Multidimensional Scaling (MDS):
MDS is widely used in the social and behavioral sciences. Its goal roughly is to
take a multivariate data set and represent it in a low dimensional Euclidean
space so as to minimize any distortion of the data. Often this is a representation in 2 dimensions. At its first
meeting, the working group explored nonlinear and nonmetric versions of MDS,
fitting of various non-Euclidean representations in both the two- and three-
way cases, and the need to develop techniques that can be applied to massive
data sets. This last problem, of dealing with massive data sets, is difficult
because it will require the development of entirely new techniques, since most
of the existing ones are extremely computationally intense and so tend to limit
the size of data arrays quite severely. One promising approach involves the
random deletion of a substantial portion of the data. Preliminary results
indicated that as much as 60% could be deleted without a serious effect on the
output. Other approaches involve using
heuristic approaches to get close to the solution and then trying to refine the
output of the heuristic. This is work done by Willem Heiser and his
colleagues from Leiden University.
Since one well-known approach to fitting two-way Euclidean MDS models
involves a singular value decomposition (SVD) of a derived matrix of scalar
products, and since methods already exist for implementing the SVD on very
large matrices, one approach, taken by the (unfortunately recently deceased) Mark
Rorvig and David Dubin in some collaborative work with Douglas
Carroll involved applying methods for SVD of massive data sets to solving
this particular version of MDS in the case of extremely large matrices of
proximities, involving proximity data on a very large number of stimuli or
other objects. Various approaches are
being explored for extending such approaches to other, more complex, MDS models
and methods.
The main accomplishment of the first
meeting of this group was the development and enhancement of cross disciplinary
research efforts. Here are the highlights of these endeavors.
Larry Hubert (Psychology,
University of Illinois, Phipps Arabie and Douglas Carroll
(Graduate School of Management, Rutgers) together with Michael Brusco
(School of Business, Florida State University) are all exploring various
mathematical programming techniques to fit MDS models, including various
possible collaborative efforts.
David Dubin (Library
Science, University of Illinois) Douglas Carroll and Michael Trossett
(Math., William and Mary) are all exploring various approaches to MDS of
massive data sets including, as already alluded to, possible extensions of some
already established research in this area.
There was also the start of or
enhancement of collaborations among academic participants and industrial
scientists such as Anil Chaturvedi of Kraft Foods and Andreas Buja
of AT&T Laboratories.
This material is based upon work supported by the National Science Foundation under Grant No. 0100921