Biology Reference
In-Depth Information
Using PCA to Find Characters
PCA provides a coordinate system for shape analysis, and may be useful for finding
characters, but first we must state an important caveat: individual PCs (like individual
PWs) cannot be viewed as characters in their own right. Just as a partial warp score is a
projection onto a single axis, so is a principal component score, and just as a similarity on
one PW does not indicate a similarity in shape, similarity on a single PC might not demon-
strate a sufficiently general (or detailed) similarity. Like PWs, PCs are context-dependent,
and thus we would not expect an individual PC to be a character any more than an indi-
vidual partial warp is.
Despite the similarities we just pointed out, analyses by PC and PW are not strictly
comparable
there is a major difference between them. PCs are orthogonal directions of
variation rather than orthogonal components of bending energy, and variation is biologi-
cally relevant to the problem at hand while bending energy is useful only in that it is used
by the method for depicting the results. PCs have a biological meaning, as orthogonal
dimensions of variance, even though that is not equivalent to the meaning of a character.
They are not likely to be characters in their own right because they are directions of varia-
tion that are constrained to be orthogonal (by definition), not directions of evolutionary
change. Directions of evolutionary change are likely to be oblique to the PCs
they are
within the space spanned by the PCs, but they need not lie along an axis nor must they be
orthogonal.
Although PCs are not likely to be characters, we may still find PCA useful for exploring
similarities and differences. The scatter-plots allow us to see the variation among taxa, and
their overlap, and both are important for finding characters. However, just as we need to
interpret partial warps in combination, so we also need to interpret PCs in combination.
Just because two or more species overlap in their PC1 scores does not mean that they are
similar with respect to all features described by PC1. They may differ in some, so that PC1
splits the difference between them and the other PCs describe what is specific to their
deviations from PC1. Taxa located in different quadrants of a scatter-plot may differ con-
siderably in shape, depending on the proportion of the variance described by each PC and
on how the PCs overlap in their descriptions of variation within the same regions. For
example, we can look at a case that should be familiar by this point
the first two PCs of
piranha juvenile body shape. The first, which accounts for 62% of the variance, clearly dis-
tinguishes three species (
S. manueli
,
S. elongatus
and
S. gouldingi
) from all others
(Figure 13.4
). Looking at the deformation that depicts the direction of greatest variance,
we can see that body depth contributes heavily to it. However, PC1 is not only body
depth; it also describes differences in proportions of the posterior body correlated with
body depth. Species with high scores on this axis have relatively long caudal peduncles
compared to the region between dorsal and adipose fins, as well as deep bodies, but we
cannot necessarily say that species with high scores on PC1 have long caudal peduncles if
other PCs also describe variation in posterior body proportions and scores on those PCs
differ among species with similar scores on PC1. PC2, which accounts for only 8.3% of the
variance, also describes variation in caudal body proportions and, on this component, spe-
cies with high scores have very short caudal peduncles relative to more anterior region.
Consequently, species with high scores on both components have a short caudal peduncle
