Aditya Narayanan
Operations
Roll No. 14125
Group B
Multidimensional scaling (MDS) is
a set of related statistical techniques often used in information visualization
for exploring similarities or dissimilarities in data. MDS is a special case of
ordination. An MDS algorithm starts with a matrix of item–item similarities, and
then assigns a location to each item in N-dimensional space, where N is
specified beforehand. For sufficiently small N, the resulting locations may be
displayed in a graph or 3D visualisation.
MDS algorithms fall into a
taxonomy, depending on the meaning of the input matrix:
Classical multidimensional
scaling
It is also known as Principal
Coordinates Analysis, 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.
Metric multidimensional scaling
It is 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
It is an extension of metric
multidimensional scaling, in which the target space is an arbitrary smooth
non-Euclidean space. In case when 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.
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