Multidimensional
scaling (MDS)
An MDS algorithm starts with a matrix of
item-item similarities, then assigns a location of each item in a low-dimensional
space, suitable for graphing or 3D visualization. Multidimensional scaling
(MDS) is a set of related statistical techniques often used in data
visualization for exploring similarities or dissimilarities in data.
MDS is a special case of ordination.
MDS algorithms fall into a taxonomy, depending on the meaning of the
input matrix:
 Classical
multidimensional scaling
also known as 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.
Generalized multidimensional scaling (GMDS)
A superset
of metric MDS that allows for the target distances to be non-Euclidean.
Non-metric multidimensional scaling
In contrast to metric MDS, non-metric MDS both finds a non-parametric
monotonic relationship between the dissimilarities in the item-item
matrix and the Euclidean distance between items, and the location of
each item in the low-dimensional space. The relationship is typically
found using isotonic regression.
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