Biology Reference
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dilation). Accordingly, we divide affine deformations into two sets: (1) implicit uniform terms, which do not alter
shape and are used in superimposing forms but are not explicitly recorded; and (2) explicit uniform terms, which
do alter shape and therefore are typically reported as components of the deformation. All uniform terms must be
known to model a deformation correctly (Chapter 5).
Fiber In geometric morphometrics, the set of all points in pre-shape space representing all possible rigid rotations
of a landmark configuration that has been centered and scaled to unit centroid size; in other words, the set of
pre-shapes that have the same shape. Fibers are collapsed to a point in shape space (Chapter 4).
Form Size-plus-shape of an object; form includes all the geometric information not removed by rotation and trans-
lation. Form is also called Size-and-shape.
Full Procrustes distance (
D F ) The distance between two landmark configurations when, after partial Procrustes
superimposition, the shape superimposed on the reference is rescaled to cs
) further to minimize the sum
of squared distances between their corresponding landmarks. See also Partial Procrustes distance, Procrustes dis-
tance (Chapter 4).
Full Procrustes superimposition The superimposition that yields the Full Procrustes distance of a shape from the
reference, achieved by reducing centroid size to cos(
5
cos(
ρ
). See also Full Procrustes distance (Chapter 4).
ρ
F
-test One of a variety of test statistics, formed as a ratio of the variance explained by a model or a factor to the
unexplained or error variance estimate. In some cases, the observed F -value is compared to an applicable analytic
model, otherwise permutation tests are commonly used to determine the associated p-value.
Generalized distance See D.
Generalized least squares superimposition A generalized superimposition method that uses a least squares fitting
criterion, meaning that the parameters are estimated to minimize the sum of squared distances over all landmarks
over all specimens. Usually, in geometric morphometrics, GLS refers specifically to a generalized least squares
Procrustes superimposition
a different approach is used in generalized resistant-fit methods (Chapter 4).
Generalized least squares Procrustes superimposition (GLS) A generalized superimposition minimizing the par-
tial Procrustes distance over all shapes in the sample, using a least squares fitting function. This is the method
usually used in geometric morphometrics; it is now usually termed Generalized Procrustes Analysis (Chapter 4).
General Linear Models (GLM) A generalization of the ideas behind MANOVA, MANCOVA and regression
models, to encompass any model that is linear in its fitted parameters. The statistical significance of the covariates
and factors in the model are typically assessed using F -tests (Chapter 9).
Generalized Procrustes Analysis (GPA) A Procrustes-based analysis using generalized superimposition to esti-
mate iteratively the mean and then superimpose all specimens on it. This is now the standard term for a
Generalized least squares Procrustes superimposition. See Generalized superimposition.
Generalized superimposition The superimposition of a set of specimens onto their mean. This involves an itera-
tive approach because the mean cannot be calculated without superimposing specimens, which cannot be super-
imposed on the mean before the mean is calculated (an alternative approach is used in ordinary Procrustes
analysis). See also Consensus configuration (Chapter 4).
Geodesic distance The shortest distance between points in a space. On a flat planar surface, this is the length of
the straight line joining the points
i.e. the Euclidean distance. On curved surfaces, this distance is the length of
an arc.
Great circle The intersection of the surface of a sphere and a plane passing through its center. A great circle
divides the surface of the sphere in half. On the surface of the sphere, the shortest distance between two points
lies along the great circle that passes through those points. If the Earth were perfectly spherical, the equator and
all lines of latitude would be great circles.
Great circle distance The arc length of the segment of the great circle connecting two points on the surface of a
sphere; this is the geodesic distance between those points, the shortest distance between the points in the space of
the surface of the sphere.
Homology (1) In biology, similarity due to common evolutionary origin. (2) In morphometrics, the correspon-
dences between landmarks, sometimes imputed by a mathematical function, called a “homology function” (e.g.
see Bookstein et al., 1985). Homology is the primary criterion for selecting landmarks (Chapter 2).
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