Linear Unidimensional Scaling in the L
2
-Norm: Basic
Optimization Methods Using MATLAB |
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Authors: | L J Hubert P Arabie J J Meulman |
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Institution: | (1) University of Illinois, Department of Psychology,;(2) Rutgers University, Faculty of Management,;(3) Leiden University, Department of Education, Data Theory Group, |
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Abstract: | L2
-norm: (1)
dynamic programming; (2) an iterative quadratic assignment improvement
heuristic; (3) the Guttman update strategy as modified by Pliner's technique
of smoothing; (4) a nonlinear programming reformulation by Lau, Leung, and
Tse. The methods are all implemented through (freely downloadable) MATLAB
m-files; their use is illustrated by a common data set carried throughout. For
the computationally intensive dynamic programming formulation that can a
globally optimal solution, several possible computational improvements are
discussed and evaluated using (a) a transformation of a given m-function with
the MATLAB Compiler into C code and compiling the latter; (b) rewriting an
m-function and a mandatory MATLAB gateway directly in Fortran and compiling
into a MATLAB callable file; (c) comparisons of the acceleration of raw
m-files implemented under the most recent release of MATLAB Version 6.5 (and compared to the absence of such
acceleration under the previous MATLAB Version 6.1). Finally, and in contrast
to the combinatorial optimization task of identifying a best unidimensional
scaling for a given proximity matrix, an approach is given for the
confirmatory fitting of a given unidimensional scaling based only on a fixed
object ordering, and to nonmetric unidensional scaling that incorporates an
additional optimal monotonic transformation of the proximities. |
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Keywords: | |
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