Principal Manifolds for Data Visualization and Dimension Reduction (eBook)

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