Information Technology Reference
In-Depth Information
(see Chapter 10); Cox and Cox (2001) or Borg and Groenen (2005) may be consulted
for overviews. In Chapter 3 we treated PCA as a matrix approximation method, but it
may be brought into the purview of this chapter as a method for minimizing the criterion
(5.1) where now
δ
ij
is constrained to be confined to distances between points arising
from projecting
X
onto a low-dimensional space.
In MDS the distances in
D
∗
may be observed directly or may be calculated from
sample values given in a data matrix
X
. In the former case information on which to base
a biplot does not exist (see Chapter 10) but it does in the latter case.
In this chapter we shall see how biplots may be found in some cases of metric MDS.
Gower
et al
. (1999) give an example of how to use the regression method to construct
linear biplots with nonmetric scaling, the nonlinearity being absorbed by nonlinear cali-
bration of the axes. Gower and Hand (1996) outlined, without implementation, a method
for constructing so-called coherent nonlinear biplots, in the context of minimizing the
metrics
stress
and
sstress
. Here, we shall be concerned with the regression method and
nonlinear biplots.
It is important to understand that MDS methods are concerned with fitting
δ
ij
to
d
ij
;thatis,
δ
ij
predicts
d
ij
, whereas biplots are concerned with the approximation of
X
to
X
with associated graphical visualizations. This ambiguity of purpose raises some
awkward issues: for example, the axis predictivities derived in Chapters 3 and 4 for
the approximation of
X
to
X
rest upon certain orthogonality relationships; these are
unavailable in the MDS context of approximating distances.
5.2 The regression method
In PCA, we saw in (2.3) that
X
:
n
×
p
=
U
V
is approximated by the rank
r
matrix
X
[
r
]
=
U
JV
=
UJ
V
=
UJ
JV
=
XVJV
,
so that the plotted points are given by
XV
r
=
Z
, say. If we calculate the regression
X
=
Z
of the data
X
on
Z
we obtain 'regression' coefficients
={
(
V
r
)
X
XV
r
}
−
1
(
V
r
)
X
X
=
(
r
)
−
1
r
(
V
r
)
=
(
V
r
)
,
(5.3)
where
X
X
V
, with
V
r
defined as the matrix consisting of the first
r
columns of
V
(see Section 3.2) and
2
V
=
=
V
V
r
defined as the
r
×
r
diagonal matrix formed from
the first
r
diagonal elements of
.
In Section 3.5 we saw how to add new (linear) biplot axes to a PCA biplot using the
regression method. The regression method also forms the basis for deriving the expression
for calculating the axis predictivity of the newly added axis. Moreover, in Sections 4.5
and 4.6 we used the regression method for adding new biplot axes together with their
axis predictivities in the case of a CVA biplot. The regression method has much wider
applicability: the columns of (
V
r
)
in (5.3) are precisely the directions of the PCA biplot
axes. This suggests that in other types of MDS with fitted coordinates
Z
,wemayuse
regression to provide linear biplot axes by setting
=
(
Z
Z
)
−
1
Z
X
.Then
x
k
=
Z
γ
k
,
where
γ
k
is the
k
th column of
, predicts the
k
th variable (i.e. the
k
th column of
X
.) It
follows that the unit marker on the
k
th axis is given by
γ
k
/
γ
k
γ
k
.
(5.4)
