Information Technology Reference
In-Depth Information
V2
2
1
q
c
d
u
r
j
v
t
h
p
e
0
1
m
n
g
f
s
i
2
1
b
V4
0.5
k
w
a
1
3
1
V5
1
2
2
1.5
V1
3
2
4
V6
V3
Figure 3.21
PCA biplot of data given in Table 3.4.
Ta b l e 3 . 5
Two-dimensional adequacies associated with the six variables of the
artificially constructed two-dimensional data set in Table 3.4.
Va r i a b l e
Va r 1
Va r 2
Va r 3
Va r 4
Va r 5
Va r 6
Adequacy
0.252
0.326
0.111
0.114
0.428
0.770
measures the degree to which the corresponding approximation agrees with the corre-
sponding true element in
X
. Since both
X
X
and
XX
yield orthogonal decompositions
of the sums of squares, we can define
axis predictivity
as the diagonal elements of the
matrix
:
p
×
p
given by
X
X
)
[diag
(
X
X
)
]
−
1
=
diag
(
V
JV
)
[diag
(
V
V
)
]
−
1
,
=
diag
(
(3.19)
and similarly
sample predictivity
as the diagonal elements of the matrix
:
n
×
n
given
by
X X
)
[diag
(
XX
)
]
−
1
=
diag
(
U
JU
)
[diag
(
U
U
)
]
−
1
=
diag
(
.
(3.20)
The diagonal elements of
are our measures of the predictive powers of each
variable and clearly cannot exceed unity. We term the
i
th element the axis predictivity
