Biology Reference
In-Depth Information
Ordinary Procrustes analysis (OPA) An approach to superimposition in which one landmark configuration is fit-
ted to another, which differs from a Generalized superimposition in that it involves only two forms. This
approach has rarely been used once iterative methods became available for generalized superimpositions. See
also Generalized Procrustes Analysis (GPA), Generalized superimposition, Consensus form (Chapter 4).
Ordination Ordering specimens along one or more axes based on some criterion (e.g. from youngest to oldest, or
shortest to tallest). Ordination methods include principal components analysis and canonical variates analysis; the
scores on the axes provide a basis for ordering specimens (Chapter 6).
Orthogonal Perpendicular (at right angles to each other). Two vectors are orthogonal if the angle between them is
90
; when they are, their dot product is zero.
Orthonormal Perpendicular and of unit length; vectors are orthonormal if they are mutually orthogonal and of
unit length.
Orthonormal basis See Basis. An orthonormal basis is comprised of a set of mutually perpendicular basis vectors
normalized so that their magnitudes are all set to one.
Outline A curve around the perimeter of an object (or around a distinct part of it).
Partial least squares analysis A method of exploring patterns of covariance or correlation between two blocks of
variables measured on the same set of specimens. A singular value decomposition is used to determine the pair
of vectors (each a linear combination of variables within one of the blocks) that expresses the greatest proportion
of the covariance between blocks. See also Singular value decomposition, Singular warps (Chapter 7).
Partial Procrustes distance (
D
p
) The distance between two landmark configurations when both shapes are cen-
tered, fixed to unit centroid size, and rotated to minimize the sum of squared distances between their correspond-
ing landmarks. See also Full Procrustes distance, Procrustes distance (Chapter 4).
Partial Procrustes superimposition The superimposition that yields the Partial Procrustes distance of a shape
from the reference, achieved by fixing centroid size at 1. See also Partial Procrustes distance (Chapter 4).
Partial warps The term partial warps sometimes refers solely to the components of the non-uniform deformation,
which are computed as eigenvectors of the bending-energy matrix projected onto the
X
,
Y
-plane of the data (they
are projections of principal warps), ordered from least to most bending energy. These eigenvectors provide an
orthonormal basis for the non-uniform part of a deformation. Sometimes “partial warps” also includes the com-
ponents of the uniform deformation, as the zeroth partial warp
in which case the scores on this component are
included among the partial warp scores (Chapter 5).
Partial warp scores Coefficients indicating the position of an individual, relative to the reference, along partial
warps. They are calculated by taking the dot product between the partial warps and the data for a specimen.
When appropriate scores on the uniform component are also included among the partial warps scores, the sum
of the squared scores equals the squared partial Procrustes distance of that specimen from the reference. This full
set of scores can be used as shape variables in any conventional statistical analysis because they are based on the
appropriate distance measure and have the same number of coordinates as degrees of freedom. See also Non-uni-
form deformation, Partial warps, Principal warps, Uniform deformation (Chapter 5).
Permutation test An approach to statistical testing that involves permuting (rather than randomly sampling)
observed values. Unlike many bootstrapping methods, the resampling done is without replacement, each speci-
men appears only once in the permutation set. See also Bootstrap test, Jackknife test, Monte Carlo simulations
(Chapters 8, 9).
Phylogenetic generalized least squares A method of accounting for the expected non-independence of observa-
tions on taxa by incorporating the phylogenetic variance
covariance matrix (a function of branch lengths between
taxa) in statistical analyses (Chapter 10).
Phylogenetic independent contrasts A method of computing independent evolutionary changes from observa-
tions of taxa that are related by varying degrees of common ancestry. Sometimes called simply independent con-
trasts, these are net differences between daughter nodes (including tips) of the phylogenetic tree, weighted by
branch lengths between the nodes (Chapter 10).
Pinocchio effect A large change concentrated at one landmark, with little or none at others; a highly localized
change. In the presence of the Pinocchio effect, Procrustes superimpositions imply that the shape difference is
