Providing an easy explanation of the fundamentals, methods, and applications of chemometrics
- Acts as a practical guide to multivariate data analysis techniques
- Explains the methods used in Chemometrics and teaches the reader to perform all relevant calculations
- Presents the basic chemometric methods as worksheet functions in Excel
- Includes Chemometrics Add In for download which uses Microsoft Excel® for chemometrics training
- Online downloads includes workbooks with examples
Table of Contents
Preface xvii
PART I INTRODUCTION 1
1 What is Chemometrics? 3
1.1 Subject of Chemometrics, 3
1.2 Historical Digression, 5
2 What the Book Is About? 8
2.1 Useful Hints, 8
2.2 Book Syllabus, 9
2.3 Notations, 10
3 Installation of Chemometrics Add-In 11
3.1 Installation, 11
3.2 General Information, 14
4 Further Reading on Chemometrics 15
4.1 Books, 15
4.1.1 The Basics, 15
4.1.2 Chemometrics, 16
4.1.3 Supplements, 16
4.2 The Internet, 17
4.2.1 Tutorials, 17
4.3 Journals, 17
4.3.1 Chemometrics, 17
4.3.2 Analytical, 18
4.3.3 Mathematical, 18
4.4 Software, 18
4.4.1 Specialized Packages, 18
4.4.2 General Statistic Packages, 19
4.4.3 Free Ware, 19
PART II THE BASICS 21
5 Matrices and Vectors 23
5.1 The Basics, 23
5.1.1 Matrix, 23
5.1.2 Simple Matrix Operations, 24
5.1.3 Matrices Multiplication, 25
5.1.4 Square Matrix, 26
5.1.5 Trace and Determinant, 27
5.1.6 Vectors, 28
5.1.7 Simple Vector Operations, 29
5.1.8 Vector Products, 29
5.1.9 Vector Norm, 30
5.1.10 Angle Between Vectors, 30
5.1.11 Vector Representation of a Matrix, 30
5.1.12 Linearly Dependent Vectors, 31
5.1.13 Matrix Rank, 31
5.1.14 Inverse Matrix, 31
5.1.15 Pseudoinverse, 32
5.1.16 Matrix–Vector Product, 33
5.2 Advanced Information, 33
5.2.1 Systems of Linear Equations, 33
5.2.2 Bilinear and Quadratic Forms, 34
5.2.3 Positive Definite Matrix, 34
5.2.4 Cholesky Decomposition, 34
5.2.5 Polar Decomposition, 34
5.2.6 Eigenvalues and Eigenvectors, 35
5.2.7 Eigenvalues, 35
5.2.8 Eigenvectors, 35
5.2.9 Equivalence and Similarity, 36
5.2.10 Diagonalization, 37
5.2.11 Singular Value Decomposition (SVD), 37
5.2.12 Vector Space, 38
5.2.13 Space Basis, 39
5.2.14 Geometric Interpretation, 39
5.2.15 Nonuniqueness of Basis, 39
5.2.16 Subspace, 40
5.2.17 Projection, 40
6 Statistics 42
6.1 The Basics, 42
6.1.1 Probability, 42
6.1.2 Random Value, 43
6.1.3 Distribution Function, 43
6.1.4 Mathematical Expectation, 44
6.1.5 Variance and Standard Deviation, 44
6.1.6 Moments, 44
6.1.7 Quantiles, 45
6.1.8 Multivariate Distributions, 45
6.1.9 Covariance and Correlation, 45
6.1.10 Function, 46
6.1.11 Standardization, 46
6.2 Main Distributions, 46
6.2.1 Binomial Distribution, 46
6.2.2 Uniform Distribution, 47
6.2.3 Normal Distribution, 48
6.2.4 Chi-Squared Distribution, 50
6.2.5 Student’s Distribution, 52
6.2.6 F-Distribution, 53
6.2.7 Multivariate Normal Distribution, 54
6.2.8 Pseudorandom Numbers, 55
6.3 Parameter Estimation, 56
6.3.1 Sample, 56
6.3.2 Outliers and Extremes, 56
6.3.3 Statistical Population, 56
6.3.4 Statistics, 57
6.3.5 Sample Mean and Variance, 57
6.3.6 Sample Covariance and Correlation, 58
6.3.7 Order Statistics, 59
6.3.8 Empirical Distribution and Histogram, 60
6.3.9 Method of Moments, 61
6.3.10 The Maximum Likelihood Method, 62
6.4 Properties of the Estimators, 62
6.4.1 Consistency, 62
6.4.2 Bias, 63
6.4.3 Effectiveness, 63
6.4.4 Robustness, 63
6.4.5 Normal Sample, 64
6.5 Confidence Estimation, 64
6.5.1 Confidence Region, 64
6.5.2 Confidence Interval, 65
6.5.3 Example of a Confidence Interval, 65
6.5.4 Confidence Intervals for the Normal Distribution, 65
6.6 Hypothesis Testing, 66
