Biology Reference
In-Depth Information
and simplified to:
SA
1
2
λ
1
A
1
5
0
(6.33)
and further rearranged so that:
SA
1
5
λ
1
A
1
(6.34)
This leads to the following substitutions and rearrangements of
Equation (6.26)
:
s
Y
1
5
A
1
SA
1
5
A
1
λ
1
A
1
5
λ
1
A
1
A
1
5
λ
(6.35)
1
Thus, the eigenvalue
λ
1
is the variance of
Y
1
.
Interpretation of Results
As we stated above, PCA is nothing more than a rotation of the original data; it is sim-
ply a descriptive tool. The utility of PCA lies in the fact that many (if not all) of the fea-
tures measured in a study will exhibit covariances because they interact during, and are
influenced by, common processes. Below, we use an analysis of jaw shape in a population
of tree squirrels to demonstrate how PCA can be used to reveal relationships among traits.
Fifteen landmarks were digitized on the lower jaws of 31 squirrels (
Figure 6.5
). These
landmarks capture information about the positions of the cheek teeth (2
5), the incisor
(1, 14 and 13), muscle attachment areas (6, 9
12, 15) and the articulation surface of the jaw
joint (7 and 8). The 31 specimens include 23 adults and 8 juveniles (individuals lacking
one or more of the adult teeth).
Figure 6.6
shows the landmark configurations of all 31 specimens, after partial
Procrustes superimposition. This plot does not tell us much beyond the fact that there is
shape variation in the sample. We can infer from the areas of the scatters for individual
landmarks that there is not much variation in the relative positions of the cheek teeth. In
contrast, many of the ventral landmarks have noticeably larger scatters, suggesting that
their positions relative to the teeth are more variable.
FIGURE 6.5
Outline drawing of the
lower jaw of
6
the fox squirrel,
Sciurus
7
niger
,
showing the
locations of
15
8
landmarks.
2
5
3
4
9
1
10
15
14
13
12
11
