Information Technology Reference
In-Depth Information
Moreover,
XX
=
XVJV
X
+
XV
(
I
−
J
)
V
X
.
(3.13)
We immediately have that
XX
=
X X
+
(
X
−
X
)(
X
−
X
)
.
(3.14)
The orthogonality result (3.14) is termed
Type A orthogonality for a data matrix
X
.
The standard result about the orthogonality of the fitted and residual sums of squares
follows immediately from the traces of either
X
X
or
XX
but the above has shown that
orthogonality applies to the
individual
terms of both matrices. In particular, it applies to
the diagonal elements of
X
X
and
XX
, equation (3.12) showing that we may apply it to
each of the columns of
X
and (3.13) that we may apply it to the individual rows of
X
.
Similar results pertain to the off-diagonal elements but we do not need them.
The matrices
V
and
needed above can also be computed from the SVD of
X
X
,
namely
X
X
=
V
2
V
=
V
V
.
(3.15)
The residuals are given by
X
(
I
−
VJ
)
with sum of squares
2
X
)(
X
−
X
)
}=
tr
{
XV
(
I
−
J
)
V
X
}
||
X
(
I
−
VJ
)
||
=
tr
{
(
X
−
from (3.13). Therefore the sum of squares for the residuals is
p
j
=
r
+
1
λ
j
.
tr
(
X
X
)
−
tr
(
X
XVJV
)
=
tr
(
−
J
)
=
(3.16)
The orthogonal analysis of variance ('Total sum of squares'
=
'Fitted sum of squares'
+
'Residual sum of squares') justifies a measure of overall
quality
of the display given
by the ratio
j
=
1
λ
j
j
=
1
λ
j
=
j
=
1
σ
2
tr
(
J
)
tr
(
)
j
quality
=
=
j
=
1
σ
j
.
(3.17)
2
Overall quality is only part of the story, and we may also want to know about the
quality of the variables as represented in
r
dimensions. Just as
XVJ
projects the samples,
so
IVJ
projects the unit points on every coordinate axis, to give
p
points that may be
regarded as representing the variables. Because
V
is an orthogonal matrix, its rows have
unit sums of squares and it follows that the sums of squares of the rows of
VJ
measure
the adequacy of representation for each variable. Algebraically this is given by the
p
×
p
diagonal matrix diag(
VJV
)
. We define the measure
adequacy
as
r
v
ij
=
i
th diagonal element of diag
(
VJV
).
adequacy
=
(3.18)
j
=
1
This is a measure of the adequacy of the representation for the
i
th variable. Adequacy
is used under different terminology in factor analysis, but happens also to be used, we
