Biology Reference
In-Depth Information
Procrustes distance); and the discrepancy between these distances increases as
increases
and the distance in the tangent plane approaches infinity. The other approach to projecting
from one space onto another is to project along lines that are orthogonal to the new space.
Point
ρ
onto the tangent plane, and this projec-
tion produces distances from the reference in the tangent plane that are
less
than the
Procrustes distance. As in the stereographic projection, the magnitude of the discrepancy
between the distances increases as
E
represents the orthogonal projection of
B
increases, but in the orthogonal projections, distances
in the tangent plane asymptotically approach the maximum equal to the radius of the
shape space.
The different projection methods do have different statistical properties (
Rohlf 1999,
2000; Slice 2001
). Most of the applications discussed in this text will be using partial
Procrustes superimposition, which is an orthogonal projection from the oriented pre-shape
space hemisphere onto the linear tangent space (see
Figure 4
,
Rohlf, 1999
or
Figure 4.18
of
the text).
ρ
Selecting the Reference Configuration
Many of the steps involved in placing target configurations in shape space, or in the
Euclidean space tangent to it, are functions of the reference shape (although the strict defi-
nition of Kendall's shape space does not require a reference, we use this approach to
describe one way to construct such a space). For example, in the construction of a shape
space, each target configuration is rotated to the orientation that minimizes its distance
from the reference. Also, in the construction of Kendall's shape space, the scaling of each
target configuration is a function of its distance from the reference. Moreover, the tangent
space is tangent to shape space at the reference. Perhaps most important, the discrepancies
between distances in the tangent space and those in shape space increase as a function of
distance between target and reference. Thus, the choice of reference can have important
consequences.
Most interesting biological questions will be concerned with differences among more
than two specimens. The inferences based on analyses of multiple specimens will be based
on all of the distances among specimens, not just their distances from the reference.
Accordingly, the choice of a reference must consider the effects of that choice on approxi-
mating distances among target specimens, not just distances of target specimens from the
reference. Not only will distances from the reference be distorted, so too will the distances
among target specimens, and this distortion will also be a function of their distances from
the reference. If these distortions are large, inferences based on distances in the Euclidean
tangent space will be unreliable.
One possible reference is the average shape of the entire sample (computed using meth-
ods discussed in Chapter 5). This approach has the advantage that it minimizes the aver-
age distance from the reference, which minimizes the average distortions of interspecimen
distances projected to the tangent plane (
Bookstein, 1996; Rohlf, 1998
). However,
Marcus
et al. (2000)
analyzed differences in skull shape among representatives of several mamma-
lian orders and found that most Procrustes distances are closely approximated by the
Euclidean distance in the tangent space. The principal exceptions were the distances from
