Biology Reference
In-Depth Information
FIGURE 6.4
Graphical
interpretation of PC scores,
continued. The angles
α
2
indicate the relationship
of PC1 to the original axes
X
1
and
X
2
. Thus,
S
1
can be com-
puted from the coordinates of the star on
X
1
and
X
2
and
the cosines of the angles between PC1 and the original
axes.
S
2
can be computed from the coordinates of the star
on
X
1
and
X
2
and the cosines of the angles between PC2
and the original axes.
α
1
and
PC1
PC2
S
1
S
2
α
2
α
1
X
1
A
2
X
2
A
1
X
1
Y
1
52
1
(6.3)
A
1
X
1
A
2
1
Y
1
A
2
5
2
X
2
(6.4)
Then we make two substitutions (
M
A
1
/A
2
and
B
Y
1
/A
2
) to produce:
52
5
X
2
5
MX
1
1
B
(6.5)
Thus, the formula for the PC is, indeed, the formula for a straight line.
Algebraic Description of PCA
We begin this description of PCA by repeating the starting conditions and the con-
straints we want to impose on the new variable. We have a set of observations of
P
traits
on
N
individuals, where
P
is the number of shape variables (not the number of land-
marks). The data comprise
P
variances and
P(P
1)/2 covariances in the sample. We want
to compute a new set of
P
variables (PCs) with variances that sum to the same total as that
computed from the variances and covariances of the original variables, and we also want
the covariances of all the PCs to be zero. In addition, we want PC1 to describe the largest
possible portion of variance, and we want each subsequent PC to describe the largest
possible portion of the variation that was not described by the preceding components.
The full set of observations can be written as the matrix
X:
2
2
4
3
5
X
11
X
12
X
13
X
1
P
?
X
21
X
22
X
23
X
2
P
?
X
5
X
3
P
^ ^ ^ & ^
X
N
1
X
N
2
X
N
3
X
31
X
32
X
33
(6.6)
?
?
X
NP
