Biology Reference
In-Depth Information
Disparity, morphological disparity (
) Phenotypic variety, usually morphological. Several metrics can be used
to measure disparity, but the one most commonly used in studies of continuous variables is:
MD
P
j
5
1
D
j
ð
MD
5
N
1
Þ
2
where
D
j
is the distance of species
j
from the overall centroid (i.e. the grand mean calculated over
N
groups, e.g.
species) (Chapters 10, 11).
Distance A function measuring the separation between points. Within any space there are multiple possible dis-
tances. For this reason, it is necessary to specify the type of distance used. See also D, D
2
, Euclidean distance,
Generalized distance, Geodesic distance, Great circle distance, Partial Procrustes distance, Full Procrustes dis-
tance, Mahalanobis' distance (Chapter 4).
Dot product (Also called inner product.) Given two vectors
A
5
{
A
1
,
A
2
,
A
3
...
A
N
}, B
{
B
1
,
B
2
,
B
3
...
B
N
}, the dot
5
product of A and B is:
A
3
B
3
1
...:
1
A
B
5
A
1
B
1
1
A
2
B
2
1
A
N
B
N
and
A
B
5
j
A
jj
B
j
cos
ðθÞ
where
j
A
j
is the magnitude of A,
j
B
j
is the magnitude of B, and
θ
is the angle between A and B. If the magnitude
of A is 1, then A
), which is the component of B along the direction specified by A. The dot prod-
uct is used to calculate scores on coordinate axes, by projecting the data onto those axes (this is how partial warp
scores and scores on principal components are calculated). It is also used to find the vector correlation,
R
V
,
between two vectors (that correlation is the cosine of the angle between vectors).
Edge registration See Baseline registration.
Eigenvalues See Eigenvectors.
Eigenvectors Eigenvectors are the non-zero vectors, A, satisfying the eigenvector equation:
ð
X
2
λ
I
Þ
A
5
B
5
j
B
j
cos(
θ
0
The values of
that satisfy this equation are eigenvalue
s
of X. Eigenvectors are orthogonal to one another,
and provide the smallest necessary set of axes for a vector space (i.e. they provide a basis for that space). The
eigenvectors of a variance
λ
covariance matrix are called principal components; the eigenvalue corresponding to
each axis gives the variance associated with it. The eigenvectors of the bending-energy matrix are the principal
warps; the eigenvalue corresponding to each axis gives the bending energy associated with it. See also Basis
(Chapters 5, 6).
Element of a matrix A number in a matrix, typically referenced by the symbol designating the matrix with sub-
scripts indicating its row and column; for example,
X
4,5
refers to the element on the fourth row and fifth column
of the matrix X.
Euclidean distance The square root of the summed squared distances along all orthogonal axes. A Euclidean dis-
tance does not change when the axes of the space are rotated (in contrast to a Manhattan distance, which is sim-
ply the sum of the distances). See also D, D
2
, Distance, Generalized distance, Geodesic distance, Great circle
distance, Procrustes distance, Full Procrustes distance, Partial Procrustes distance (Chapter 4).
Euclidean space A coordinate space in which the metric is a Euclidean distance.
Exchangeable A concept used in designing permutation tests. If the null hypothesis used in the test states that a
property, such as group membership or size, is not a statistically significant factor, then that property is said to be
exchangeable under that hypothesis. If age is exchangeable in a given test, then the age of the specimens would
be permuted as part of the test. If age is significant, then this would be evident from the permutation test, as the
observed statistic would be unlikely (i.e. have a low probability, p) based on the distribution of the statistic over
the permuted data, and we would reject the null hypothesis as having a low probability of being true.
Explicit uniform term, explicit uniform component A uniform component describes affine or uniform deforma-
tions. Some of these do not alter shape (i.e. rotation, translation and rescaling) whereas others do (i.e. shear and
