Information Technology Reference
In-Depth Information
Ta b l e 3 . 1 0
Means of species
O. bullata, O. porosa
and
O. kenyensis
for each of six
variables.
VesD
VesL
FibL
RayH
RayW
NumVes
O.bul
98
.
10
412
.
00
1185
.
40
375
.
35
32
.
30
14
.
30
O.ken
137
.
29
401
.
71
1568
.
86
446
.
14
37
.
29
9
.
14
O.por
129
.
30
342
.
40
1051
.
70
398
.
20
39
.
40
14
.
80
Ta b l e 3 . 1 1
Overall quality (%) of the Figure 3.23 biplot in each of the six dimensions.
Dim 1
Dim 2
Dim 3
Dim 4
Dim 5
Dim 6
Quality
47.38
64.37
79.93
88.87
96.14
100.00
The call
PCA.predictivities(Ocotea.data[,3:8], scaled.mat = TRUE,
X.new.samples = new.samples)
returns the measures of fit given in Tables 3.11 - 3.14.
From Table 3.11 we see that the Figure 3.23 biplot has an overall quality of 64%.
Adding another dimension would increase the quality to 80% and constructing a three-
dimensional biplot can thus be considered. Inspecting the axis predictivities in Table 3.12
shows that
Ve s L
has a poor predictivity of 0.29, while
FibL
and
RayW
have predic-
tivities exceeding 0.79. Judging from the three-dimensional predictivities it seems that
all variables have satisfactory predictivities in three dimensions. Turning attention to the
individual samples, it is clear from Table 3.13 that some of the predictions for some
sample points could be made very accurately from the Figure 2.23 biplot (e.g.
S1
,
S2
,
S3
,
S35
), while others have a very low sample predictivity (e.g.
S37
). It is clear from
Table 3.14 that interpolating the sample mean of
O. kenyensis
into the biplot results in
a point whose predictions for the six variables can be very accurately inferred from the
two-dimensional PCA biplot. For the two other interpolated new points adding a third
dimension is desirable. In the next chapter we will see how a canonical variate analysis
biplot allows us to represent the means for all three species exactly in two dimensions.
3.5 Adding new axes to a PCA biplot and defining
their predictivities
We saw in Section 2.7 how to use the regression method for obtaining coordinates for
constructing a biplot axis for a new variable. Let
x
∗
:
n
1 denote a centred (and scaled,
if required) vector of sample values for a new variable. Note that this new variable is
centred and scaled using its own values analogous to the centring and scaling of the
original matrix
X
. It follows from (2.20) that the coordinates for adding a new biplot
axis representing
x
∗
are given by the first
r
nonzero elements (which we denote by
b
r
)
of
J
×
−
1
U
x
∗
=
as defined in Section 3.2. This added variable does not
play any role in providing the scaffolding axes for constructing the PCA biplot but its axis
predictivity can be defined as follows. For predictivity, we have to predict the values
of
x
∗
given the
r
-dimensional PCA coordinates
XVJ
. The usual orthogonal multiple
Jb
with
J
and
