Information Technology Reference
In-Depth Information
We may check these results by verifying that the relevant cross products vanish. That
this is indeed so follows for Type B orthogonality from
(
I
−
H
)
XL
X
)
L
=
L
(
M
)
−
1
JM
X
(
I
−
H
)
X
(
I
−
MJM
−
1
(
I
−
H
)(
X
−
)
L
=
L
(
M
)
−
1
JM
WM
(
I
−
J
)
M
−
1
L
=
L
(
M
)
−
1
J
(
I
−
J
)
M
−
1
L
=
0
.
Type A orthogonality is the result of
XLL
(
X
−
X
)
(
I
−
H
)
=
(
I
−
H
)(
XMJM
−
1
)
MM
(
I
−
(
M
)
−
1
JM
)
X
(
I
−
H
)
=
(
I
−
H
)
XMJ
(
I
−
J
)
M
X
(
I
−
H
)
=
0
.
(
I
−
H
)
Thus, as for the between-class measures, Type A and Type B orthogonality are satisfied
within classes. Also
L
may be eliminated from the Type B result in (4.21a). The Type
B result is trivial because
L
X
(
I
−
H
)
XL
=
L
WL
=
I
, so it simplifies to
I
=
VJV
+
V
(
I
-
J
)
V
or, on eliminating
L
,to
W
=
(
M
)
−
1
JM
−
1
+
(
M
)
−
1
(
I
−
J
)
M
−
1
.
From (4.21a), after eliminating
L
we may define within-group axis predictivity as the
diagonal elements of the matrix
W
:
p
×
p
where
X
(
I
−
H
)
X
)
[diag
(
X
(
I
−
H
)
X
)
]
−
1
X
(
I
−
H
)
X
)
[diag
(
W
)
]
−
1
W
=
diag
(
=
diag
(
.
(4.22)
Result (4.22) gives an overall measure of predictivity for each variable (column) of
(
I
-
H
)
X
. A version could be constructed to give the axis predictivities within each
group separately. From (4.21b), for Type A, we define within-group sample predictivity
as the diagonal elements of the matrix
W
:
n
×
n
where
XW
−
1
X
(
I
−
H
))
{
diag((
I
−
H
)
XW
−
1
X
(
I
−
H
))
}
−
1
W
=
diag((
I
−
H
)
.
(4.23)
Notice once again that the matrix
X
in (4.22) as well as in (4.23) requires the appropriate
matrix
M
obtained from the currently defined matrix
C
.
Although the means of the
K
classes are exactly represented in
m
, or fewer, dimen-
sions, the individual samples are generally not exactly represented in fewer than
p
dimensions. The within-group sample predictivities can be less than unity in dimen-
sions
m
+
1,
...
,
p
−
1. Section 4.2 shows that there is some degree of arbitrariness in the
singular vectors corresponding to dimensions
m
+
1,
...
,
p
, making it invalid to examine
each of the arbitrary dimensions individually. The best that can be done is to com-
bine all the dimensions to give an overall within-group predictivity for the residual
space orthogonal to the space of the group means. This can be done by subtracting the
(
m
−
1)-dimensional predictivities from unity (see (4.20)).
