Identification of Nonlinear Physiological

Significant advances have been made in the field since the previous classic texts were written. This text brings the available knowledge up to date. * Enables the reader to use a wide variety of nonlinear system identification techniques. * Offers a thorough treatment of the underlying theory. * Provides a MATLAB toolbox containing implementation of the latest identification methods together with an extensive set of problems using realistic data sets.

David T. Westwick is an assistant professor in the Department of Electrical and Computer Engineering at the University of Calgary. <p> Robert E. Kearney is professor and Chair of the Department of Biomedical Engineering at McGill University. A recipient of the IEEE Millenium Medal, he is a Fellow of the IEEE and former President of the IEEE Engineering in Medicine and Biology Society.

Preface

xi

1 Introduction

1

(12)

1.1 Signals

1

(2)

1.1.1 Domain and Range

2

(1)

1.1.2 Deterministic and Stochastic Signals

2

(1)

1.1.3 Stationary and Ergodic Signals

3

(1)

1.2 Systems and Models

3

(5)

1.2.1 Model Structure and Parameters

4

(1)

1.2.2 Static and Dynamic Systems

5

(1)

1.2.3 Linear and Nonlinear Systems

6

(1)

1.2.4 Time-Invariant and Time-Varying Systems

7

(1)

1.2.5 Deterministic and Stochastic Systems

7

(1)

1.3 System Modeling

8

(1)

1.4 System Identification

8

(3)

1.4.1 Types of System Identification Problems

9

(2)

1.4.2 Applications of System Identification

11

(1)

1.5 How Common are Nonlinear Systems?

11

(2)

2 Background

13

(26)

2.1 Vectors and Matrices

13

(1)

2.2 Gaussian Random Variables

14

(2)

2.2.1 Products of Gaussian Variables

15

(1)

2.3 Correlation Functions

16

(9)

2.3.1 Autocorrelation Functions

16

(2)

2.3.2 Cross-Correlation Functions

18

(2)

2.3.3 Effects of Noise

20

(1)

2.3.4 Estimates of Correlation Functions

21

(1)

2.3.5 Frequency Domain Expressions

22

(1)

2.3.6 Applications

23

(2)

2.3.7 Higher-Order Correlation Functions

25

(1)

2.4 Mean-Square Parameter Estimation

25

(4)

2.4.1 Linear Least-Squares Regression

26

(1)

2.4.2 Properties of Estimates

27

(2)

2.5 Polynomials

29

(6)

2.5.1 Power Series

29

(1)

2.5.2 Orthogonal Polynomials

30

(1)

2.5.3 Hermite Polynomials

31

(1)

2.5.4 Tchebyshev Polynomials

32

(1)

2.5.5 Multiple-Variable Polynomials

33

(2)

2.6 Notes and References

35

(1)

2.7 Problems

36

(1)

2.8 Computer Exercises

36

(3)

3 Models of Linear Systems

39

(18)

3.1 Linear Systems

39

(1)

3.2 Nonparametric Models

40

(6)

3.2.1 Time Domain Models

41

(2)

3.2.2 Frequency Domain Models

43

(3)

3.3 Parametric Models

46

(6)

3.3.1 Parametric Frequency Domain Models

46

(2)

3.3.2 Discrete-Time Parametric Models

48

(4)

3.4 State-Space Models

52

(2)

3.4.1 Example: Human Ankle Compliance-Discrete-Time, State-Space Model

54

(1)

3.5 Notes and References

54

(1)

3.6 Theoretical Problems

55

(1)

3.7 Computer Exercises

56

(1)

4 Models of Nonlinear Systems

57

(46)

4.1 The Volterra Series

57

(10)

4.1.1 The Finite Volterra Series

59

(3)

4.1.2 Multiple-Input Systems

62

(2)

4.1.3 Polynomial Representation

64

(1)

4.1.4 Convergence Issues()

65

(2)

4.2 The Wiener Series

67

(6)

4.2.1 Orthogonal Expansion of the Volterra Series

68

(2)

4.2.2 Relation Between the Volterra and Wiener Series

70

(1)

4.2.3 Example: Peripheral Auditory Model-Wiener Kernels

71

(2)

4.2.4 Nonwhite Inputs

73

(1)

4.3 Simple Block Structures

73

(14)

4.3.1 The Wiener Model

73

(4)

4.3.2 The Hammerstein Model

77

(2)

4.3.3 Sandwich or Wiener-Hammerstein Models

79

(4)

4.3.4 NLN Cascades

83

(4)

4.3.5 Multiple-Input Multiple-Output Block Structured Models

87

(1)

4.4 Parallel Cascades

87

(4)

4.4.1 Approximation Issues()

89

(2)

4.5 The Wiener-Bose Model

91

(9)

