Biology Reference
In-Depth Information
Shape variable A general term for any variable expressing the shape of an object, including ratios, angles, shape
coordinates obtained by a superimposition method, or vectors of coefficients obtained from partial warp analysis,
principal components analysis, regression, etc. Shape variables are invariant under translation, scaling and
rotation.
Shear An affine (or uniform) deformation that leaves the
Y
-coordinate fixed while the
X
-coordinate is displaced
along the
X
-axis by a multiple of
Y
. Under a shear, the point (
X
,
Y
) maps to (
X
AY
,
Y
)
,
where
A
is the magni-
tude of the shear. Visually, this looks like altering a square by sliding the top side to the left or right, without
altering its height or the lengths of the top and bottom (Chapter 5).
Singular axes Orthonormal vectors produced by singular value decomposition. See Singular value decomposi-
tion (Chapter 7).
Singular value In a singular value decomposition, a quantity expressing a relationship between two singular
axes; an element
1
λ
i
of the diagonal matrix S. In partial least squares analysis, each singular value represents the
covariance explained by the corresponding pair of singular axes. See Singular value decomposition (Chapter 7).
Singular value decomposition (SVD) A mathematical technique for taking an
M
N
matrix A (where
N
is
3
greater than or equal to
M
) and decomposing it into three matrices:
USV
T
A
5
where U is an
M
N
matrix whose columns are orthonormal vectors, S is an
N
N
diagonal matrix with on-diag-
3
3
onal elements
λ
i
are called the
singular values
of the decomposition, and the columns of U and V are called the
singular vectors
or
singular axes
corresponding to a given singular value. In partial least squares analysis, A is the matrix of covariances between
the two blocks, the columns of U are linear combinations of the variables in one of the two data sets, the columns
of V are linear combinations of the variables in the other data set, and each
λ
i
, and V is an
N
N
matrix whose columns are orthonormal vectors. The values
3
λ
i
is the portion of the total covariance
explained by the corresponding pair of singular axes (Chapter 7).
Singular warps Sometimes used in geometric morphometrics for singular axes computed from shape data (partial
warp scores or residuals of a Procrustes superimposition), so that the singular axes describe patterns of differ-
ences in shape. See Singular value decomposition (Chapter 7).
Size Any positive real valued function g(X), where X is a configuration or set of points, such that g(
A
X)
A
g(X),
where
A
is any positive, real scalar value. In other words, multiplying every element in X by
A
multiplies g(X)by
A
. There are a wide variety of measures of size, including lengths measured between landmarks, sums or differ-
ences of interlandmark distances, square roots of area, etc. The size measure used in geometric morphometrics is
centroid size. See also Centroid size (Chapters 3, 4).
Size-and-shape All the geometric information remaining in an object (such as a landmark configuration) after dif-
ferences in location and rotational effects are removed. See Procrustes SP, Form.
Space A set of objects (or measurements thereof) that satisfies some definition. For example, a space might be
defined as the set of all four-landmark configurations measured in two dimensions.
Statistic Any mathematical function based on an analysis of all measured individuals, e.g. the mean, standard
deviation, variance, maximum, minimum, and range. The true value of the statistic in the population is called the
parameter, which we are trying to estimate from our sample (Chapter 8).
Superimposition A method for matching two landmark configurations (or matrices) prior to further analysis,
sometimes also called a registration. A number of different optimality criteria may be used. See also Bookstein
coordinates, Procrustes superimposition (also Full Procrustes superimposition and Partial Procrustes superim-
position, RFTRA, Sliding baseline registration (Chapters 3, 4).
Strain See Principal strain.
Tangent space The linear vector space tangent to a curved space. In geometric morphometrics, the Euclidean
space tangent to Kendall's shape space. In the tangent space, distances between shapes are linear functions, which
allows for analysis of shape variation by ordinary multivariate statistical methods. When the linear approximation
to the curved surface is accurate (when all shapes in a study are close to the point of tangency), distances in the
tangent space approximate distances in the curved space. The point of tangency between Kendall's shape space
and the tangent space is the reference form. See also Kendall's shape space, Reference form (Chapter 4).
5
