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Concept
Multidimensional scaling
Summary
Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of objects or individuals" into a configuration of points mapped into an abstract Cartesian space.
More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction.
Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot.
Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis.
MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:
It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, which is given by
where denote vectors in N-dimensional space, denotes the scalar product between and , and are the elements of the matrix defined on step 2 of the following algorithm, which are computed from the distances.
Steps of a Classical MDS algorithm:
Classical MDS uses the fact that the coordinate matrix can be derived by eigenvalue decomposition from . And the matrix can be computed from proximity matrix by using double centering.
Set up the squared proximity matrix
Apply double centering: using the centering matrix , where is the number of objects, is the identity matrix, and is an matrix of all ones.
Official source
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