Foreword ix
Preface xi
Nomenclature and Conventions xiii
1 Introduction 1
1.1 What is multi-way analysis? 1
1.2 Conceptual aspects of multi-way data analysis 1
1.3 Hierarchy of multivariate data structures in chemistry 5
1.4 Principal component analysis and PARAFAC 11
1.5 Summary 12
2 Array definitions and properties 13
2.1 Introduction 13
2.2 Rows, columns and tubes; frontal, lateral and horizontal slices 13
2.3 Elementary operations 15
2.4 Linearity concepts 21
2.5 Rank of two-way arrays 22
2.6 Rank of three-way arrays 28
2.7 Algebra of multi-way analysis 32
2.8 Summary 34
Appendix 2.A 34
3 Two-way component and regression models 35
3.1 Models for two-way one-block data analysis: component models 35
3.2 Models for two-way two-block data analysis: regression models 46
3.3 Summary 53
Appendix 3.A: some PCA results 54
Appendix 3.B: PLS algorithms 55
4 Three-way component and regression models 57
4.1 Historical introduction to multi-way models 57
4.2 Models for three-way one-block data: three-way component models 59
4.3 Models for three-way two-block data: three-way regression models 76
4.4 Summary 83
Appendix 4.A: alternative notation for the PARAFAC model 84
Appendix 4.B: alternative notations for the Tucker3 model 86
5 Some properties of three-way component models 89
5.1 Relationships between three-way component models 89
5.2 Rotational freedom and uniqueness in three-way component models 98
5.3 Properties of Tucker3 models 106
5.4 Degeneracy problem in PARAFAC models 107
5.5 Summary 109
6 Algorithms 111
6.1 Introduction 111
6.2 Optimization techniques 111
6.3 PARAFAC algorithms 113
6.4 Tucker3 algorithms 119
6.5 Tucker2 and Tucker1 algorithms 123
6.6 Multi-linear partial least squares regression 124
6.7 Multi-way covariates regression models 128
6.8 Core rotation in Tucker3 models 130
6.9 Handling missing data 131
6.10 Imposing non-negativity 135
6.11 Summary 136
Appendix 6.A: closed-form solution for the PARAFAC model 136
Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition 144
7 Validation and diagnostics 145
7.1 What is validation? 145
7.2 Test-set and cross-validation 147
7.3 Selecting which model to use 154
7.4 Selecting the number of components 156
7.5 Residual and influence analysis 166
7.6 Summary 173
8 Visualization 175
8.1 Introduction 175
8.2 History of plotting in three-way analysis 179
8.3 History of plotting in chemical three-way analysis 180
8.4 Scree plots 180
8.5 Line plots 184
8.6 Scatter plots 190
8.7 Problems with scatter plots 192
8.8 Image analysis 201
8.9 Dendrograms 202
8.10 Visualizing the Tucker core array 204
8.11 Joint plots 205
8.12 Residual plots 216
8.13 Leverage plots 216
8.14 Visualization of large data sets 216
8.15 Summary 219
9 Preprocessing 221
9.1 Background 221
9.2 Two-way centering 228
9.3 Two-way scaling 232
9.4 Simultaneous two-way centering and scaling 238
9.5 Three-way preprocessing 239
9.6 Summary 244
Appendix 9.A: other types of preprocessing 245
Appendix 9.B: geometric view of centering 247
Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data 249
Appendix 9.D: rank reduction and centering 250
Appendix 9.E: centering data with missing values 251
Appendix 9.F: incorrect centering introduces artificial variation 251
Appendix 9.G: alternatives to centering 254
10 Applications 257
10.1 Introduction 257
10.2 Curve resolution of fluorescence data 259
10.3 Second-order calibration 276
10.4 Multi-way regression 285
10.5 Process chemometrics 288
10.6 Exploratory analysis in chromatography 302
10.7 Exploratory analysis in environmental sciences 312
10.8 Exploratory analysis of designed data 323
10.9 Analysis of variance of data with complex interactions 340
Appendix 10.A: an illustration of the generalized rank annihilation method 346
Appendix 10.B: other types of second-order calibration problems 347
Appendix 10.C: the multiple standards calibration model of the second-order calibration example 349
References 351
Index 371
