Information Technology Reference
In-Depth Information
difficulties associated with the visual inspection of inner products from pairs of points
(see Section 2.3). Methods (ii) and (iii) merely interchange the roles of rows and columns,
so it suffices to discuss (ii). In (ii), rows are represented by
p
points and the columns
by
q
axes. As with PCA, axes may be calibrated and the devices of axis lambda-scaling
(Section 2.3) and axis shifts (Section 2.4) are available for improving presentation. In
addition, the axis calibrations may be adjusted to include and mark the relevant main
effects. To assess how well the columns of the
r
-dimensional approximation account for
the full representation in
Z
we may calculate
Z
Z
)
}{
diag
(
Z
Z
)
}
−
1
Column predictivities
={
diag
(
(6.9)
and associate these values with the axes. Similarly, for (iii) we may calculate
Z Z
)
}
−
1
Row predictivities
={
diag
(
)
}{
diag
(
ZZ
.
(6.10)
Note that (6.9) and (6.10) remain valid for all of (i), (ii) and (iii), but (ii) provides a
convenient means for displaying (6.9) and (iii) a convenient means for displaying (6.10).
6.5 Interpolating new rows or columns
The requirement to interpolate a new row or column into
X
is rather unlikely; usually it
would be simpler and better to redo the whole analysis on the enlarged
X
. However, it is
not an impossible requirement, so here we give the details of how to do it. The process
proceeds similarly to PCA, but modified to handle the main effects.
Given a new row
z
of
Z
, we merely have to calculate
z
VJ
,
z
V
−
1
/
2
J
or
z
V
−
1
J
according to how we have scaled the row coordinates in the display of
Z
.When
-scaling
has been used, this too must be taken into account. Thus, we have only to decide how
to calculate
z
, given a new row
x
of
X
. Clearly we have to eliminate the main effect
associated with the new row and also to adjust for the main effects associated with the
q
columns. From (6.7) this gives
z
=
λ
x
(
1
p
1
X
I
−
Q
)
−
(
I
−
Q
).
(6.11)
Because
V
is a singular vector of
Z
, which has zero row and column totals, it follows
that
1
VJ
=
0
and thence
QVJ
=
0
. Thus, when calculating
z
VJ
, (6.11) implies that
x
−
p
1
X
VJ
.
z
VJ
=
1
(6.12)
It follows that the predictivity of the new row
x
:1
×
q
is given by
z
VJV
z
z
z
,
which, on using (6.11) and (6.12), may be written as
1
1
p
1
X
)
(
x
−
p
1
X
)
VJV
(
x
−
p
1
X
)
.
(6.13)
1
1
(
x
−
p
1
X
)(
I
−
Q
)(
x
−
Note that predictivity is a function of
z
and does not depend on the particular factorization
used to give the biplot display.
