Sammon mapping or Sammon projection is an algorithm that
maps
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
a high-dimensional space to a space of lower dimensionality (see
multidimensional scaling
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 n objects or individuals" into a configurati ...
) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection.
It is particularly suited for use in
exploratory data analysis
In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but pr ...
.
The method was proposed by John W. Sammon in 1969.
It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as
principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
, which also makes it more difficult to use for classification applications.
Denote the distance between ith and jth objects in the original space by
, and the distance between their projections by
.
Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error:
:
The minimization can be performed either by
gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
, as proposed initially, or by other means, usually involving iterative methods.
The number of iterations needs to be experimentally determined and convergent solutions are not always guaranteed.
Many implementations prefer to use the first Principal Components as a starting configuration.
The Sammon mapping has been one of the most successful nonlinear metric multidimensional scaling methods since its advent in 1969, but effort has been focused on algorithm improvement rather than on the form of the stress function.
The performance of the Sammon mapping has been improved by extending its stress function using left
Bregman divergence
In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. W ...
and right Bregman divergence.
See also
*
Prefrontal cortex basal ganglia working memory
*
State–action–reward–state–action
*
Constructing skill trees
References
External links
HiSee – an open-source visualizer for high dimensional dataA C# based program with code on CodeProject
Functions and mappings
Dimension reduction
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