Age K. Smilde received his MSc in Econometrics at the University of Groningen in 1986. He moved to the Department of Pharmacy in the same city where he did his PhD in Analytical Chemistry. His PhD was on "Multivariate Calibration of Reversed Phase Chromatographic Systems", and he received his degree in 1990. In the year 1992, he visited the Center for Process Analytical Chemistry (Seattle, USA), where he worked together with Prof. Bruce Kowalski. He is the Eastern Analytical Symposium 2006 Award Recipient for Achievements in Chemo metrics. In 1996 he was chairman of the Gordon Research Conference on Statistics in Chemistry and Chemical Engineering in Oxford (UK). Together with Rasmus Bro and Paul Geladi he wrote the book Multiway Analysis: Applications in the Chemical Sciences. Rasmus Bro (born 1965) studied mathematics and analytical chemistry at the Technical University of Denmark and received his M.Sc. in 1994. In 1998 he obtained his Ph.D. (Cum Laude) in multiway analysis from the University of Amsterdam, The Netherlands. In 2000 he received the third Elsevier Chemo metrics Award for noteworthy accomplishments in the field of chemo metrics by younger scientists, and in 2004 he received the Eastern Analytical Symposium Award for Achievements in Chemo metrics. He has authored more than 100 peer-reviewed scientific papers, 2 books on chemo metrics, and more than 20 proceedings, book contributions, reviews, and patents.Paul Geladi currently works at the Department of Food Science, Stellenbosch University. Paul does research in Analytical Chemistry, Chemo-informatics and Electrochemistry. Their most recent publication is 'Covalently electro grafted carb ox phenyl layers onto gold surface serving as a platform for the construction of an immune sensor for detection of methotrexate.

• Foreword ixPreface xiNomenclature and Conventions xiii1 Introduction 11.1 What is multi-way analysis? 11.2 Conceptual aspects of multi-way data analysis 11.3 Hierarchy of multivariate data structures in chemistry 51.4 Principal component analysis and PARAFAC 111.5 Summary 122 Array definitions and properties 132.1 Introduction 132.2 Rows, columns and tubes; frontal, lateral and horizontal slices 132.3 Elementary operations 152.4 Linearity concepts 212.5 Rank of two-way arrays 222.6 Rank of three-way arrays 282.7 Algebra of multi-way analysis 322.8 Summary 34Appendix 2.A 343 Two-way component and regression models 353.1 Models for two-way one-block data analysis: component models 353.2 Models for two-way two-block data analysis: regression models 463.3 Summary 53Appendix 3.A: some PCA results 54Appendix 3.B: PLS algorithms 554 Three-way component and regression models 574.1 Historical introduction to multi-way models 574.2 Models for three-way one-block data: three-way component models 594.3 Models for three-way two-block data: three-way regression models 764.4 Summary 83Appendix 4.A: alternative notation for the PARAFAC model 84Appendix 4.B: alternative notations for the Tucker3 model 865 Some properties of three-way component models 895.1 Relationships between three-way component models 895.2 Rotational freedom and uniqueness in three-way component models 985.3 Properties of Tucker3 models 1065.4 Degeneracy problem in PARAFAC models 1075.5 Summary 1096 Algorithms 1116.1 Introduction 1116.2 Optimization techniques 1116.3 PARAFAC algorithms 1136.4 Tucker3 algorithms 1196.5 Tucker2 and Tucker1 algorithms 1236.6 Multi-linear partial least squares regression 1246.7 Multi-way covariates regression models 1286.8 Core rotation in Tucker3 models 1306.9 Handling missing data 1316.10 Imposing non-negativity 1356.11 Summary 136Appendix 6.A: closed-form solution for the PARAFAC model 136Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition 1447 Validation and diagnostics 1457.1 What is validation? 1457.2 Test-set and cross-validation 1477.3 Selecting which model to use 1547.4 Selecting the number of components 1567.5 Residual and influence analysis 1667.6 Summary 1738 Visualization 1758.1 Introduction 1758.2 History of plotting in three-way analysis 1798.3 History of plotting in chemical three-way analysis 1808.4 Scree plots 1808.5 Line plots 1848.6 Scatter plots 1908.7 Problems with scatter plots 1928.8 Image analysis 2018.9 Dendrograms 2028.10 Visualizing the Tucker core array 2048.11 Joint plots 2058.12 Residual plots 2168.13 Leverage plots 2168.14 Visualization of large data sets 2168.15 Summary 2199 Preprocessing 2219.1 Background 2219.2 Two-way centering 2289.3 Two-way scaling 2329.4 Simultaneous two-way centering and scaling 2389.5 Three-way preprocessing 2399.6 Summary 244Appendix 9.A: other types of preprocessing 245Appendix 9.B: geometric view of centering 247Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data 249Appendix 9.D: rank reduction and centering 250Appendix 9.E: centering data with missing values 251Appendix 9.F: incorrect centering introduces artificial variation 251Appendix 9.G: alternatives to centering 25410 Applications 25710.1 Introduction 25710.2 Curve resolution of fluorescence data 25910.3 Second-order calibration 27610.4 Multi-way regression 28510.5 Process chemometrics 28810.6 Exploratory analysis in chromatography 30210.7 Exploratory analysis in environmental sciences 31210.8 Exploratory analysis of designed data 32310.9 Analysis of variance of data with complex interactions 340Appendix 10.A: an illustration of the generalized rank annihilation method 346Appendix 10.B: other types of second-order calibration problems 347Appendix 10.C: the multiple standards calibration model of the second-order calibration example 349References 351Index 371

