http://www.springer.com/math/cse/book/978-3-540-73749-0



Gorban A. N., Kegl B., Wunsch D., Zinovyev A. Y. (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, Series: Lecture Notes in Computational Science and Engineering 58, Springer, Berlin — Heidelberg — New York, 2007, XXIV, 340 p. 82 illus. ISBN 978-3-540-73749-0



In 1901 Karl Pearson invented Principal Component Analysis (PCA). Since then, PCA serves as a prototype for many other tools of data analysis, visualization and dimension reduction: Independent Component Analysis (ICA), Multidimensional Scaling (MDS), Nonlinear PCA (NLPCA), Self Organizing Maps (SOM), etc.



The book starts with the quote of the classical Pearson definition of PCA and includes reviews of various methods: NLPCA, ICA, MDS, embedding and clustering algorithms, principal manifolds and SOM. New approaches to NLPCA, principal manifolds, branching principal components and topology preserving mappings are described as well. Presentation of algorithms is supplemented by case studies, from engineering to astronomy, but mostly of biological data: analysis of microarray and metabolite data. The volume ends with a tutorial PCA deciphers genome.



The book is meant to be useful for practitioners in applied data analysis in life sciences, engineering, physics and chemistry; it will also be valuable to PhD students and researchers in computer sciences, applied mathematics and statistics.



Contents



1 Developments and Applications of Nonlinear

Principal Component Analysis – a Review

Uwe Kruger, Junping Zhang, Lei Xie



2 Nonlinear Principal Component Analysis:

Neural Network Models and Applications

Matthias Scholz, Martin Fraunholz, Joachim Selbig



3 Learning Nonlinear Principal Manifolds

by Self-Organising Maps

Hujun Yin



4 Elastic Maps and Nets for Approximating

Principal Manifolds and Their Application

to Microarray Data Visualization

Alexander N Gorban, Andrei Y Zinovyev



5 Topology-Preserving Mappings for Data Visualisation

Marian PeЇna, Wesam Barbakh, Colin Fyfe



6 The Iterative Extraction Approach to Clustering

Boris Mirkin



7 Representing Complex Data Using Localized Principal

Components with Application to Astronomical Data

Jochen Einbeck, Ludger Evers, Coryn Bailer-Jones



8 Nonlinear Principal Component

Analysis, Manifolds and Projection Pursuit

St´ephane Girard, Serge Iovleff



9 Beyond The Concept of Manifolds: Principal Trees,

Metro Maps, and Elastic Cubic Complexes

Alexander N. Gorban, Neil R. Sumner, Andrei Y. Zinovyev



10 Diffusion Maps - a Probabilistic Interpretation

for Spectral Embedding and Clustering Algorithms

Boaz Nadler, Stephane Lafon, Ronald Coifman, Ioannis G. Kevrekidis



11 On Bounds for Diffusion, Discrepancy

and Fill Distance Metrics

Steven B. Damelin



12 Geometric Optimization Methods for the Analysis

of Gene Expression Data

Michel Journ´ee, Andrew E. Teschendorff, Pierre-Antoine Absil,

Simon Tavar´e, Rodolphe Sepulchre



13 Dimensionality Reduction and Microarray data

David A. Elizondo, Benjamin N. Passow, Ralph Birkenhead,

Andreas Huemer



14 PCA and K-Means Decipher Genome

Alexander N. Gorban, Andrei Y. Zinovyev



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