Glossary - Geometric Morphometrics for Biologists - page 455

distributed over all landmarks. Resistant-fit methods, such as RFTRA, were devised to avoid that implication
(Chapter 4).
Population The set of all possible individuals of a specific type, such as all members of a species, or all leaves on
a particular kind of tree. See also Sample (Chapter 8).
Position See Centroid position.
Pre-shape A centered landmark configuration, scaled to unit centroid size (Chapter 4).
Pre-shape space The set of all possible pre-shapes for a given number of landmarks with a given number of
dimensions. This is the surface of a sphere of KM 1 dimensions, where K is the number of landmarks and

M is the number of dimensions of each landmark (Chapter 4).
Principal axes The set of orthogonal axes used in modeling the change of one shape into another as an affine
transformation. This transformation can be parameterized by its effect on a circle or sphere (for two or three
dimensional shapes, respectively). In two dimensions, an affine transformation takes a circle into an ellipse and
the principal axes are the directions of the circle that undergo the greatest relative elongation or shortening
mapped onto the major and minor axes of the ellipse. The ratio of the lengths of these axes is the anisotropy, a
measure of the amount of affine shape change. Principal axes are invariant under a change in the coordinate sys-
tem. See also Principal strains (Chapters 3, 4).
Principal components analysis (PCA) A method for reducing the dimensionality of multivariate data, performed
by extracting the eigenvectors of the variance

2 M 2

covariance matrix. These eigenvectors are called principal compo-
nents. Their associated eigenvalues are the variance explained by each axis. Principal components provide an
orthonormal basis. The position of a specimen along a principal component is represented as its principal compo-
nent score, calculated by taking the dot product between that principal component and the data for that specimen
(Chapter 6).
Principal strain In an affine deformation, the ratio of the length of a principal axis in the ellipse to the original
diameter of the circle. See also Principal axes (Chapter 3).
Principal warp An eigenvector of the bending-energy matrix interpreted as a warped surface over the surface of
the X , Y -plane of the landmark coordinates. Principal warps are ordered from least to most bending energy (smal-
lest to largest eigenvalue), which corresponds to the least to most spatially localized deformation. Principal warps
differ from partial warps in that partial warps are projections of principal warps onto the X , Y -plane of the data.
See also Bending energy, Bending-energy matrix, Orthonormal basis, Partial warp, Thin-plate spline
(Chapter 5).
Probability distribution A mathematical function that describes the probability of a measurement taking on
either a particular value or a range of values, depending on whether the variable is discrete or continuous, respec-
tively (Chapter 8).
Procrustes distance This term has been used to refer to the sum of squared distances between corresponding
points of two superimposed shapes after one shape has been centered on the other and rotated to minimize that
sum of squares. When the shape being superimposed is reduced in centroid size to minimize further the differ-
ence between it and the target, the distance may be called a Full Procrustes distance ( D F ). When both sizes are
held at centroid size

1, the distance may be called a Partial Procrustes distance ( D p ). Both D F and D p are related 5

to the arc distance ( ) between configurations in the space of aligned preshapes with centroid sizes fixed at 1.

This arc length has also been called a Procrustes distance, with the others called full (or partial) Procrustes chord
distances to distinguish them from the arc length. See also Full Procrustes distance, Partial Procrustes distance
(Chapter 4).
Procrustes Form Space An approach to analyzing both size and shape. In this method, a data matrix is formed of
the coordinates in Procrustes superimposition and a column of the log centroid size values is added. This data
matrix is then analyzed using a PCA or in hypothesis testing procedures (Chapter 14).
Procrustes methods A general term referring to the superimposition of matrices based on a least squares crite-
rion. The term comes from the Greek mythological figure, Procrustes, who fitted visitors to a bed by stretching
them or amputating overhanging parts (Chapter 4). See Full Procrustes, Partial Procrustes.
Procrustes residuals Coordinates of a landmark configuration obtained by a Procrustes superimposition. They
are residuals in the sense that they indicate the deviation of each specimen from the mean (i.e. the consensus

ρ