Biology Reference
In-Depth Information
FIGURE 5.1
Ontogenetic change in body shape
of Serrasalmus gouldingi, depicted by relative displa-
cements of Bookstein shape coordinates.
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3
5
2
6
1
15
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distance between them). Even the landmarks that are not displaced relative to others must
be considered. What we need is a method for visualizing changes between landmarks
over the entire form.
That visualization is the primary purpose of the thin-plate spline. Using it, we can inter-
polate between landmarks, taking all displacements of all landmarks relative to all others
into account (
Figure 5.2
). There are many different types of interpolation functions avail-
able (see
Dryden and Mardia, 1998
), all of which attempt to estimate unknown displace-
ments between landmarks from known displacements at the landmarks. The thin-plate
spline method uses a mathematical approach to this interpolation that is optimally
smooth, producing interpolations with the fewest possible abrupt changes or differences.
Other spline methods are available, but have seen little, if any, use in geometric morpho-
metrics. Deformation grid plots based on thin-plate splines are very effective for
two-dimensional data. Effective presentation of three-dimensional data using the spline is
challenging and other methods, such as wireframes, tend to be used to represent changes
in three-dimensional data.
The other major purpose of the spline has been mentioned previously in this text.
As mentioned several times in this text, we need a set of shape variables to use in conven-
tional statistical tests, by which we mean that the number of variables matches the degrees
of freedom in the data. Specifically, when working with two-dimensional landmark data,
we need a set of variables that spans the entire space of our data but numbering only
2K
1)/2))
where K is the number of landmarks in M dimensions). The thin-plate spline method can
2
4 for two-dimensional data (more generally, numbering (KM
2
1
2
M
2
(M (M
2
