Principal Manifolds for Data Visualization and Dimension Reduction

Pirmais vāks
Alexander N. Gorban, Balázs Kégl, Donald C. Wunsch, Andrei Zinovyev
Springer Science & Business Media, 2007. gada 11. sept. - 340 lappuses

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 and K-means decipher 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.

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Atlasītās lappuses

Saturs

636 Some Applications
171
References
174
Representing Complex Data Using Localized Principal Components with Application to Astronomical Data
178
72 Localized Principal Component Analysis
181
722 Principal Curves
185
723 Further Approaches
188
73 Combining Principal Curves and Regression
189
732 The Generalization to Principal Curves
190

14 Nonlinear PCA Extensions
15
141 Principal Curves and Manifolds
16
142 Neural Network Approaches
24
143 Kernel PCA
29
15 Analysis of Existing Work
31
152 Generalization of Linear PCA?
33
153 Roadmap for Future Developments Basics and Beyond
37
16 Concluding Summary
38
References
39
Nonlinear Principal Component Analysis Neural Network Models and Applications
44
22 Standard Nonlinear PCA
47
23 Hierarchical nonlinear PCA
48
231 The Hierarchical Error Function
49
24 Circular PCA
51
25 Inverse Model of Nonlinear PCA
52
251 The Inverse Network Model
53
252 NLPCA Models Applied to Circular Data
55
253 Inverse NLPCA for Missing Data
56
254 Missing Data Estimation
57
26 Applications
58
261 Application of Hierarchical NLPCA
59
262 Metabolite Data Analysis
60
263 Gene Expression Analysis
62
27 Summary
64
References
65
Learning Nonlinear Principal Manifolds by SelfOrganising Maps
68
32 Biological Background
69
322 From Von Marsburg and Willshaws Model to Kohonens SOM
72
323 The SOM Algorithm
75
33 Theories
76
332 Topological Ordering Measures
79
34 SOMs Multidimensional Scaling and Principal Manifolds
80
342 Principal Manifolds
82
343 Visualisation Induced SOM ViSOM
84
35 Examples
86
351 Data Visualisation
87
352 Document Organisation and Content Management
88
References
91
Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization
96
412 Principal Manifolds
98
413 Elastic Functional and Elastic Nets
100
42 Optimization of Elastic Nets for Data Approximation
103
422 Missing Data Values
105
423 Adaptive Strategies
106
43 Elastic Maps
109
44 Principal Manifold as Elastic Membrane
110
45 Method Implementation
112
462 Modeling Molecular Surfaces
113
463 Visualization of Microarray Data
114
47 Discussion
125
References
127
TopologyPreserving Mappings for Data Visualisation
131
52 Clustering Techniques
132
522 KHarmonic Means
133
523 Neural Gas
135
524 Weighted KMeans
136
525 The Inverse Weighted KMeans
137
53 Topology Preserving Mappings
138
532 Topographic Product of Experts ToPoE
140
533 The Harmonic Topograpic Map
141
534 Topographic Neural Gas
143
54 Experiments
144
543 Umatrix Hit Histograms and Distance Matrix
145
544 The Quality of The Map
147
55 Conclusions
149
The Iterative Extraction Approach to Clustering
151
62 Clustering Entitytofeature Data
152
622 Additive Clustering Model and ITEX
154
623 Overlapping and Fuzzy Clustering Case
156
624 KMeans and iKMeans Clustering
157
63 ITEX Structuring and Clustering for Similarity Data
162
632 The Additive Structuring Model and ITEX
163
633 Additive Clustering Model
165
634 Approximate Partitioning
166
635 One Cluster Clustering
168
733 Using Directions Other than the Local Principal Components
192
734 A Simple Example
193
74 Application to the Gaia Survey Mission
194
742 Principal Manifold Based Approach
196
75 Conclusion
198
References
199
AutoAssociative Models Nonlinear Principal Component Analysis Manifolds and Projection Pursuit
202
82 AutoAssociative Models
203
822 A Projection Pursuit Algorithm
205
823 Theoretical Results
206
83 Examples
207
832 Additive AutoAssociative Models and Neural Networks
208
84 Implementation Aspects
209
842 Computation of Principal Directions
211
85 Illustration on Real and Simulated Data
213
References
216
Beyond The Concept of Manifolds Principal Trees Metro Maps and Elastic Cubic Complexes
219
911 Elastic Principal Graphs
221
92 Optimization of Elastic Graphs for Data Approximation
222
922 Optimal Application of Graph Grammars
223
923 Factorization and Transformation of Factors
224
93 Principal Trees Branching Principal Curves
225
932 Visualization of Data Using Metro Map TwoDimensional Tree Layout
226
Product of Principal Trees
227
94 Analysis of the Universal 7Cluster Structure of Bacterial Genomes
229
941 Brief Introduction
230
942 Visualization of the 7Cluster Structure
232
952 Principal Tree of Human Tissues
234
96 Discussion
235
Diffusion Maps a Probabilistic Interpretation for Spectral Embedding and Clustering Algorithms
238
102 Diffusion Distances and Diffusion Maps
240
1021 Asymptotics of the Diffusion Map
245
103 Spectral Embedding of Low Dimensional Manifolds
246
104 Spectral Clustering of a Mixture of Gaussians
251
105 Summary and Discussion
258
On Bounds for Diffusion Discrepancy and Fill Distance Metrics
261
112 Energy Discrepancy Distance and Integration on Measurable Sets in Euclidean Space
262
113 Set Learning via Normalized Laplacian Dimension Reduction and Diffusion Distance
266
Bounds for Discrepancy Diffusion and Fill Distance Metrics
268
References
269
Geometric Optimization Methods for the Analysis of Gene Expression Data
271
122 ICA as a Geometric Optimization Problem
272
123 Contrast Functions
274
1232 FCorrelation 14
276
1233 NonGaussianity 17
277
1234 Joint Diagonalization of Cumulant Matrices 19
278
124 Matrix Manifolds for ICA
279
125 Optimization Algorithms
280
1252 FastICA
282
1253 Jacobi Rotations
284
1262 Evaluation of the Biological Relevance of the Expression Modes
287
1263 Results Obtained on the Breast Cancer Microarray Data Set
288
127 Conclusion
290
Dimensionality Reduction and Microarray Data
293
132 Background
295
1322 Methods for Dimension Reduction
296
1323 Linear Separability
297
133 Comparison Procedure
300
1332 Dimensionality Reduction
301
1333 Perceptron Models
303
135 Conclusions
306
References
307
PCA and KMeans Decipher Genome
309
142 Required Materials
310
143 Genomic Sequence
311
1432 Sequences for the Analysis
312
145 Data Visualization
313
1452 Understanding Plots
314
146 Clustering and Visualizing Results
315
147 Task List and Further Information
317
148 Conclusion
318
Color Plates
325
Index
333
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