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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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Cited by 10 (1 self)
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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.
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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Cited by 7 (3 self)
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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.
Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes
"... Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of lowdimensional “principal object”: a principal cubic complex. This complex is a generalization of linear and nonlinear principal manifolds and includ ..."
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Cited by 2 (2 self)
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Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of lowdimensional “principal object”: a principal cubic complex. This complex is a generalization of linear and nonlinear principal manifolds and includes them as a particular case. To construct such an object, we combine a method of topological grammars with the minimization of an elastic energy defined for its embedment into multidimensional data space. The whole complex is presented as a system of nodes and springs and as a product of onedimensional continua (represented by graphs), and the grammars describe how these continua transform during the process of optimal complex construction. The simplest case of a topological grammar (“add a node”, “bisect an edge”) is equivalent to the construction of “principal trees”, an object useful in many practical applications. We demonstrate how it can be applied to the analysis of bacterial genomes and for visualization of cDNA microarray data using the “metro map” representation.
14 PCA and KMeans Decipher Genome
"... Summary. In this paper, we aim to give a tutorial for undergraduate students studying statistical methods and/or bioinformatics. The students will learn how data visualization can help in genomic sequence analysis. Students start with a fragment of genetic text of a bacterial genome and analyze its ..."
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Summary. In this paper, we aim to give a tutorial for undergraduate students studying statistical methods and/or bioinformatics. The students will learn how data visualization can help in genomic sequence analysis. Students start with a fragment of genetic text of a bacterial genome and analyze its structure. By means of principal component analysis they “discover ” that the information in the genome is encoded by nonoverlapping triplets. Next, they learn how to find gene positions. This exercise on PCA and KMeans clustering enables active study of the basic bioinformatics notions. The Appendix contains program listings that go along with this exersice.
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 ..."
Abstract
 Add to MetaCart
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.