"...covers some very important topics in modern chemical data analysis...this book is a reasonable place to start. I know of no others." (Technometrics, November 2005) "It will be a valuable addition to the library collections at universities and companies." (E-STREAMS, September 2005)"...extremely useful..." (Journal of Chemometrics, Vol 19 (2), February 2005)

Age K. Smilde received his MSc in Econometrics at the University of Groningen in 1986. He moved to the Department of Pharmacy in the same city where he did his PhD in Analytical Chemistry. His PhD was on "Multivariate Calibration of Reversed Phase Chromatographic Systems", and he received his degree in 1990. In the year 1992, he visited the Center for Process Analytical Chemistry (Seattle, USA), where he worked together with Prof. Bruce Kowalski. He is the Eastern Analytical Symposium 2006 Award Recipient for Achievements in Chemo metrics. In 1996 he was chairman of the Gordon Research Conference on Statistics in Chemistry and Chemical Engineering in Oxford (UK). Together with Rasmus Bro and Paul Geladi he wrote the book Multiway Analysis: Applications in the Chemical Sciences. Rasmus Bro (born 1965) studied mathematics and analytical chemistry at the Technical University of Denmark and received his M.Sc. in 1994. In 1998 he obtained his Ph.D. (Cum Laude) in multiway analysis from the University of Amsterdam, The Netherlands. In 2000 he received the third Elsevier Chemo metrics Award for noteworthy accomplishments in the field of chemo metrics by younger scientists, and in 2004 he received the Eastern Analytical Symposium Award for Achievements in Chemo metrics. He has authored more than 100 peer-reviewed scientific papers, 2 books on chemo metrics, and more than 20 proceedings, book contributions, reviews, and patents.Paul Geladi currently works at the Department of Food Science, Stellenbosch University. Paul does research in Analytical Chemistry, Chemo-informatics and Electrochemistry. Their most recent publication is 'Covalently electro grafted carb ox phenyl layers onto gold surface serving as a platform for the construction of an immune sensor for detection of methotrexate.