6.6.1 Hypothesis, 66
6.6.2 Hypothesis Testing, 66
6.6.3 Type I and Type II Errors, 67
6.6.4 Example, 67
6.6.5 Pearson’s Chi-Squared Test, 67
6.6.6 F-Test, 69
6.7 Regression, 70
6.7.1 Simple Regression, 70
6.7.2 The Least Squares Method, 71
6.7.3 Multiple Regression, 72
Conclusion, 73
7 Matrix Calculations in Excel 74
7.1 Basic Information, 74
7.1.1 Region and Language, 74
7.1.2 Workbook, Worksheet, and Cell, 76
7.1.3 Addressing, 77
7.1.4 Range, 78
7.1.5 Simple Calculations, 78
7.1.6 Functions, 78
7.1.7 Important Functions, 81
7.1.8 Errors in Formulas, 85
7.1.9 Formula Dragging, 86
7.1.10 Create a Chart, 87
7.2 Matrix Operations, 88
7.2.1 Array Formulas, 88
7.2.2 Creating and Editing an Array Formula, 90
7.2.3 Simplest Matrix Operations, 91
7.2.4 Access to the Part of a Matrix, 91
7.2.5 Unary Operations, 93
7.2.6 Binary Operations, 95
7.2.7 Regression, 95
7.2.8 Critical Bug in Excel 2003, 99
7.2.9 Virtual Array, 99
7.3 Extension of Excel Possibilities, 100
7.3.1 VBA Programming, 100
7.3.2 Example, 101
7.3.3 Macro Example, 103
7.3.4 User-Defined Function Example, 104
7.3.5 Add-Ins, 105
7.3.6 Add-In Installation, 106
Conclusion, 107
8 Projection Methods in Excel 108
8.1 Projection Methods, 108
8.1.1 Concept and Notation, 108
8.1.2 PCA, 109
8.1.3 PLS, 110
8.1.4 Data Preprocessing, 111
8.1.5 Didactic Example, 112
8.2 Application of Chemometrics Add-In, 113
8.2.1 Installation, 113
8.2.2 General, 113
8.3 PCA, 114
8.3.1 ScoresPCA, 114
8.3.2 LoadingsPCA, 114
8.4 PLS, 116
8.4.1 ScoresPLS, 116
8.4.2 UScoresPLS, 117
8.4.3 LoadingsPLS, 118
8.4.4 WLoadingsPLS, 119
8.4.5 QLoadingsPLS, 120
8.5 PLS2, 121
8.5.1 ScoresPLS2, 121
8.5.2 UScoresPLS2, 122
8.5.3 LoadingsPLS2, 124
8.5.4 WLoadingsPLS2, 125
8.5.5 QLoadingsPLS2, 126
8.6 Additional Functions, 127
8.6.1 MIdent, 127
8.6.2 MIdentD2, 127
8.6.3 MCutRows, 129
8.6.4 MTrace, 129
Conclusion, 130
PART IIICHEMOMETRICS 131
9 Principal Component Analysis (PCA) 133
9.1 The Basics, 133
9.1.1 Data, 133
9.1.2 Intuitive Approach, 134
9.1.3 Dimensionality Reduction, 136
9.2 Principal Component Analysis, 136
9.2.1 Formal Specifications, 136
9.2.2 Algorithm, 137
9.2.3 PCA and SVD, 137
9.2.4 Scores, 138
9.2.5 Loadings, 139
9.2.6 Data of Special Kind, 140
9.2.7 Errors, 140
9.2.8 Validation, 143
9.2.9 Decomposition “Quality”, 143
9.2.10 Number of Principal Components, 144
9.2.11 The Ambiguity of PCA, 145
9.2.12 Data Preprocessing, 146
9.2.13 Leverage and Deviation, 146
9.3 People and Countries, 146
9.3.1 Example, 146
9.3.2 Data, 147
9.3.3 Data Exploration, 147
9.3.4 Data Pretreatment, 148
9.3.5 Scores and Loadings Calculation, 149
9.3.6 Scores Plots, 151
9.3.7 Loadings Plot, 152
9.3.8 Analysis of Residuals, 153
Conclusion, 153
10 Calibration 156
10.1 The Basics, 156
10.1.1 Problem Statement, 156
10.1.2 Linear and Nonlinear Calibration, 157
10.1.3 Calibration and Validation, 158
10.1.4 Calibration “Quality”, 160
10.1.5 Uncertainty, Precision, and Accuracy, 162
10.1.6 Underfitting and Overfitting, 163
10.1.7 Multicollinearity, 164
10.1.8 Data Preprocessing, 166
10.2 Simulated Data, 166
10.2.1 The Principle of Linearity, 166
10.2.2 “Pure” Spectra, 166
10.2.3 “Standard” Samples, 166
10.2.4 X Data Creation, 167
10.2.5 Data Centering, 168
10.2.6 Data Overview, 168
10.3 Classic Calibration, 169
10.3.1 Univariate (Single Channel) Calibration, 169
10.3.2 The Vierordt Method, 172
10.3.3 Indirect Calibration, 174
10.4 Inverse Calibration, 176
10.4.1 Multiple Linear Calibration, 177
10.4.2 Stepwise Calibration, 178
10.5 Latent Variables Calibration, 180
10.5.1 Projection Methods, 180
10.5.2 Latent Variables Regression, 184
