Biology Reference
In-Depth Information
that the actual anatomical landmarks are really no more variable than in
Figure 3.6A and
B
changing the baseline does nothing but rotate and rescale the data. Although the con-
sequences for our perception of the shape differences can be dramatic, particularly when
it makes the data seem inordinately noisy, those consequences can be understood as
the result of a change in perspective. In general, it is easiest to interpret results when the
baseline lies along an organismal body axis. Even though results can be interpreted in a
baseline-invariant way, the interpretations still refer to sides of the triangle. It is most con-
venient when at least one side is a conventional and familiar reference. Thus, even though
we
can
interpret shape changes without reference to organismal body axes, we might still
wish to orient our findings with respect to them. This motivates choosing a baseline along
one of those axes.
SIZE
To this point we have talked only about shape. In the course of obtaining shape coordi-
nates, we lost no information about shape, but we removed all the information about size.
Specifically, we removed it by rescaling the baseline to a length of one. We can restore the
information about size by using a measure that captures the notion of scale
the property
that changes when an image is enlarged or reduced. There are several other meanings
of size, including a simple measure of the length of an organism (e.g. snout
vent length),
or area, volume, weight or even a linear combination of all measured quantities that
captures the positive correlations among them (as such as the first principal component).
In geometric morphometrics, we use a specific concept of size, one related to geometric
scale. One reason for choosing such a measure is that it is geometrically independent
of shape, at least under some models of error (
Bookstein, 1991
). To clarify this idea of
geometric independence, consider what happens when every dimension is enlarged by
the same proportion so that the organism gets larger without altering its shape. In this
case, each coordinate is moved away from the center in proportion to its original distance
from the center. The size variable that captures this radial notion of scale is centroid size,
graphically illustrated in
Figure 3.7
.
FIGURE 3.7
A geometric depiction of the calcula-
tion of centroid size: the square root of the summed
squared lengths of distances of landmarks from the
centroid: L1, L2, L3.
