Biology Reference
In-Depth Information
FIGURE 5.2
Ontogenetic change in body shape of
S. gouldingi, depicted both by relative displacements
of Bookstein shape coordinates and by the thin-plate
spline.
4
3
5
2
1
6
14
15
7
16
8
13
12
11
10
9
be used to produce 2k
7 in three dimensions) orthogonal basis vectors, which
allow us to express any shape change in the k landmarks using only 2k
2
4 (or 3k
2
7
coordinates, called partial warp scores, as described in detail later. There is no loss of
information in this process, and no changes in distances (in the linear tangent space)
between specimens caused by this change from landmark locations relative to the refer-
ence to locations along the axes produced by the thin-plate spline method. This is a
dimensionality reduction, from 2k to 2k
2
4or3k
2
7, with no loss of information,
leaving us with variables well suited to conventional, analytic multivariate statistical
techniques.
The situation becomes more complex when we start to use semilandmarks, as semiland-
marks have lost 1 degree of freedom per semilandmark after alignment. If we have a two-
dimensional system with k landmarks, and l semilandmarks, there will be 2k
2
4, or 3k to 3k
2
1
l
2
4
degrees of freedom in the data, but 2k
1
2l coordinates. These data can be expressed as
