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Principal Graphs and Manifolds
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Author(s): Alexander N. Gorban (University of Leicester, UK)and Andrei Y. Zinovyev (Institut Curie, France)
Copyright: 2010
Pages: 32
Source title:
Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques
Source Author(s)/Editor(s): Emilio Soria Olivas (University of Valencia, Spain), José David Martín Guerrero (University of Valencia, Spain), Marcelino Martinez-Sober (University of Valencia, Spain), Jose Rafael Magdalena-Benedito (University of Valencia, Spain)and Antonio José Serrano López (University of Valencia, Spain)
DOI: 10.4018/978-1-60566-766-9.ch002
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Abstract
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 k-means 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 k-means 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.
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