposition
Resources available electronically, see Web
resources
Robust methods, 119
Rotation, 52, 74, 389
Rotational effects, 111
Row vector, 76
Saliences, 263
Sample, 190
Sampling with replacement, 196
Scalar, 209
Scale, 55
Scaling, 74, 388–9
Scaling factor, 425
Scatter-plot, 54
Scree plot, 168
SemiLand program, 406
Semilandmarks:
superimposition
differential weighting of landmarks and
semilandmarks, 399, 403
equal weighting of landmarks and
semilandmarks, 397–9
sliding to minimize bending energy,
399–401
sliding to perpendicular alignment on
reference, 401–2
sensitivity analysis, 403–4
statistical analysis, 404–5
software, 406
Shape:
definition of, geometric, 11, 72–3
space, 83–7, see also Kendall’s shape space,
pre-shape space, tangent space
variables, 5, 23–152, see also Bookstein
shape coordinates, superimposition,
partial warps
see also Shape change, shape, theory of
Shape change:
anisotropy of, 62
as a deformation, 131–2
covariance with separately measured factors,
see Regression
ontogenetic, see Ontogenetic trajectory
visualization of, 150–1
Shape coordinates, see Bookstein shape
coordinates, superimpositions
Shape space, 83–7, see also Kendall’s shape
space, tangent space
Shape, theory of, 72–104
configuration matrix, 76–9
configuration space, 77
position or location, 77–8
size, 78–9
definition of shape, 11, 72–5
dimensions and degrees of freedom of, 97–8
pre-shape space, 79–83
fibers in, 81–3
shape of, 79–80
reference configuration, 96–7
rotation to maximal similarity, 87
shape spaces, 83–7
tangent spaces, 95–6
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442 INDEX
Shape variable, 5, see also Bookstein shape
coordinates, superimposition, partial
warps
Shear, 137–42
ShuffleAllometry program, 258
Simple3D program, 406
Singular axes, 262
Singular value, 263
Singular value decomposition, 262–3
Singular warps, 262
Size, 11–13, 51–3, 55–6, 230, see also Centroid
size
Sliding baseline registration, 109–13
uniform transformations, calculation from,
142–3
Sliding semilandmarks, 399–402
Software, 19–20
Bookstein shape coordinates, 69
canonical variates analysis, 184–6
comparing directions of regression vectors,
257–9
comparing subspaces of variation, 317
customizing symbols for scatter-plots, 186–7
disparity, 314–17
MANOVA, 223–8
partial least squares analysis, 285–9
principal components analysis
three-dimensional landmarks, 406
two-dimensional landmarks, 181–4
regression
three-dimensional landmarks, 406
two-dimensional landmarks, 255–9
semilandmarks, 406
sources on the web, 19
standardization by regression, 257
traditional morphometric variables,
computing from landmarks, 358–60
superimposition
three-dimensional data, 406
two-dimensional data, 121, 127–8
variation, 314–17
see also individual programs
SpaceAngle program, 317
Standard6 program, 257
Standardization by regression, 242–4
Strain, see Principal strain
Strauss–Sadler estimate of range, 307
Student’s t-test, 58, 197–9, 211–12, 222,
see also Hotelling’s T
2
test
Superimposition, 105–8, 387–390
methods
generalized least squares Procrustes,
113–19, 387–390
resistant-fit, 119–21
sliding baseline registration, 109–13
three-dimensional landmarks, 387–390
resolving apparent inconsistencies among
methods, 124–6
software, 127–8, 406
SuperPoser program, 121, 127
Symmetric tensors, 61
t-test, see Student’s t-test
Tangent space, 95–6
Target form, 137–8, 147
Taxonomic discrimination, 365–7
Taxonomy, 364
TBox program, 226
Thin-plate spline, 129–52
non-uniform components from
three-dimensional landmarks, 392
non-uniform components from
two-dimensional landmarks, 143–50
physical metaphor, 133–4
shape change as deformation, 131–2
software, 151–2
uniform (affine) components, 134–43
visualization of shape change, 150–1
ThreeDPCA6 program, 406
ThreeDRegress6 program, 406
TIFF format for image files, 45
tpsDig program, 48–9, 406
tpsrelw program, 406
TradMorphGen program, 359–60
Translation, 51, 52, 57, 74, 77, 80, 84, 387–8,
see also Centered configuration matrix
Transpose of a matrix, 162
Transpositional allometry, 341
Triangles, shape coordinates, 52–5
Truss measurement scheme, 4–5
Two-point registration, 51–72, see also
Bookstein shape coordinates
TwoGroup program, 69, 71–2, 224–6
Type 1 landmark, 31–2
Type 2 landmark, 31–2
Type 3 landmark, 31–2
Type I error, 193
Type II error, 193
Uniform (affine) transformations, 134–43
Conceptual framework for, 137–8
Mathematical derivation of, 138–42
shear and compression/dilation terms,
137–42
six component model, 136
sliding baseline registration and, 142
Bookstein shape coordinates and, 142
RFTRA, and, 143
component scores, 181, 185
Variables, 190
selection of, 23–4
Variance, 162, 214
Variance-covariance matrices, 263, 295, 298
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INDEX 443
Variance transfer, 105–9
Variation, 295–6, see also Disparity
metrics, 296–7
structure, analysis of
nearest-neighbor analysis, 303–8
common subspace analysis, 308–14
see also Principal components analysis
VecCompare program, 257–8
VecDisplay program, 258–9
Vectors:
eigenvectors, 163–4, 167
inner (dot) product of, 251
see also Linear combinations, partial warps,
principal components analysis
Web resources, 19
Wilks’ lambda, 58, 178, 219, 236
Windows metafiles, 45
Index 4/6/2004 17: 31 page 444