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Principal graphs and manifolds
 IN “HANDBOOK OF RESEARCH ON MACHINE LEARNING APPLICATIONS AND TRENDS: ALGORITHMS, METHODS AND TECHNIQUES
, 2008
"... In many physical statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found ‘lines and planes of closest fit to system of po ..."
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In many physical statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found ‘lines and planes of closest fit to system of points’. The famous kmeans algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the ‘middle ’ of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and kmeans principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.
1 PRINCIPAL MANIFOLDS AND GRAPHS IN PRACTICE: FROM MOLECULAR BIOLOGY TO DYNAMICAL SYSTEMS
"... We present several applications of nonlinear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen‟s selforganizing maps, a class of artificial neural networ ..."
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We present several applications of nonlinear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen‟s selforganizing maps, a class of artificial neural networks. On several examples we show advantages of using nonlinear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and nonlinear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of highthroughput data in molecular biology, from analysis of dynamical systems.