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Digital Signal Processing with Kernel Methods

ISBN: 978-1-118-61179-1
672 pages
January 2018
Digital Signal Processing with Kernel Methods (1118611799) cover image

Description

A realistic and comprehensive review of joint approaches to machine learning and signal processing algorithms, with application to communications, multimedia, and biomedical engineering systems

Digital Signal Processing with Kernel Methods reviews the milestones in the mixing of classical digital signal processing models and advanced kernel machines statistical learning tools. It explains the fundamental concepts from both fields of machine learning and signal processing so that readers can quickly get up to speed in order to begin developing the concepts and application software in their own research.

Digital Signal Processing with Kernel Methods provides a comprehensive overview of kernel methods in signal processing, without restriction to any application field. It also offers example applications and detailed benchmarking experiments with real and synthetic datasets throughout. Readers can find further worked examples with Matlab source code on a website developed by the authors. 

  • Presents the necessary basic ideas from both digital signal processing and machine learning concepts
  • Reviews the state-of-the-art in SVM algorithms for classification and detection problems in the context of signal processing
  • Surveys advances in kernel signal processing beyond SVM algorithms to present other highly relevant kernel methods for digital signal processing

An excellent book for signal processing researchers and practitioners, Digital Signal Processing with Kernel Methods will also appeal to those involved in machine learning and pattern recognition. 

