Kégl / Zinovyev Principal Manifolds for Data Visualization and Dimension Reduction

1. Auflage 2007
ISBN: 978-3-540-73750-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 340 Seiten

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. Presentation of algorithms is supplemented by case studies. The volume ends with a tutorial PCA deciphers genome.

Autoren/Hrsg.

Kégl, Balázs

Zinovyev, Andrei

Inhaltsverzeichnis

1;Preface;5
1.1;References;12
2;Contents;13
3;List of Contributors;20
4;1 Developments and Applications of Nonlinear Principal Component Analysis – a Review;24
4.1;1.1 Introduction;24
4.2;1.2 PCA Preliminaries;26
4.3;1.3 Nonlinearity Test for PCA Models;29
4.4;1.4 Nonlinear PCA Extensions;38
4.5;1.5 Analysis of Existing Work;54
4.6;1.6 Concluding Summary;61
4.7;References;62
5;2 Nonlinear Principal Component Analysis: Neural Network Models and Applications;67
5.1;2.1 Introduction;67
5.2;2.2 Standard Nonlinear PCA;70
5.3;2.3 Hierarchical nonlinear PCA;71
5.4;2.4 Circular PCA;74
5.5;2.5 Inverse Model of Nonlinear PCA;75
5.6;2.6 Applications;81
5.7;2.7 Summary;87
5.8;Availability of Software;88
5.9;References;88
6;3 Learning Nonlinear Principal Manifolds by Self- Organising Maps;91
6.1;3.1 Introduction;91
6.2;3.2 Biological Background;92
6.3;3.3 Theories;99
6.4;3.4 SOMs, Multidimensional Scaling and Principal Manifolds;103
6.5;3.5 Examples;109
6.6;References;114
7;4 Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization;119
7.1;4.1 Introduction and Overview;119
7.2;4.2 Optimization of Elastic Nets for Data Approximation;126
7.3;4.3 Elastic Maps;132
7.4;4.4 Principal Manifold as Elastic Membrane;133
7.5;4.5 Method Implementation;135
7.6;4.6 Examples;135
7.7;4.7 Discussion;148
7.8;References;150
8;5 Topology-Preserving Mappings for Data Visualisation;154
8.1;5.1 Introduction;154
8.2;5.2 Clustering Techniques;155
8.3;5.3 Topology Preserving Mappings;161
8.4;5.4 Experiments;167
8.5;5.5 Conclusions;172
8.6;References;172
9;6 The Iterative Extraction Approach to Clustering;174
9.1;6.1 Introduction;174
9.2;6.2 Clustering Entity-to-feature Data;175
9.3;6.3 ITEX Structuring and Clustering for Similarity Data;185
9.4;Conclusion;197
9.5;References;197
10;7 Representing Complex Data Using Localized Principal Components with Application to Astronomical Data;201
10.1;7.1 Introduction;201
10.2;7.2 Localized Principal Component Analysis;204
10.3;7.3 Combining Principal Curves and Regression;212
10.4;7.4 Application to the Gaia Survey Mission;217
10.5;7.5 Conclusion;221
10.6;References;222
11;8 Auto-Associative Models, Nonlinear Principal Component Analysis, Manifolds and Projection Pursuit;225
11.1;8.1 Introduction;225
11.2;8.2 Auto-Associative Models;226
11.3;8.3 Examples;230
11.4;8.4 Implementation Aspects;232
11.5;8.5 Illustration on Real and Simulated Data;236
11.6;References;239
12;9 Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes;242
12.1;9.1 Introduction and Overview;242
12.2;9.2 Optimization of Elastic Graphs for Data Approximation;245
12.3;9.3 Principal Trees (Branching Principal Curves);248
12.4;9.4 Analysis of the Universal 7-Cluster Structure of Bacterial Genomes;252
12.5;9.5 Visualization of Microarray Data;255
12.6;9.6 Discussion;258
12.7;References;258
13;10 Diffusion Maps - a Probabilistic Interpretation for Spectral Embedding and Clustering Algorithms;261
13.1;10.1 Introduction;261
13.2;10.2 Diffusion Distances and Diffusion Maps;263
13.3;10.3 Spectral Embedding of Low Dimensional Manifolds;269
13.4;10.4 Spectral Clustering of a Mixture of Gaussians;274
13.5;10.5 Summary and Discussion;281
13.6;References;281
14;11 On Bounds for Diffusion, Discrepancy and Fill Distance Metrics;284
14.1;11.1 Introduction;284
14.2;11.2 Energy, Discrepancy, Distance and Integration on Measurable Sets in Euclidean Space;285
14.3;11.3 Set Learning via Normalized Laplacian Dimension Reduction and Diffusion Distance;289
14.4;11.4 Main Result: Bounds for Discrepancy, Diffusion and Fill Distance Metrics;291
14.5;References;292
15;12 Geometric Optimization Methods for the Analysis of Gene Expression Data;294
15.1;12.1 Introduction;294
15.2;12.2 ICA as a Geometric Optimization Problem;295
15.3;12.3 Contrast Functions;297
15.4;12.4 Matrix Manifolds for ICA;302
15.5;12.5 Optimization Algorithms;303
15.6;12.6 Analysis of Gene Expression Data by ICA;307
15.7;12.7 Conclusion;313
15.8;References;313
16;13 Dimensionality Reduction and Microarray Data;316
16.1;13.1 Introduction;316
16.2;13.2 Background;318
16.3;13.3 Comparison Procedure;323
16.4;13.4 Results;326
16.5;13.5 Conclusions;329
16.6;References;330
17;14 PCA and K-Means Decipher Genome;332
17.1;14.1 Introduction;332
17.2;14.2 Required Materials;333
17.3;14.3 Genomic Sequence;334
17.4;14.4 Converting Text to a Numerical Table;335
17.5;14.5 Data Visualization;336
17.6;14.6 Clustering and Visualizing Results;338
17.7;14.7 Task List and Further Information;340
17.8;14.8 Conclusion;341
17.9;References;341
17.10;Appendix. Program listings;343
18;Color Plates;347
19;Index;355
20;Editorial Policy;357
21;General Remarks;358