Biology Reference
In-Depth Information
that list
does
affect shape. Also, the list of operations that do not alter shape is useful
because we know that we are free to use those operations when we compare shapes
mathematically.
The entire theory of geometric morphometrics follows from the definition of shape, so
we need to develop it further. First, we need a more precise definition of a landmark.
When we discussed the criteria for choosing them in Chapter 2, we emphasized that the
criterion of homology has mathematical as well as biological implications. The mathemati-
cal implication follows from the formal definition of a landmark (
Dryden and Mardia,
1998
):
A landmark is a point of correspondence on each object that matches between and within populations.
The concept of matching encoded in that passage is not necessarily one of biological
homology, but the idea of correspondence is essential to the mathematical theory of shape.
If the landmarks do not correspond, we cannot compare shapes.
Another crucial idea is that of a
configuration of landmarks
; the full set of landmarks
recorded for each specimen. All comparisons of shapes are between matching configura-
tions of landmarks, not between individual landmarks (analyzed separately). An individ-
ual landmark is not an object of comparison because it does not satisfy the definition of
shape. The objects of comparison are entire configurations comprised of
K
landmarks
(where
K
refers to the number of landmarks), each of which has
M
coordinates (i.e.
M
2
for planar shapes). For example, in the case of the piranhas introduced in the second chap-
ter,
K
5
2. Whatever the number of landmarks and coordinates, our analyses
and conclusions are based on the
entire set
. Thus, if we have 16 landmarks with two coor-
dinates apiece, we have one shape
5
16 and
M
5
not 32 variables. No one landmark (and no one coor-
dinate) is a shape variable in its own right. Instead, we view each shape as the entire
configuration and we analyze samples of entire configurations.
This is a very different view of measurement (and variables) from that commonly
encountered in traditional morphometrics, where a single measurement might be viewed
as a variable, meriting analysis in its own right. It is common to analyze measurements
separately and to draw biological conclusions from them individually. Sometimes, the con-
clusions based on one measurement conflict with conclusions based on another, and the
inference often drawn in such situations is that the processes are trait-specific. In geomet-
ric morphometrics, individual measurements are not traits or even variables. Rather, a
shape variable is the entire vector of coefficients representing the complete difference in
landmark configurations between samples or, alternatively, the entire vector of coefficients
measuring the covariance between the landmark configurations and some other variable
(e.g. size).
This view of shape as a configuration of landmarks is central to the theory of geometric
morphometrics. Recognizing that, and conforming to the requirements it imposes on ana-
lytic methods, is crucial. It may seem biologically unreasonable to treat an entire shape as
a single entity, but the pay-off for doing so is the guarantee that our results do not depend
on arbitrary choices we happened to make in the course of an analysis. The reward for fol-
lowing what might seem like a rigid set of rules is the rigor and power of these methods,
as well as the visual appeal of the graphics.
