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d
d
= d
c
(d
ii
′
,
d
ii
′
)
d
ii
′
(
d
ii
′
+
d
ii
′
,
d
ii
′
+
d
ii
′
)
1
2
d
0
c
d
ii
′
d
= −d +
c
Figure 10.1
The vertical axis gives observed distances
d
ii
and the horizontal axis
distances
δ
ii
fitted in
r
dimensions. These determine the points (
δ
ii
,d
ii
)
indicated by
the red filled circles. The projection (indicated by the green arrow) of (
δ
ii
,d
ii
)
onto the
d
=
δ
'regression' line determines the point
1
2
(
d
ii
+
δ
ii
,
d
ii
+
δ
ii
)
which is indicated in
green. The squared residual length associated with this green point is
1
2
2
.The
(
d
ii
−
δ
ii
)
stress
criterion is twice the sum of squares of these residuals.
by horizontal projection, as shown, so the residuals in this direction do not depend on
the absolute values of the
d
ii
, only on their ordering. Now the
stress
criterion may be
modified to:
n
ˆ
2
ˆ
2
i
(δ
ii
−
δ
ii
)
=||
−
||
.
i
<
The above gives a very brief overview of how nonmetric MDS criteria are defined and
relate to their metric counterparts. There are several variants. For example, the monotonic
regression function may be expressed in terms of monotonic spline functions, thus giving
a continuous monotonic regression. There is also an issue concerning what to do about
tied values of
d
ii
; should constraints be imposed so that they map into tied values of
δ
ii
or not? See Borg and Groenen (2005) for a detailed discussion of this issue.
Figures 10.1 and 10.2 illustrate how the respective criteria are defined but say nothing
about how to minimize them. The numerical minimization of
stress
and
sstress
, either
