Information Technology Reference
In-Depth Information
Using a similar argument leading to the definition of axis adequacies in the case of
PCA (see (3.18), we define axis adequacy for CVA as the diagonal elements of the
p
×
p
matrix
MJM
)
MM
)
]
−
1
diag
(
[diag
(
.
(4.18)
Furthermore, for axis predictivity we proceed similar to before defining the diagonal
elements of the matrix
:
p
×
p
where
X
C X
)
[diag
(
X
CX
)
]
−
1
,
=
diag
(
(4.19)
justified by (4.15). This applies to the predictivities of the original variables; a similar
expression could be derived from (4.13) for the predictivities of the canonical variables.
From (4.17) and (4.19) we have that
tr[
(
X
CX
)
]
tr
Overall quality
=
,
X
CX
(
)
expressing overall quality as a weighted mean of the individual axis predictivities.
Class predictivity is defined by the diagonal elements of the matrix
:
K
×
K
where
=
diag
(
C
1
/
2
XW
−
1
X
C
1
/
2
)
[diag
(
C
1
/
2
XW
−
1
X
C
1
/
2
)
]
−
1
(4.20)
pertaining
to
the
canonical
means
and
justified
by
(4.14).
Result
(4.20)
simpli-
fies to
XW
−
1
X
)
[diag
(
XW
−
1
X
)
]
−
1
=
diag
(
for the weighted case after eliminating
C
.
The above measures refer to class predictivity, rather than sample predictivity; infor-
mation on the individual samples has not been used. For the samples, we look at
predictivities within classes as discussed below, that is, the predictivities of the indi-
vidual samples corrected for the class means, (
I
-
H
)
X
. These transform into canonical
variables (
I
-
H
)
XL
with the equivalent partition to (4.11),
XL
+
(
I
−
H
)(
X
−
X
)
L
,
(
I
−
H
)
XL
=
(
I
−
H
)
where
X
=
XMJM
−
1
. These are standard orthogonal projections, so Type A and Type B
orthogonality are satisfied: for Type B,
L
X
(
I
−
H
)
XL
=
L
X
(
I
−
H
)
XL
+
L
(
X
−
X
)
(
I
−
H
)(
X
−
X
)
L
,
(4.21a)
and for Type A,
XLL
X
(
I
−
H
)
+
(
I
−
H
)(
X
−
X
)(
X
−
X
)
(
I
−
H
).
(4.21b)
(
I
−
H
)
XLL
X
(
I
−
H
)
=
(
I
−
H
)