10.5.3 Implementation of Latent Variable Calibration, 185
10.5.4 Principal Component Regression (PCR), 186
10.5.5 Projection on the Latent Structures-1 (PLS1), 188
10.5.6 Projection on the Latent Structures-2 (PLS2), 191
10.6 Methods Comparison, 193
Conclusion, 197
11 Classification 198
11.1 The Basics, 198
11.1.1 Problem Statement, 198
11.1.2 Types of Classes, 199
11.1.3 Hypothesis Testing, 199
11.1.4 Errors in Classification, 200
11.1.5 One-Class Classification, 200
11.1.6 Training and Validation, 201
11.1.7 Supervised and Unsupervised Training, 201
11.1.8 The Curse of Dimensionality, 201
11.1.9 Data Preprocessing, 201
11.2 Data, 202
11.2.1 Example, 202
11.2.2 Data Subsets, 203
11.2.3 Workbook Iris.xls, 204
11.2.4 Principal Component Analysis, 205
11.3 Supervised Classification, 205
11.3.1 Linear Discriminant Analysis (LDA), 205
11.3.2 Quadratic Discriminant Analysis (QDA), 210
11.3.3 PLS Discriminant Analysis (PLSDA), 214
11.3.4 SIMCA, 217
11.3.5 k-Nearest Neighbors (kNN), 223
11.4 Unsupervised Classification, 225
11.4.1 PCA Again (Revisited), 225
11.4.2 Clustering by K-Means, 225
Conclusion, 229
12 Multivariate Curve Resolution 230
12.1 The Basics, 230
12.1.1 Problem Statement, 230
12.1.2 Solution Ambiguity, 232
12.1.3 Solvability Conditions, 234
12.1.4 Two Types of Data, 235
12.1.5 Known Spectrum or Profile, 236
12.1.6 Principal Component Analysis (PCA), 236
12.1.7 PCA and MCR, 237
12.2 Simulated Data, 237
12.2.1 Example, 237
12.2.2 Data, 238
12.2.3 PCA, 238
12.2.4 The HELP Plot, 240
12.3 Factor Analysis, 241
12.3.1 Procrustes Analysis, 241
12.3.2 Evolving Factor Analysis (EFA), 244
12.3.3 Windows Factor Analysis (WFA), 246
12.4 Iterative Methods, 249
12.4.1 Iterative Target Transform Factor Analysis (ITTFA), 249
12.4.2 Alternating Least Squares (ALS), 250
Conclusion, 252
PART IV SUPPLEMENTS 255
13 Extension Of Chemometrics Add-In 257
13.1 Using Virtual Arrays, 257
13.1.1 Simulated Data, 257
13.1.2 Virtual Array, 259
13.1.3 Data Preprocessing, 259
13.1.4 Decomposition, 260
13.1.5 Residuals Calculation, 260
13.1.6 Eigenvalues Calculation, 262
13.1.7 Orthogonal Distances Calculation, 263
13.1.8 Leverages Calculation, 264
13.2 Using VBA Programming, 265
13.2.1 VBA Advantages, 265
13.2.2 Virtualization of Real Arrays, 265
13.2.3 Data Preprocessing, 266
13.2.4 Residuals Calculation, 267
13.2.5 Eigenvalues Calculation, 268
13.2.6 Orthogonal Distances Calculation, 269
13.2.7 Leverages Calculation, 270
Conclusion, 271
14 Kinetic Modeling of Spectral Data 272
14.1 The “Grey” Modeling Method, 272
14.1.1 Problem Statement, 272
14.1.2 Example, 274
14.1.3 Data, 274
14.1.4 Soft Method of Alternating Least Squares (Soft-ALS), 275
14.1.5 Hard Method of Alternating Least Squares (Hard-ALS), 277
14.1.6 Using Solver Add-In, 279
Conclusions, 282
15 MATLAB®: Beginner’s Guide 283
15.1 The Basics, 283
15.1.1 Workspace, 283
15.1.2 Basic Calculations, 285
15.1.3 Echo, 285
15.1.4 Workspace Saving: MAT-Files, 286
15.1.5 Diary, 286
15.1.6 Help, 287
15.2 Matrices, 287
15.2.1 Scalars, Vectors, and Matrices, 287
15.2.2 Accessing Matrix Elements, 289
15.2.3 Basic Matrix Operations, 289
15.2.4 Special Matrices, 290
15.2.5 Matrix Calculations, 292
15.3 Integrating Excel and MATLAB®, 294
15.3.1 Configuring Excel, 294
15.3.2 Data Exchange, 294
15.4 Programming, 295
15.4.1 M-Files, 295
15.4.2 Script File, 296
15.4.3 Function File, 297
15.4.4 Plotting, 298
15.4.5 Plot Printing, 300
15.5 Sample Programs, 301
15.5.1 Centering and Scaling, 301
15.5.2 SVD/PCA, 301
15.5.3 PCA/NIPALS, 302
15.5.4 PLS1, 303
15.5.5 PLS2, 304
Conclusion, 306
Afterword. The Fourth Paradigm 307
Index 311