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Table of Contents

About the Authors xiii

Preface xvii

Acknowledgements xxi

List of Abbreviations xxiii

Part I Fundamentals and Basic Elements 1

1 From Signal Processing to Machine Learning 3

1.1 A New Science is Born: Signal Processing 3

1.1 A New Science is Born: Signal Processing 3

1.1.1 Signal Processing Before Being Coined 3

1.1.2 1948: Birth of the Information Age 4

1.1.3 1950s: Audio Engineering Catalyzes Signal Processing 4

1.2 From Analog to Digital Signal Processing 5

1.2.1 1960s: Digital Signal Processing Begins 5

1.2.2 1970s: Digital Signal Processing Becomes Popular 6

1.2.3 1980s: Silicon Meets Digital Signal Processing 6

1.3 Digital Signal Processing Meets Machine Learning 7

1.3.1 1990s: New Application Areas 7

1.3.2 1990s: Neural Networks, Fuzzy Logic, and Genetic Optimization 7

1.4 Recent Machine Learning in Digital Signal Processing 8

1.4.1 Traditional Signal Assumptions Are No Longer Valid 8

1.4.2 Encoding Prior Knowledge 8

1.4.3 Learning and Knowledge from Data 9

1.4.4 From Machine Learning to Digital Signal Processing 9

1.4.5 From Digital Signal Processing to Machine Learning 10

2 Introduction to Digital Signal Processing 13

2.1 Outline of the Signal Processing Field 13

2.1.1 Fundamentals on Signals and Systems 14

2.1.2 Digital Filtering 21

2.1.3 Spectral Analysis  24

2.1.4 Deconvolution 28

2.1.5 Interpolation 30

2.1.6 System Identification 31

2.1.7 Blind Source Separation 36

2.2.3 Sparsity, Compressed Sensing, and Dictionary Learning 44

2.3 Multidimensional Signals and Systems 48

2.3.1 Multidimensional Signals 49

2.3.2 Multidimensional Systems 51

2.4 Spectral Analysis on Manifolds 52

2.4.1 Theoretical Fundamentals 52

2.4.2 Laplacian Matrices 54

2.5 Tutorials and Application Examples 57

2.5.1 Real and Complex Signal Processing and Representations 57

2.5.2 Convolution, Fourier Transform, and Spectrum 63

2.5.3 Continuous-Time Signals and Systems 67

2.5.4 Filtering Cardiac Signals 70

2.5.5 Nonparametric Spectrum Estimation 74

2.5.6 Parametric Spectrum Estimation 77

2.5.7 Source Separation 81

2.5.8 Time–Frequency Representations and Wavelets 84

2.5.9 Examples for Spectral Analysis on Manifolds 87

2.6 Questions and Problems 94

3 Signal Processing Models 97

3.1 Introduction 97

3.2 Vector Spaces, Basis, and Signal Models 98

3.2.1 Basic Operations for Vectors 98

3.2.2 Vector Spaces 100

3.2.3 Hilbert Spaces 101

3.2.4 Signal Models 102

3.2.5 Complex Signal Models 104

3.2.6 Standard Noise Models in Digital Signal Processing 105

3.2.7 The Role of the Cost Function 107

3.2.8 The Role of the Regularizer 109

3.3 Digital Signal Processing Models 111

3.3.1 Sinusoidal Signal Models 112

3.3.2 System Identification Signal Models 113

3.3.3 Sinc Interpolation Models 116

3.3.4 Sparse Deconvolution 120

3.3.5 Array Processing 121

3.4 Tutorials and Application Examples 122

3.4.1 Examples of Noise Models 123

3.4.2 Autoregressive Exogenous System Identification Models 132

3.4.3 Nonlinear System Identification Using Volterra Models 138

3.4.4 Sinusoidal Signal Models 140

3.4.5 Sinc-based Interpolation 144

3.4.6 Sparse Deconvolution 152

3.4.7 Array Processing 157

3.5 Questions and Problems 160

3.A MATLABsimpleInterp Toolbox Structure 161

4 Kernel Functions and Reproducing Kernel Hilbert Spaces 165

4.1 Introduction 165

4.2 Kernel Functions and Mappings 169

4.2.1 Measuring Similarity with Kernels 169

4.2.2 Positive-Definite Kernels 169

4.2.3 Reproducing Kernel in Hilbert Space and Reproducing Property 170

4.2.4 Mercer’s Theorem 173

4.3 Kernel Properties 174

4.3.1 Tikhonov’s Regularization 175

4.3.2 Representer Theorem and Regularization Properties 176

4.3.3 Basic Operations with Kernels 178

4.4 Constructing Kernel Functions 179

4.4.1 Standard Kernels 179

4.4.2 Properties of Kernels 180

4.4.3 Engineering Signal Processing Kernels 181

4.5 Complex Reproducing Kernel in Hilbert Spaces 184

4.6 Support Vector Machine Elements for Regression and Estimation 186

4.6.1 Support Vector Regression Signal Model and Cost Function 186

4.6.2 Minimizing Functional 187

4.7 Tutorials and Application Examples 191

4.7.1 Kernel Calculations and Kernel Matrices 191

4.7.2 Basic Operations with Kernels 194

4.7.3 Constructing Kernels 197

4.7.4 Complex Kernels 199

4.7.5 Application Example for Support Vector Regression Elements 202

4.8 Concluding Remarks 205

4.9 Questions and Problems 205

Part II Function Approximation and Adaptive Filtering 209

5 A Support Vector Machine Signal Estimation Framework 211

5.1 Introduction 211

5.2 A Framework for Support Vector Machine Signal Estimation  213

5.3 Primal Signal Models for Support Vector Machine Signal Processing 216

