Biology Reference
In-Depth Information
Hypersphere The generalization of a three-dimensional sphere to more than three dimensions. In three dimen-
sions, points on the surface of a sphere of radius
R
that
is centered at
the origin satisfy the equation
X
2
R
2
.
Implicit uniform terms See Explicit uniform terms.
Induced correlation A correlation induced by dividing two values by a third which is common to both. The
induced correlation between the (rescaled) variables is not present in the original variables.
Inner product See Dot product.
Invariant A quantity is invariant under a mathematical operation or transformation when it is not changed by
that operation. For example, centroid size is invariant under translation, centroid position is not.
Isometric In general, a transformation that leaves distances between points unaltered. In morphometrics, isometry
usually means that shape is uncorrelated with size. In statistical tests of allometry, isometry is the null hypothesis
(Chapters 8, 11).
Isotropic A property is said to be isotropic if it is uniform in all directions, i.e. if it does not differ as a function of
direction. When an error is isotropic, it is equal in all directions, and there is no correlation among errors.
Isotropic is the opposite of anisotropic.
Jackknife test An approach to statistical testing that involves resampling the original observations to generate an
empirical distribution. Jackknifing is carried out by omitting one specimen at a time. See also Bootstrap test,
Permutation test (Chapter 8).
Kendall's shape space The space in which the distance between landmark configurations is the Procrustes dis-
tance. This space is constructed by using operations that do not alter shape to minimize differences between all
configurations of landmarks that have the same values of
K
(number of landmarks) and
M
(number of coordi-
nates of a landmark). Kendall's shape space is the curved surface of a hypersphere, so conventional statistical
analyses are conducted in a Euclidean tangent space (Chapter 4).
Landmark Biologically,
Y
2
Z
2
1
1
5
landmarks are discrete, homologous anatomical
loci; mathematically,
landmarks are
points of correspondence, matching within and between populations (Chapter 2).
Landmark configuration The positions (coordinates) of a set of landmarks representing a single object, containing
information about size, shape, location and orientation. The number of landmarks is typically represented by
K
,
and the dimensionality of the landmarks (number of coordinates) is typically represented by
M
. Therefore, if
there are 16 landmarks, each with an
X
- and
Y
-coordinate, then
K
2
(Chapter 4).
Least squares A method of choosing parameters that minimizes the summed square differences over all indivi-
duals (and variables) (Chapters 4, 7, 8).
Linear A function f(
X
) is linear if it depends only on the first power of
X
; e.g. f(
X
)
5
16 and
M
5
(
X
)
2
2(
X
) is linear, but f(
X
)
5
5
is not.
Linear combination A vector produced by multiplying and summing coefficients of one or more vectors. For
example, given the vector X
T
X
N
} and A
T
5
{
X
1
,
X
2
...
5
{A
1
,
A
2
...
A
N
}, then Y
5
A
1
X
1
1
A
2
X
2
1...
A
N
X
N
is a
A
T
X.
Linear transformation A transformation producing a set of new vectors that are linear combinations of the origi-
nal variables. See Linear combination.
Linear vector space The set of all linear combinations of a set of vectors. The space spans all possible linear com-
binations of the basis vectors, as well as all sums or differences of any linear combination of those basis vectors.
The two-dimensional Cartesian plane is the linear vector space formed by the linear combinations of two vectors
of unit length, one along the
X
-axis, the other along the
Y
-axis.
Mahalanobis' distance (D
2
) The squared distance between two means divided by the pooled sample variance
linear combination of the vectors. We can write this as Y
5
covariance matrices. This is a generalized statistical distance, adjusting for correlations among variables. See also
D, Generalized distance.
MANCOVA Multivariate analysis of covariance. A method for testing the hypothesis that samples do not differ
in their means when the effects of a covariate are taken into account. See also General Linear Models (GLM),
ANOVA, ANCOVA and MANOVA (Chapters 8, 9).
