Preface xiii
Chapter 1 Introduction 1
1.1. Modern Analytical Chemistry 1
1.1.1. Developments in Modern Chemistry 1
1.1.2. Modern Analytical Chemistry 2
1.1.3. Multidimensional Dataset 3
1.2. Chemometrics 5
1.2.1. Introduction to Chemometrics 5
1.2.2. Instrumental Response and Data Processing 8
1.2.3. White, Black, and Gray Systems 9
1.3. Chemometrics-Based Signal Processing Techniques 10
1.3.1. Common Methods for Processing Chemical Data 10
1.3.2. Wavelets in Chemistry 11
1.4. Resources Available on Chemometrics and Wavelet Transform 12
1.4.1. Books 12
1.4.2. Online Resources 14
1.4.3. Mathematics Software 15
Chapter 2 One-dimensional Signal Processing Techniques in Chemistry 23
2.1. Digital Smoothing and Filtering Methods 23
2.1.1. Moving-Window Average Smoothing Method 24
2.1.2. Savitsky-Golay Filter 25
2.1.3. Kalman Filtering 32
2.1.4. Spline Smoothing 36
2.2. Transformation Methods of Analytical Signals 39
2.2.1. Physical Meaning of the Convolution Algorithm 39
2.2.2. Multichannel Advantage in Spectroscopy and Hadamard Transformation 41
2.2.3. Fourier Transformation 44
2.2.3.1. Discrete Fourier Transformation and Spectral Multiplex Advantage 45
2.2.3.2. Fast Fourier Transformation 48
2.2.3.3. Fourier Transformation as Applied to Smooth Analytical Signals 50
2.2.3.4. Fourier Transformation as Applied to Convolution and Deconvolution 52
2.3. Numerical Differentiation 54
2.3.1. Simple Difference Method 54
2.3.2. Moving-Window Polynomial Least-Squares Fitting Method 55
2.4. Data Compression 57
2.4.1. Data Compression Based on B-Spline Curve Fitting 57
2.4.2. Data Compression Based on Fourier Transformation 64
2.4.3. Data Compression Based on Principal-Component Analysis 64
Chapter 3 Two-dimensional Signal Processing Techniques in Chemistry 69
3.1. General Features of Two-Dimensional Data 69
3.2. Some Basic Concepts for Two-Dimensional Data from Hyphenated Instrumentation 70
3.2.1. Chemical Rank and Principal-Component Analysis (PCA) 71
3.2.2. Zero-Component Regions and Estimation of Noise Level and Background 75
3.3. Double-Centering Technique for Background Correction 77
3.4. Congruence Analysis and Least-Squares Fitting 78
3.5. Differentiation Methods for Two-Dimensional Data 80
3.6 Resolution Methods for Two-Dimensional Data 81
3.6.1. Local Principal-Component Analysis and Rankmap 83
3.6.2. Self-Modeling Curve Resolution and Evolving Resolution Methods 85
3.6.2.1. Evolving Factor Analysis (EFA) 88
3.6.2.2. Window Factor Analysis (WFA) 90
3.6.2.3. Heuristic Evolving Latent Projections (HELP) 94
Chapter 4 Fundamentals of Wavelet Transform 99
4.1. Introduction to Wavelet Transform and Wavelet Packet Transform 100
4.1.1. A Simple Example: Haar Wavelet 103
4.1.2. Multiresolution Signal Decomposition 108
4.1.3. Basic Properties of Wavelet Function 112
4.2. Wavelet Function Examples 113
4.2.1. Meyer Wavelet 113
4.2.2. B-Spline (Battle--Lemarie) Wavelets 114
4.2.3. Daubechies Wavelets 116
4.2.4. Coiflet Functions 117
4.3. Fast Wavelet Algorithm and Packet Algorithm 118
4.3.1. Fast Wavelet Transform 119
4.3.2. Inverse Fast Wavelet Transform 122
4.3.3. Finite Discrete Signal Handling with Wavelet Transform 125
4.3.4. Packet Wavelet Transform 132
4.4. Biorthogonal Wavelet Transform 134
4.4.1. Multiresolution Signal Decomposition of Biorthogonal Wavelet 134
4.4.2. Biorthogonal Spline Wavelets 136
