Biology Reference
In-Depth Information
small ones, they cannot be used as characters for at least two reasons. The first is
obvious (in hindsight at least): partial warps have a spatial scale only in so far as
spatial scale refers to the relative proximity of landmarks with the largest contrasting
displacements. In reality, every partial warp spans the entire organism and extends to
infinity as the implied deformation asymptotically approaches zero. Moreover, an indi-
vidual partial warp (PW) describes only part of a small-scale anatomical feature. No
matter how anatomically localized the change, its description by PWs will entail scores
on the largest scale PW that reflects the best fit of that PW to the data. Scores on pro-
gressively smaller scale PWs each reflect their fit to the
difference
between the data and
the sum of the higher scale PWs. Consequently, partitioning a change by PWs does not
correspond to partitioning it by anatomy or by characters. Even if a difference between
taxa did (fortuitously) closely match a single lower scale PW, its description would be
composed of scores on higher and lower scale PWs that represent that single, spatially
coherent change only when taken together. Furthermore, having a high score on a local-
ized PW does not mean that there is a localized change. Instead, it may simply mean
that in this particular region a large scale anatomical change is not well described by
the large scale PWs and the localized PW supplements that description. If the smaller
scale PW is taken out of context of the larger-scale PWs, we cannot make anatomical
sense of the one at smaller scale. Two taxa that have identical values for a small-scale
PW might differ anatomically
in that same region
because the differences between the
taxa cannot be seen without looking at all PWs.
All that may be obvious to readers who worked through the first several chapters, but
to clarify the point (and for those who jumped straight here) we can re-examine the exam-
ple that we found most promising at the time
the ontogenetic change in scores on one
PW (Figure 13.2). Two of the taxa, which were used as outgroups (
Pygopristis denticulata
and
Serrasalmus gouldingi
), have statistically significant ontogenetic change on that PW (in
both
X
and
Y
directions), whereas the three
Pygocentrus
do not. We would not normally be
concerned about similarities among outgroups, but this example shows that similarities
implied by individual PWs are not found in complete descriptions. That
P. denticulata
and
S. gouldingi
have anything in common in their development of that region is not at all
obvious when looking at more complete descriptions of the five ontogenies (Figure 13.3).
They are similar to each other, and differ from the three
Pygocentrus
, only in that they
undergo an ontogenetic change in the caudal peduncle region that is not fully described
by PWs at higher spatial scales. However,
P. denticulata
and
S. gouldingi
are not similar to
each other in the changes described by the higher spatial scales (and neither are the three
Pygocentrus
). Being similar in one PW does not mean being similar in shape (or ontogeny
of shape) in a particular anatomical region. When looking at one PW we lose the context
supplied by all the others, and PWs are all context-dependent. Therefore, we cannot
describe what happens within any one region of the body without placing every PW in
context of every other. Even judged by what the method was supposed to do, it fails; it
does not provide an objective, non-arbitrary method for decomposing changes (except in a
purely geometric sense).
The second reason partial warps cannot be used as characters, which is related to the
one above but important in a broader context, is that interpretations based on individual
variables violate the fundamental principles of geometric shape analysis
that results be
