Chau Chemometrics From Basics to Wavelet Transform
.pdfChemometrics
CHEMICAL ANALYSIS
A SERIES OF MONOGRAPHS ON ANALYTICAL CHEMISTRY
AND ITS APPLICATIONS
Edited by
J. D. WINEFORDNER
VOLUME 164
Chemometrics
From Basics to Wavelet Transform
FOO-TIM CHAU
Hong Kong Polytechnic University
YI-ZENG LIANG
Central South University
JUNBIN GAO
University of New England
XUE-GUANG SHAO
University of Science and Technology of China
Copyright ? 2004 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Chemometrics: from basics to wavelet transform/Foo-tim Chau . . . [et al.]. p. cm. --- (Chemical analysis; 1075)
Includes bibliographical references. ISBN 0-471-20242-8 (acid-free paper)
1. Instrumental analysis---Data processing. 2. Instrumental analysis---Automation. 3. Wavelets (Mathematics) I. Chau, Foo-tim. II. Chemical analysis; v. 1075.
QD79.I5C44 2004 543 .07 0285---dc21
2003002429
Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1
CONTENTS
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 |
v |
|
vi |
contents |
|
|
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 |
|
contents |
vii |
|
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 |
viii |
contents |
|
|
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 |
contents |
ix |
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 |
x |
contents |
|
|
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 |