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Global Optimization Methods for Multidimensional Scaling Applied to Mobile Communications Patrick J.F. Groenen1 , Rudolf Mathar2 , and Javier Trejos3 1
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Data Theory Group, Department of Education, Leiden University, Leiden, The Netherlands Institute of Statistics, Aachen University of Technology, Aachen, Germany CIMPA, Escuela de Matematic´ a, Universidad de Costa Rica, San Jos´e, Costa Rica
Abstract. The purpose of this paper is to present a short overview of recent developments of global optimization in least squares multidimensional scaling. Three promising candidates —the genetic algorithm, simulated annealing, and distance smoothing— are discussed in more detail and compared on a data set arising in mobile communication.
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Introduction
In recent years, there has been a growing interest in multidimensional scaling (MDS). Several new exciting applications have arisen in various disciplines. For example, MDS has been applied to model atoms in large molecules (Havel (1991); Glunt, Hayden, and Raydan (1993)) and MDS has been incorporated into multivariate analysis by Meulman (1986, 1992). In this paper, we shall use data from yet another application of MDS emerging in the area of mobile telecommunication. This data set is characterized by a large number of objects (typically around 1000 objects) and many missing data (more than 90%). More details are given in the next section. Before we continue, let us define the aim of least-squares MDS in words: try to reconstruct given dissimilarities δij between pairs of objects i and j as Euclidean distances between rows i and j of a configuration matrix X as closely as possible in the least-squares sense. This objective can be formalized as minimizing the Stress loss function 2