4.5.1 Similarity Transformations and Uniqueness

92

(2)

4.5.2 Approximation Issues()

94

(1)

4.5.3 Volterra Kernels of the Wiener-Bose Model

94

(1)

4.5.4 Wiener Kernels of the Wiener-Bose Model

95

(2)

4.5.5 Relationship to the Parallel Cascade Model

97

(3)

4.6 Notes and References

100

(1)

4.7 Theoretical Problems

100

(1)

4.8 Computer Exercises

101

(2)

5 Identification of Linear Systems

103

(22)

5.1 Introduction

103

(4)

5.1.1 Example: Identification of Human Joint Compliance

103

(2)

5.1.2 Model Evaluation

105

(2)

5.2 Nonparametric Time Domain Models

107

(8)

5.2.1 Direct Estimation

107

(1)

5.2.2 Least-Squares Regression

108

(1)

5.2.3 Correlation-Based Methods

109

(6)

5.3 Frequency Response Estimation

115

(4)

5.3.1 Sinusoidal Frequency Response Testing

115

(1)

5.3.2 Stochastic Frequency Response Testing

116

(1)

5.3.3 Coherence Functions

117

(2)

5.4 Parametric Methods

119

(3)

5.4.1 Regression

119

(1)

5.4.2 Instrumental Variables

120

(1)

5.4.3 Nonlinear Optimization

121

(1)

5.5 Notes and References

122

(1)

5.6 Computer Exercises

122

(3)

6 Correlation-Based Methods

125

(44)

6.1 Methods for Functional Expansions

125

(24)

6.1.1 Lee-Schetzen Cross-Correlation

125

(15)

6.1.2 Colored Inputs

140

(4)

6.1.3 Frequency Domain Approaches

144

(5)

6.2 Block-Structured Models

149

(18)

6.2.1 Wiener Systems

150

(5)

6.2.2 Hammerstein Models

155

(7)

6.2.3 LNL Systems

162

(5)

6.3 Problems

167

(1)

6.4 Computer Exercises

167

(2)

7 Explicit Least-Squares Methods

169

(38)

7.1 Introduction

169

(1)

7.2 The Orthogonal Algorithms

169

(18)

7.2.1 The Orthogonal Algorithm

171

(2)

7.2.2 The Fast Orthogonal Algorithm

173

(7)

7.2.3 Variance of Kernel Estimates

180

(2)

7.2.4 Example: Fast Orthogonal Algorithm Applied to Simulated Fly Retina Data

182

(4)

7.2.5 Application: Dynamics of the Cockroach Tactile Spine

186

(1)

7.3 Expansion Bases

187

(11)

7.3.1 The Basis Expansion Algorithm

190

(1)

7.3.2 The Laguerre Expansion

191

(1)

7.3.3 Limits on α

192

(2)

7.3.4 Choice of α and P

194

(1)

7.3.5 The Laguerre Expansion Technique

195

(1)

7.3.6 Computational Requirements

195

(1)

7.3.7 Variance of Laguerre Kernel Estimates

195

(1)

7.3.8 Example: Laguerne Expansion Kernels of the Fly Retina Model

196

(2)

7.4 Principal Dynamic Modes

198

(7)

7.4.1 Example: Principal Dynamic Modes of the Fly Retina Model

200

(1)

7.4.2 Application: Cockroach Tactile Spine

201

(4)

7.5 Problems

205

(1)

7.6 Computer Exercises

205

(2)

8 Iterative Least-Squares Methods

207

(44)

8.1 Optimization Methods

207

(16)

8.1.1 Gradient Descent Methods

208

(1)

8.1.2 Identification of Block-Structured Models

209

(3)

8.1.3 Second-Order Optimization Methods

212

(4)

8.1.4 Jacobians for Other Block Structures

216

(3)

8.1.5 Optimization Methods for Parallel Cascade Models

219

(1)

8.1.6 Example: Using a Separable Volterra Network

220

(3)

8.2 Parallel Cascade Methods

223

(23)

8.2.1 Parameterization Issues

226

(2)

8.2.2 Testing Paths for Significance

228

(2)

8.2.3 Choosing the Linear Elements

230

(12)

8.2.4 Parallel Wiener Cascade Algorithm

242

(1)

8.2.5 Longer Cascades

242

(1)

8.2.6 Example: Parallel Cascade Identification

243

(3)

8.3 Application: Visual Processing in the Light-Adapted Fly Retina

246

(3)

8.4 Problems

249

(1)

8.5 Computer Exercises

250

(1)

References

251

(8)

Index

259

(3)

IEEE Press Series in Biomedical Engineering

262

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Identification of Nonlinear Physiological Systems

Summary

Author Biography

Table of Contents

Supplemental Materials