Linear Unidimensional Scaling in the L ₂ -Norm: Basic Optimization Methods Using MATLAB

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.