Information Technology Reference
In-Depth Information
Expressions (4.13) and (4.14) pertain to the canonical means, whereas we are con-
cerned with predictivities for the original means. However, because
L
is nonsingular, it
may be eliminated from the Type B result in (4.13), implying that
X
C X
+
(
X
−
X
)
C
(
X
−
X
).
X
CX
=
(4.15)
Result (4.15) shows that Type B orthogonality remains available for the original means
but since the matrix
L
cannot be eliminated from (4.14), Type A orthogonality is not
available for the original variables. Because
LL
=
W
−
1
, the interpretation of Type B
orthogonality in (4.14) is merely that Mahalanobis squared distances may be split into
components in the approximation and residual spaces (Pythagoras).
The PCA measures of fit discussed in Section 3.3 and (4.15) enable us to define the
following measures of fit for the class means in CVA. From (4.13) we may define
tr
(
L
X
C XL
)
tr
(
L
X
CXL
)
Overall quality
=
.
Now,
tr
(
L
X
C XL
)
=
tr
(
V
L
X
C XLV
)
=
tr
(
M
X
C XM
)
=
tr
(
JM
X
CXMJ
)
=
tr
(
J
J
)
=
tr
(
J
).
The denominator is obtained in the same way but with
J
replaced by
I
. Thus
j
=
1
λ
j
j
=
1
λ
j
tr
)
tr
(
)
(
J
Overall quality
=
=
,
(4.16)
where, of course, at most the first
m
=
min(
K
-1,
p
)
eigenvalues obtained from (4.4)
may be nonzero.
It follows that when
r
>
m
, overall quality is always unity and
X
=
X
.Themea-
sure (4.16) is the equivalent of the PCA measure (see (3.17)) and in CVA is concerned
with the quality of fit in the canonical variables. Therefore, we denote the overall qual-
ity (4.16) as
Overall quality
Canvar
. We may be interested in a similar measure based
directly on the original variables derived from the Type B orthogonality (4.15). This
gives tr
(
X
C X
)/
tr
(
X
CX
)
.Now
X
C X
)
=
tr
{
M
−
1
J
[
M
X
CXM
]
JM
−
1
}=
tr
{
M
−
1
J
[
]
JM
−
1
}=
tr
{
(
M
M
)
−
1
J
J
}
.
tr
(
The denominator is obtained in the same way but with
J
replaced by
I
. Thus, we have
j
=
1
λ
j
m
jj
j
=
1
λ
j
m
jj
,
X
C X
tr
)
tr
(
X
CX
)
=
(
Overall quality
Origvar
=
(4.17)
where
m
jj
represents the
j
th diagonal value of (
M
M
)
−
1
. This would agree with (4.16)
if we had
m
jj
=
1forall
j
.