4.4.3. A Computing Example 137
4.5. Two-Dimensional Wavelet Transform 140
4.5.1. Multidimensional Wavelet Analysis 140
4.5.2. Implementation of Two-Dimensional Wavelet Transform 141
Chapter 5 Application of Wavelet Transform In Chemistry 147
5.1. Data Compression 148
5.1.1. Principle and Algorithm 149
5.1.2. Data Compression Using Wavelet Packet Transform 155
5.1.3. Best-Basis Selection and Criteria for Coefficient Selection 158
5.2. Data Denoising and Smoothing 166
5.2.1. Denoising 167
5.2.2. Smoothing 173
5.2.3. Denoising and Smoothing Using Wavelet Packet Transform 179
5.2.4. Comparison between Wavelet Transform and Conventional Methods 182
5.3. Baseline/Background Removal 183
5.3.1. Principle and Algorithm 184
5.3.2. Background Removal 185
5.3.3. Baseline Correction 191
5.3.4. Background Removal Using Continuous Wavelet Transform 191
5.3.5. Background Removal of Two-Dimensional Signals 196
5.4. Resolution Enhancement 199
5.4.1. Numerical Differentiation Using Discrete Wavelet Transform 200
5.4.2. Numerical Differentiation Using Continuous Wavelet Transform 205
5.4.3. Comparison between Wavelet Transform and other Numerical Differentiation Methods 210
5.4.4. Resolution Enhancement 212
5.4.5. Resolution Enhancement by Using Wavelet Packet Transform 220
5.4.6. Comparison between Wavelet Transform and Fast Fourier Transform for Resolution Enhancement 221
5.5. Combined Techniques 225
5.5.1. Combined Method for Regression and Calibration 225
5.5.2. Combined Method for Classification and Pattern Recognition 227
5.5.3. Combined Method of Wavelet Transform and Chemical Factor Analysis 228
5.5.4. Wavelet Neural Network 230
5.6. An Overview of the Applications in Chemistry 232
5.6.1. Flow Injection Analysis 233
5.6.2. Chromatography and Capillary Electrophoresis 234
5.6.3. Spectroscopy 238
5.6.4. Electrochemistry 244
5.6.5. Mass Spectrometry 246
5.6.6. Chemical Physics and Quantum Chemistry 248
5.6.7. Conclusion 249
Appendix Vector and Matrix Operations and Elementary MATLAB 257
A.1. Elementary Knowledge in Linear Algebra 257
A.1.1. Vectors and Matrices in Analytical Chemistry 257
A.1.2. Column and Row Vectors 259
A.1.3. Addition and Subtraction of Vectors 259
A.1.4. Vector Direction and Length 260
A.1.5. Scalar Multiplication of Vectors 261
A.1.6. Inner and Outer Products between Vectors 262
A.1.7. The Matrix and Its Operations 263
A.1.8. Matrix Addition and Subtraction 264
A.1.9. Matrix Multiplication 264
A.1.10. Zero Matrix and Identity Matrix 264
A.1.11. Transpose of a Matrix 265
A.1.12. Determinant of a Matrix 265
A.1.13. Inverse of a Matrix 266
A.1.14. Orthogonal Matrix 266
A.1.15. Trace of a Square Matrix 267
A.1.16. Rank of a Matrix 268
A.1.17. Eigenvalues and Eigenvectors of a Matrix 268
A.1.18. Singular-Value Decomposition 269
A.1.19. Generalized Inverse 270
A.1.20. Derivative of a Matrix 271
A.1.21. Derivative of a Function with Vector as Variable 271
A.2. Elementary Knowledge of MATLAB 273
A.2.1. Matrix Construction 275
A.2.2. Matrix Manipulation 275
A.2.3. Basic Mathematical Functions 276
A.2.4. Methods for Generating Vectors and Matrices 278
A.2.5. Matrix Subscript System 280
A.2.6. Matrix Decomposition 286
A.2.6.1. Singular-Value Decomposition (SVD) 286
A.2.6.2. Eigenvalues and Eigenvectors (eig) 287
A.2.7. Graphic Functions 288
Index 293
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