5.3.1 Nonparametric Spectrum and System Identification 218

5.3.2 Orthogonal Frequency Division Multiplexing Digital Communications 220

5.3.3 Convolutional Signal Models 222

5.3.4 Array Processing 225

5.4 Tutorials and Application Examples 227

5.4.1 Nonparametric Spectral Analysis with Primal Signal Models 227

5.4.2 System Identification with Primal Signal Model 𝛾-filter 228

5.4.3 Parametric Spectral Density Estimation with Primal Signal Models 230

5.4.4 Temporal Reference Array Processing with Primal Signal Models 231

5.4.5 Sinc Interpolation with Primal Signal Models 233

6 Reproducing Kernel Hilbert Space Models for Signal Processing 241

6.1 Introduction 241

6.2 Reproducing Kernel Hilbert Space Signal Models 242

6.2.1 Kernel Autoregressive Exogenous Identification 244

6.2.2  Kernel Finite Impulse Response and the 𝛾-Filter 247

6.2.3 Kernel Array Processing with Spatial Reference 248

6.2.4 Kernel Semiparametric Regression 249

6.3 Tutorials and Application Examples 258

6.3.1 Nonlinear System Identification with Support Vector Machine–Autoregressive and Moving Average 258

6.3.2 Nonlinear System Identification with the 𝛾-filter 260

6.3.3 Electric Network Modeling with Semiparametric Regression 264

6.3.4 Promotional Data 272

6.3.5 Spatial and Temporal Antenna Array Kernel Processing 275

6.4 Questions and Problems 279

7 Dual Signal Models for Signal Processing 281

7.1 Introduction 281

7.2 Dual Signal Model Elements 281

7.3 Dual Signal Model Instantiations 283

7.3.1 Dual Signal Model for Nonuniform Signal Interpolation 283

7.3.2 Dual Signal Model for Sparse Signal Deconvolution 284

7.3.3 Spectrally Adapted Mercer Kernels 285

7.4 Tutorials and Application Examples 289

7.4.1 Nonuniform Interpolation with the Dual Signal Model 290

7.4.2  Sparse Deconvolution with the Dual Signal Model 292

7.4.3 Doppler Ultrasound Processing for Fault Detection 294

7.4.4 Spectrally Adapted Mercer Kernels 296

7.4.5 Interpolation of Heart Rate Variability Signals 304

7.4.6 Denoising in Cardiac Motion-Mode Doppler Ultrasound Images 309 m

7.4.7 Indoor Location from Mobile Devices Measurements 316

7.4.8 Electroanatomical Maps in Cardiac Navigation Systems 322

7.5 Questions and Problems 331

8 Advances in Kernel Regression and Function Approximation 333

8.1 Introduction 333

8.2 Kernel-Based Regression Methods 333

8.2.1 Advances in Support Vector Regression 334

8.2.2 Multi-output Support Vector Regression 338

8.2.3 Kernel Ridge Regression 339

8.2.4 Kernel Signal-To-Noise Regression 341

8.2.5 Semisupervised Support Vector Regression 343

8.2.6 Model Selection in Kernel Regression Methods 345

8.4.1 Comparing Support Vector Regression, Relevance Vector Machines, and Gaussian Process Regression 360

8.4.2 Profile-Dependent Support Vector Regression 362

8.4.3 Multi-output Support Vector Regression 364

8.4.4 Kernel Signal-to-Noise Ratio Regression 366

8.4.5 Semisupervised Support Vector Regression 368

8.4.6 Bayesian Nonparametric Model 369

8.4.7 Gaussian Process Regression 370

8.4.8 Relevance Vector Machines 379

8.5 Concluding Remarks 382

8.6 Questions and Problems 383

9 Adaptive Kernel Learning for Signal Processing 387

9.1 Introduction 387

9.2 Linear Adaptive Filtering 387

9.2.1 Least Mean Squares Algorithm 388

9.2.2 Recursive Least-Squares Algorithm 389

9.3 Kernel Adaptive Filtering 392

9.4 Kernel Least Mean Squares 392

9.4.1 Derivation of Kernel Least Mean Squares 393

9.4.2 Implementation Challenges and Dual Formulation 394

9.5.3 Prediction of the Mackey–Glass Time Series with Kernel Recursive Least Squares 401

9.5.4 Beyond the Stationary Model 402

9.5.5 Example on Nonlinear Channel Identification and Reconvergence 405

9.6 Explicit Recursivity for Adaptive Kernel Models 406

9.6.1 Recursivity in Hilbert Spaces 406

9.6.2 Recursive Filters in Reproducing Kernel Hilbert Spaces 408

9.7 Online Sparsification with Kernels 411

9.7.1 Sparsity by Construction 411

9.7.2 Sparsity by Pruning 413

9.8 Probabilistic Approaches to Kernel Adaptive Filtering 414

9.8.1 Gaussian Processes and Kernel Ridge Regression 415

9.8.2 Online Recursive Solution for Gaussian Processes Regression 416

9.8.3 Kernel Recursive Least Squares Tracker 417

9.8.4 Probabilistic Kernel Least Mean Squares 418

9.9 Further Reading 418

9.9.1 Selection of Kernel Parameters 418

9.9.2 Multi-Kernel Adaptive Filtering 419

9.9.3 Recursive Filtering in Kernel Hilbert Spaces 419

9.10 Tutorials and Application Examples 419

9.10.1 Kernel Adaptive Filtering Toolbox 420

9.10.2 Prediction of a Respiratory Motion Time Series 421

9.10.3 Online Regression on the KIN h eK Dataset 423

9.10.4 The Mackey–Glass Time Series 425

