Biology Reference
In-Depth Information
Cartesian coordinates Coordinates that specify the location of a point as displacements along fixed, mutually per-
pendicular axes. The axes intersect at the origin, or zero point, of all axes. Two Cartesian coordinates are needed
to specify positions in a plane (flat surface); three are required to specify positions in a three-dimensional space.
These coordinates are called “Cartesian” after the philosopher Descartes, a pioneer in the field of analytic
geometry.
Centered A matrix is centered when its centroid is at the origin of a Cartesian coordinate system; i.e. at (0, 0) of a
two-dimensional system or at (0, 0, 0) of a three-dimensional system (Chapter 4).
Centroid See Centroid position.
Centroid position The position of the averaged coordinates of a configuration of landmarks. The centroid posi-
tion has the same number of coordinates as each of the landmarks. The
X
-component of the centroid position is
the average of the
X
-coordinates of all landmarks of an individual configuration. Similarly, the
Y
-component is
the average of the
Y
-coordinates of all landmarks of an individual configuration. It is common to place the cen-
troid position at (0, 0), because this often simplifies other computations (Chapter 4).
Centroid size (CS) A measure of geometric scale, calculated as the square root of the summed squared distances
of each landmark from the centroid of the landmark configuration. This is the size measure used in geometric
morphometrics. It is favored because centroid size is uncorrelated with shape in the absence of allometry, and
also because centroid size is used in the definition of the Procrustes distance (Chapter 4).
Coefficient A number multiplying a function. For example, in the equation
Y
mX
,
m
is the coefficient for the
5
slope, which is the function that relates
X
and
Y
(Chapters 8, 9).
Column vector A vector whose entries are arranged in a column. Contrast to a Row vector.
Complex numbers A number consisting of both a real and an imaginary part. An imaginary number is a real
number multiplied by
i
, where
i
is
p
2
iY
, where
X
and
Y
are real
numbers. In that notation,
X
is said to be the real part of
Z
and
Y
is the imaginary part. A complex number is
often used to represent a vector in two dimensions. The mathematics of two-dimensional vectors and complex
numbers are similar, so it is sometimes useful to perform calculations or derivations in complex number form.
Configuration see Landmark configuration.
Configuration matrix A matrix representing the configuration of
K
landmarks, each of which has
M
dimensions.
A configuration matrix is a
K
1
. A complex number is written as
Z
X
5
1
M
matrix in which each row represents a landmark and each column represents
one Cartesian coordinate of that landmark;
M
3
2 for landmarks of two-dimensional configurations (planar
5
shapes), and
M
3 for landmarks of three-dimensional configurations. Two configuration matrices can differ in
location, size and orientation, as well as shape (Chapter 4).
Configuration space The set of all possible configuration matrices describing all possible configurations of
K
landmarks with
M
coordinates (all with the same values of
K
and
M
). Because there are
K
5
3
M
elements in the
configuration matrices, there are
K
3
M
dimensions in the configuration space. In statistical analyses, the configu-
ration space accounts for
K
M
degrees of freedom because that is the number of independent pieces of informa-
tion (e.g. landmark coordinates) needed to specify a particular configuration (Chapter 4).
Consensus configuration The mean (average) configuration of landmarks in a sample of configurations. Usually,
this is calculated after superimposing coordinates. See also Generalized Procrustes superimposition, Reference
form (Chapter 4).
Contraction A mathematical mapping that “shrinks” a configuration along one axis. A contraction along the
X
-axis would map the point (
X
,
Y
) to the point (
AX
,
Y
), where
A
is less than one. A contraction along the
Y
-axis
would map (
X
,
Y
)to(
X
,
AY
). Expansion or dilation is the opposite of contraction (
A
3
1).
Coordinates The set of values that specify the location of a point along a set of axes (see Cartesian coordinates).
Correlation A measure of the association between two or more variables. In morphometrics, correlation is most
often measured using Pearson's product-moment correlation, which is the covariance divided by the product of
the variances:
.
P
ð
Y
mean
Þ
P
ð
X
2
X
mean
Þð
Y
2
R
XY
5
q
P
ð
2
2
X
2
X
mean
Þ
Y
2
Y
mean
Þ
