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Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. It refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. An MDS algorithm aims to place each object in ''N''-dimensional space such that the between-object distances are preserved as well as possible. Each object is then assigned coordinates in each of the ''N'' dimensions. The number of dimensions of an MDS plot ''N'' can exceed 2 and is specified a priori. Choosing ''N''=2 optimizes the object locations for a two-dimensional scatterplot. ==Types== MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: ; Classical multidimensional scaling: Also known as Principal Coordinates Analysis, Torgerson Scaling or Torgerson–Gower scaling. Takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called ''strain''.〔 ; Metric multidimensional scaling: A superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called ''stress'', which is often minimized using a procedure called stress majorization. ; Non-metric multidimensional scaling: In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. The relationship is typically found using isotonic regression. ; *Louis Guttman's smallest space analysis (SSA) is an example of a non-metric MDS procedure. ; Generalized multidimensional scaling: An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multidimensional scaling」の詳細全文を読む スポンサード リンク
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