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that is minimized over X.hew
ij
are weights that can be chosen to reflect variabil-
ity, measurement error, or missing data. his is precisely the objective function (
.
)
derived from the general framework of force-directed techniques previously intro-
duced and discussed.
A number of variations of (
.
) have appeared in the literature. In McGee (
),
the loss function has weights δ
−
ij
. he loss function is interpreted as the amount of
physical work that must be done on elastic springs to stretch or compress them from
an initial length δ
ij
to a final length d
ij
. On the other hand, the following choice of
weights w
ij
δ
−
ij
is discussed in Sammon (
).
Minimization of the loss function (
.
) can be accomplished either by an iterative
majorization algorithm (Borgand Groenen
;De Leeuw and Michailidis
)or
by a steepest descent method (Buja and Swayne
). he latter method is used in
the implementation of MDS in the GGobi visualization system (Swayne et al.,
).
A
-DMDS solution for the sleeping bag data is shown in Fig.
.
.It can be seen that
the solution spreadsthe objects in the data setfairly uniformly in the plane, andedge
crossings are avoided.
Wediscussnextafairlyrecent application ofMDS.Inmanyinstances, thedata ex-
hibit nonlinearities, i.e., they lie on a low-dimensional manifold of some curvature.
his has led to several approaches that still rely on the embedding (MDS) approach
for visualization purposes but appropriately alter the input distances
=
δ
ij
.Apop-
Figure
.
.
MDS representation of sleeping bag data set based on χ
-distances. Due to the discrete
nature of the data, multiple objects are mapped onto the same location, as shown in the plot.Further,
for reference purposes, the categories to which the sleeping bags belong have been added to the plot at
the centroids of the object points