• Foreword ixPreface xiNomenclature and Conventions xiii1 Introduction 11.1 What is multi-way analysis? 11.2 Conceptual aspects of multi-way data analysis 11.3 Hierarchy of multivariate data structures in chemistry 51.4 Principal component analysis and PARAFAC 111.5 Summary 122 Array definitions and properties 132.1 Introduction 132.2 Rows, columns and tubes; frontal, lateral and horizontal slices 132.3 Elementary operations 152.4 Linearity concepts 212.5 Rank of two-way arrays 222.6 Rank of three-way arrays 282.7 Algebra of multi-way analysis 322.8 Summary 34Appendix 2.A 343 Two-way component and regression models 353.1 Models for two-way one-block data analysis: component models 353.2 Models for two-way two-block data analysis: regression models 463.3 Summary 53Appendix 3.A: some PCA results 54Appendix 3.B: PLS algorithms 554 Three-way component and regression models 574.1 Historical introduction to multi-way models 574.2 Models for three-way one-block data: three-way component models 594.3 Models for three-way two-block data: three-way regression models 764.4 Summary 83Appendix 4.A: alternative notation for the PARAFAC model 84Appendix 4.B: alternative notations for the Tucker3 model 865 Some properties of three-way component models 895.1 Relationships between three-way component models 895.2 Rotational freedom and uniqueness in three-way component models 985.3 Properties of Tucker3 models 1065.4 Degeneracy problem in PARAFAC models 1075.5 Summary 1096 Algorithms 1116.1 Introduction 1116.2 Optimization techniques 1116.3 PARAFAC algorithms 1136.4 Tucker3 algorithms 1196.5 Tucker2 and Tucker1 algorithms 1236.6 Multi-linear partial least squares regression 1246.7 Multi-way covariates regression models 1286.8 Core rotation in Tucker3 models 1306.9 Handling missing data 1316.10 Imposing non-negativity 1356.11 Summary 136Appendix 6.A: closed-form solution for the PARAFAC model 136Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition 1447 Validation and diagnostics 1457.1 What is validation? 1457.2 Test-set and cross-validation 1477.3 Selecting which model to use 1547.4 Selecting the number of components 1567.5 Residual and influence analysis 1667.6 Summary 1738 Visualization 1758.1 Introduction 1758.2 History of plotting in three-way analysis 1798.3 History of plotting in chemical three-way analysis 1808.4 Scree plots 1808.5 Line plots 1848.6 Scatter plots 1908.7 Problems with scatter plots 1928.8 Image analysis 2018.9 Dendrograms 2028.10 Visualizing the Tucker core array 2048.11 Joint plots 2058.12 Residual plots 2168.13 Leverage plots 2168.14 Visualization of large data sets 2168.15 Summary 2199 Preprocessing 2219.1 Background 2219.2 Two-way centering 2289.3 Two-way scaling 2329.4 Simultaneous two-way centering and scaling 2389.5 Three-way preprocessing 2399.6 Summary 244Appendix 9.A: other types of preprocessing 245Appendix 9.B: geometric view of centering 247Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data 249Appendix 9.D: rank reduction and centering 250Appendix 9.E: centering data with missing values 251Appendix 9.F: incorrect centering introduces artificial variation 251Appendix 9.G: alternatives to centering 25410 Applications 25710.1 Introduction 25710.2 Curve resolution of fluorescence data 25910.3 Second-order calibration 27610.4 Multi-way regression 28510.5 Process chemometrics 28810.6 Exploratory analysis in chromatography 30210.7 Exploratory analysis in environmental sciences 31210.8 Exploratory analysis of designed data 32310.9 Analysis of variance of data with complex interactions 340Appendix 10.A: an illustration of the generalized rank annihilation method 346Appendix 10.B: other types of second-order calibration problems 347Appendix 10.C: the multiple standards calibration model of the second-order calibration example 349References 351Index 371

"...covers some very important topics in modern chemical data analysis...this book is a reasonable place to start. I know of no others." (Technometrics, November 2005) "It will be a valuable addition to the library collections at universities and companies." (E-STREAMS, September 2005)"...extremely useful..." (Journal of Chemometrics, Vol 19 (2), February 2005)

Age K. Smilde received his MSc in Econometrics at the University of Groningen in 1986. He moved to the Department of Pharmacy in the same city where he did his PhD in Analytical Chemistry. His PhD was on "Multivariate Calibration of Reversed Phase Chromatographic Systems", and he received his degree in 1990. In the year 1992, he visited the Center for Process Analytical Chemistry (Seattle, USA), where he worked together with Prof. Bruce Kowalski. He is the Eastern Analytical Symposium 2006 Award Recipient for Achievements in Chemo metrics. In 1996 he was chairman of the Gordon Research Conference on Statistics in Chemistry and Chemical Engineering in Oxford (UK). Together with Rasmus Bro and Paul Geladi he wrote the book Multiway Analysis: Applications in the Chemical Sciences. Rasmus Bro (born 1965) studied mathematics and analytical chemistry at the Technical University of Denmark and received his M.Sc. in 1994. In 1998 he obtained his Ph.D. (Cum Laude) in multiway analysis from the University of Amsterdam, The Netherlands. In 2000 he received the third Elsevier Chemo metrics Award for noteworthy accomplishments in the field of chemo metrics by younger scientists, and in 2004 he received the Eastern Analytical Symposium Award for Achievements in Chemo metrics. He has authored more than 100 peer-reviewed scientific papers, 2 books on chemo metrics, and more than 20 proceedings, book contributions, reviews, and patents.Paul Geladi currently works at the Department of Food Science, Stellenbosch University. Paul does research in Analytical Chemistry, Chemo-informatics and Electrochemistry. Their most recent publication is 'Covalently electro grafted carb ox phenyl layers onto gold surface serving as a platform for the construction of an immune sensor for detection of methotrexate.

