Metric multidimensional scaling

Fig. 38.6. Three-dimensional configuration according to a non-metric multidimensional scaling applied to 24 types of bread differing in appearance as assessed by 12 panellists.

Kruskal, J. B. (1964) Non-metric multidimensional scaling A numerical method. Psychometrika 29, 115-129. 

Taguchi Y, Oono Y. Relational patterns of gene expression via non-metric multidimensional scaling analysis. Bioinformatics 2005 21 730-40. 

A close analogy exists between PCoA and PCA, the difference lying in the source of the data. In the former they appear as a square distance table, while in the latter they are defined as a rectangular measurement table. The result of PCoA also serves as a starting point for multidimensional scaling (MDS) which attempts to reproduce distances as closely as possible in a low-dimensional space. In this context PCoA is also referred to as classical metric scaling. In MDS, one minimizes the stress between observed and reconstructed distances, while in PCA one maximizes the variance reproduced by successive factors. 

Let s first suppose we knew that we can consider things on a pairwise basis. The attraction of metric distance, principal component analysis, multidimensional scaling, clustering, and dendrograms is that they consider the mere N(N - 2)12 parameters for an N parameter problem. The approaches are powerful and persuasive. But we just illustrated above that this is not justified in general. You just never get to miss the more complicated relationships because you never get to see them. 

Fig. 16.4. Phylogenetic trees or cladograms based on the sequence differences (SEQ) and three-dimensional structural differences (STR) of immunoglobulin fragments. The structural distance metric is a function of both the rms distance difference between superposed structures and the number of topologically equivalent positions in each pairwise comparison (taken from [11]). The lower part of the diagram shows a multidimensional scaling analysis based on structural differences. The constant (C) and variable (V) domains cluster together for the light (L) and heavy (H) chains of the immunoglobulin fragments...

Alternate projection methods aim to reduce the data dimensionality by optimizing the representation in the lower-dimension space so that the distances between points in the projected space are as similar as possible to the distances between the corresponding points in the original space. We will describe here a class of methods known as multidimensional scaling (MDS). The aim of these methods is to project data from a pseudo-metric space (i.e., one characterized by a dissimilarity measure) onto a metric space. Such methods are especially useful for preprocessing non-metric data in order to use algorithms valid only for metric input. 

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