Abstract
Two types of solutions for MDS are discussed. If the proximities are Euclidean distances, classical MDS yields an easy algebraic solution. In most MDS applications, iterative methods are needed, because they admit many types of data and distances. They use a two-phase optimization algorithm, moving the points in MDS space in small steps while holding the data and their transforms fixed, and vice versa, until convergence is reached.
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This means that the centroid of the MDS configuration becomes the origin. The coordinates of \(\mathbf {X}\), thus, should sum to 0 in each column of \(\mathbf {X}\). This does not carry any consequences for the distances of \(\mathbf {X}\); that is, any other point could also serve as the origin. However, one point must be picked as an origin to compute scalar products.
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smacof is an acronym for “Scaling by MAjorizing a COmplicated Function” (De Leeuw and Heiser 1980). The optimization method used by smacof is called “Majorization” (De Leeuw 1977, Groenen 1993). The basic idea of this method is that a complicated goal function (i.e., Stress within the MDS context) is approximated in each iteration by a less complicated function which is easier to optimize. For more details on how this method is used to solve MDS problems, see De Leeuw and Mair (2009) or Borg and Groenen (2005).
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References
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Borg, I., Groenen, P.J.F., Mair, P. (2018). MDS Algorithms. In: Applied Multidimensional Scaling and Unfolding. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-73471-2_9
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DOI: https://doi.org/10.1007/978-3-319-73471-2_9
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