9.10.5 Explicit Recursivity on Reproducing Kernel in Hilbert Space and Electroencephalogram Prediction 427

9.10.6 Adaptive Antenna Array Processing 428

9.11 Questions and Problems 430

Part III Classification, Detection, and Feature Extraction 433

10 Support Vector Machine and Kernel Classification Algorithms 435

10.1 Introduction 435

10.2 Support Vector Machine and Kernel Classifiers 435

10.2.1 Support Vector Machines 435

10.2.2 Multiclass and Multilabel Support Vector Machines 441

10.2.3 Least-Squares Support Vector Machine 447

10.2.4 Kernel Fisher’s Discriminant Analysis 448

10.3 Advances in Kernel-Based Classification 452

10.3.1 Large Margin Filtering 452

10.3.2 Semisupervised Learning 454

10.3.3 Multiple Kernel Learning 460

10.3.4 Structured-Output Learning 462

10.3.5 Active Learning 468

10.4 Large-Scale Support Vector Machines 477

10.4.1 Large-Scale Support Vector Machine Implementations 477

10.4.2 Random Fourier Features 478

10.4.3 Parallel Support Vector Machine 480

10.4.4 Outlook 483

10.5 Tutorials and Application Examples 485

10.5.1 Examples of Support Vector Machine Classification 485

10.5.2 Example of Least-Squares Support Vector Machine 492

10.5.3 Kernel-Filtering Support Vector Machine for Brain–Computer Interface Signal Classification 493

10.5.4 Example of Laplacian Support Vector Machine 494

10.5.5 Example of Graph-Based Label Propagation 498

10.5.6 Examples of Multiple Kernel Learning 498

10.6 Concluding Remarks 501

10.7 Questions and Problems 502

11 Clustering and Anomaly Detection with Kernels 503

11.1 Introduction 503

11.2 Kernel Clustering 506

11.2.1 Kernelization of the Metric 506

11.2.2 Clustering in Feature Spaces 508

11.3 Domain Description Via Support Vectors 514

11.3.1 Support Vector Domain Description 514

11.3.2 One-Class Support Vector Machine 515

11.3.3 Relationship Between Support Vector Domain Description and Density Estimation 516

11.3.4 Semisupervised One-Class Classification 517

11.4 Kernel Matched Subspace Detectors 518

11.4.1 Kernel Orthogonal Subspace Projection 518

11.4.2 Kernel Spectral Angle Mapper 520

11.5 Kernel Anomaly Change Detection 522

11.5.1 Linear Anomaly Change Detection Algorithms 522

11.5.2 Kernel Anomaly Change Detection Algorithms 523

11.6 Hypothesis Testing with Kernels 525

11.6.1 Distribution Embeddings 526

11.6.3 Maximum Mean Discrepancy 527

11.6.3 One-Class Support Measure Machine 528

11.7 Tutorials and Application Examples 529

11.7.1 Example on Kernelization of the Metric 529

11.7.2 Example on Kernel k-Means 530

11.7.3 Domain Description Examples 531

11.7.4 Kernel Spectral Angle Mapper and Kernel Orthogonal Subspace Projection Examples 534

11.7.5 Example of Kernel Anomaly Change Detection Algorithms 536

11.7.6 Example on Distribution Embeddings and Maximum Mean Discrepancy 540

11.8 Concluding Remarks 541

11.9 Questions and Problems 542

12 Kernel Feature Extraction in Signal Processing 543

12.1 Introduction 543

12.2 Multivariate Analysis in Reproducing Kernel Hilbert Spaces 545

12.2.1 Problem Statement and Notation 545

12.2.2  Linear Multivariate Analysis 546

12.2.3 Kernel Multivariate Analysis 549

12.2.4 Multivariate Analysis Experiments 551

12.3 Feature Extraction with Kernel Dependence Estimates 555

12.3.1 Feature Extraction Using Hilbert–Schmidt Independence Criterion 556

12.3.2  Blind Source Separation Using Kernels 563

12.4 Extensions for Large-Scale and Semisupervised Problems 570

12.4.2 Efficiency with the Incomplete Cholesky Decomposition 570

12.4.3  Efficiency with Random Fourier Features 570

12.4.3 Sparse Kernel Feature Extraction 571

12.4.4 Semisupervised Kernel Feature Extraction 573

12.5 Domain Adaptation with Kernels 575

12.5.1 Kernel Mean Matching 578

12.5.2 Transfer Component Analysis 579

12.5.3 Kernel Manifold Alignment 581

12.5.4 Relations between Domain Adaptation Methods 585

12.5.5 Experimental Comparison between Domain Adaptation Methods

12.6 Concluding Remarks 587

12.7 Questions and Problems 588

References 589

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Author Information

José Luis Rojo-Álvarez, PhD, is a Full Professor in the Department of Signal Theory and Communications at the University Rey Juan Carlos, Spain.

Manel Martínez-Ramón, PhD, is a Professor in the Department of Electrical and Computer Engineering at the University of New Mexico, Albuquerque, The United States of America.

Jordi Muñoz-Marí, PhD, is an Associate Professor in the Department of Electronics Engineering at the Universitat de València, Spain.

Gustau Camps-Valls, PhD, is an Associate Professor in the Department of Electronics Engineering at the Universitat de València, Spain.

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