• Foreword ixPreface xiNomenclature and Conventions xiii1 Introduction 11.1 What is multi-way analysis? 11.2 Conceptual aspects of multi-way data analysis 11.3 Hierarchy of multivariate data structures in chemistry 51.4 Principal component analysis and PARAFAC 111.5 Summary 122 Array definitions and properties 132.1 Introduction 132.2 Rows, columns and tubes; frontal, lateral and horizontal slices 132.3 Elementary operations 152.4 Linearity concepts 212.5 Rank of two-way arrays 222.6 Rank of three-way arrays 282.7 Algebra of multi-way analysis 322.8 Summary 34Appendix 2.A 343 Two-way component and regression models 353.1 Models for two-way one-block data analysis: component models 353.2 Models for two-way two-block data analysis: regression models 463.3 Summary 53Appendix 3.A: some PCA results 54Appendix 3.B: PLS algorithms 554 Three-way component and regression models 574.1 Historical introduction to multi-way models 574.2 Models for three-way one-block data: three-way component models 594.3 Models for three-way two-block data: three-way regression models 764.4 Summary 83Appendix 4.A: alternative notation for the PARAFAC model 84Appendix 4.B: alternative notations for the Tucker3 model 865 Some properties of three-way component models 895.1 Relationships between three-way component models 895.2 Rotational freedom and uniqueness in three-way component models 985.3 Properties of Tucker3 models 1065.4 Degeneracy problem in PARAFAC models 1075.5 Summary 1096 Algorithms 1116.1 Introduction 1116.2 Optimization techniques 1116.3 PARAFAC algorithms 1136.4 Tucker3 algorithms 1196.5 Tucker2 and Tucker1 algorithms 1236.6 Multi-linear partial least squares regression 1246.7 Multi-way covariates regression models 1286.8 Core rotation in Tucker3 models 1306.9 Handling missing data 1316.10 Imposing non-negativity 1356.11 Summary 136Appendix 6.A: closed-form solution for the PARAFAC model 136Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition 1447 Validation and diagnostics 1457.1 What is validation? 1457.2 Test-set and cross-validation 1477.3 Selecting which model to use 1547.4 Selecting the number of components 1567.5 Residual and influence analysis 1667.6 Summary 1738 Visualization 1758.1 Introduction 1758.2 History of plotting in three-way analysis 1798.3 History of plotting in chemical three-way analysis 1808.4 Scree plots 1808.5 Line plots 1848.6 Scatter plots 1908.7 Problems with scatter plots 1928.8 Image analysis 2018.9 Dendrograms 2028.10 Visualizing the Tucker core array 2048.11 Joint plots 2058.12 Residual plots 2168.13 Leverage plots 2168.14 Visualization of large data sets 2168.15 Summary 2199 Preprocessing 2219.1 Background 2219.2 Two-way centering 2289.3 Two-way scaling 2329.4 Simultaneous two-way centering and scaling 2389.5 Three-way preprocessing 2399.6 Summary 244Appendix 9.A: other types of preprocessing 245Appendix 9.B: geometric view of centering 247Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data 249Appendix 9.D: rank reduction and centering 250Appendix 9.E: centering data with missing values 251Appendix 9.F: incorrect centering introduces artificial variation 251Appendix 9.G: alternatives to centering 25410 Applications 25710.1 Introduction 25710.2 Curve resolution of fluorescence data 25910.3 Second-order calibration 27610.4 Multi-way regression 28510.5 Process chemometrics 28810.6 Exploratory analysis in chromatography 30210.7 Exploratory analysis in environmental sciences 31210.8 Exploratory analysis of designed data 32310.9 Analysis of variance of data with complex interactions 340Appendix 10.A: an illustration of the generalized rank annihilation method 346Appendix 10.B: other types of second-order calibration problems 347Appendix 10.C: the multiple standards calibration model of the second-order calibration example 349References 351Index 371

"...covers some very important topics in modern chemical data analysis...this book is a reasonable place to start. I know of no others." (Technometrics, November 2005) "It will be a valuable addition to the library collections at universities and companies." (E-STREAMS, September 2005)"...extremely useful..." (Journal of Chemometrics, Vol 19 (2), February 2005)