Principal Graphs and Manifolds

Details Principal Graphs and Manifolds Principal Graphs and Manifolds

2008-09-02

arxiv.org/abs/0809.0490v2

Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Ch. 2, Information Scien

Computer Science

Learning

36 pages, 6 figures, minor corrections

Scientific paper

10.4018/978-1-60566-766-9

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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