Abstract
The usual convergence proof of the SMACOF algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted, called the pseudodistances. When this assumption is violated, erratic convergence behavior is known to occur. Three types of circumstances in which some of the pseudodistances may become negative are outlined: nonmetric multidimensional scaling with normalization on the variance, metric multidimensional scaling including an additive constant, and multidimensional scaling under the city-block distance model. A generalization of the SMACOF method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals.
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I am grateful to Patrick Groenen and Rian van Blokland-Vogelesang for their help with some of the computations, and to the anonymous referees for their very useful comments.
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Heiser, W.J. A generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative. Psychometrika 56, 7–27 (1991). https://doi.org/10.1007/BF02294582
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DOI: https://doi.org/10.1007/BF02294582