Metric unidimensional scaling and global optimization

Abstract

Unidimensional scaling, Seriation, Local minima, Global optimization, Smoothing technique, Multidimensional scaling,

Stringing High-Dimensional Data for Functional Analysis

“…It is well-known that in UDS there is a high chance that various minimization algorithms terminate in local minima, which however is of less concern when only the rank order of the proximities matters (Borg and Groenen 2005;Hubert and Arabie 1986;Hubert et al 1997;Pliner 1996), as is the case in our application of UDS. This is confirmed by simulation results, where the UDS solutions (even where they might correspond to local minima) prove to have excellent properties under various settings.…”

An interactive multiobjective programming approach to combinatorial data analysis

“…Multiobjective programming can also have considerable utility within the context of unidimensional scaling of symmetric proximity matrices. Objective criteria for these problems include the minimization of a least-absolute error function (Simantiraki, 1996), minimization of a least-squared error function (Defays, 1978;de Leeuw & Heiser, 1977, 1980Groenen, 1993;Heiser, 1989;Pliner, 1996), and minimization of generalized least-squares functions containing constant terms (Hubert et al 1997). Like the asymmetric problems studied herein, it is often possible to develop optimal solutions to small symmetric unidimensional scaling problems using dynamic programming or other methods (e.g., branch-and-bound).…”

Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

“…Using MATLAB 6.5, we implemented the two most promising heuristic methods in this suite to obtain solutions for each of the 25 × 25 synthetic test problems in Section 4.2 and 4.3, as well as several of the empirical data matrices in Section 4.4. Specifically, 100 replications of Pliner's (1996) "smoothing technique" and 100 replications of iterative quadratic assignment (Hubert and Arabie, 1994;Hubert et al, 1997) were applied to each test problem. Table 7 presents, for each test problem, the number of times (out of 100) each heuristic method identified the optimal solution.…”

“…For example, Pliner (1996) and Groenen, Heiser, and Meulman (1999) have developed smoothing heuristics, which often provide excellent solutions to large problems. Quite a number of authors have also observed that the minimization of (1) is equivalent to a discrete optimization problem (Defays, 1978;De Leeuw and Heiser, 1977;Groenen, 1993;Groenen and Heiser, 1996;Hubert and Arabie, 1986).…”