1. Two-way Analysis of Three-way Data

2. Two-way Analysis of Two-way Data = X D Y D = X Y 23

3. Two-way Analysis of Two-way Data = D Y D = X Q Y z X 22

4. Three-way Data D3D3 D1D1 D2D2 D 21

5. = X2X2 D2D2 Y2Y2 D 2 = X 2 Q 2 Y 2 z2z2 = X1X1 D1D1 Y1Y1 D 1 = X 1 Q 1 Y 1 z1z1 = X3X3 D2D2 Y3Y3 D 3 = X 3 Q 3 Y 3 z3z3 Structure of three way data 20

6. Trilinearity D3D3 D1D1 D2D2 D3D3 D1D1 D2D2 Rank = r 1 Rank = r 2 19

7. = X2X2 D2D2 Y2Y2 D 2 = X 2 Q 2 Y 2 z2z2 = X1X1 D1D1 Y1Y1 D 1 = X 1 Q 1 Y 1 z1z1 = X3X3 D2D2 Y3Y3 D 3 = X 3 Q 3 Y 3 z3z3 Similar X matrices 18

8. D3D3 D1D1 D2D2 = X1X1 Y1Y1 z1z1 Y2Y2 z2z2 Y3Y3 z3z3 = X1X1 Q1Y1Q1Y1 Q2Y2Q2Y2 Q3Y3Q3Y3 Row wise augmented MCR 17

9. Similar Y matrices = X2X2 D2D2 Y2Y2 D 2 = X 2 Q 2 Y 2 z2z2 = X1X1 D1D1 Y1Y1 D 1 = X 1 Q 1 Y 1 z1z1 = X3X3 D2D2 Y3Y3 D 3 = X 3 Q 3 Y 3 z3z3 16

10. Column wise augmented MCR D3D3 D1D1 D2D2 = X2X2 z2z2 X1X1 z1z1 X3X3 z3z3 Y1Y1 = Q2X2Q2X2 Q1X1Q1X1 Q3X3Q3X3 Y1Y1 15

11. Similar X and Y matrices = X2X2 D2D2 Y2Y2 D 2 = X 2 Q 2 Y 2 z2z2 = X1X1 D1D1 Y1Y1 D 1 = X 1 Q 1 Y 1 z1z1 = X3X3 D2D2 Y3Y3 D 3 = X 3 Q 3 Y 3 z3z3 14

12. D3D3 D1D1 D2D2 = X1X1 Q1Y1Q1Y1 Q2Y2Q2Y2 Q3Y3Q3Y3 Row wise or column wise augmented MCR D3D3 D1D1 D2D2 = Q2X2Q2X2 Q1X1Q1X1 Q3X3Q3X3 Y1Y1 13

13. Trilinearity constraint D3D3 D1D1 D2D2 = Q2X2Q2X2 Q1X1Q1X1 Q3X3Q3X3 Y1Y1 PCA 12

14. = X2X2 D2D2 Y2Y2 D 2 = X 2 Q 2 Y 2 z2z2 = X1X1 D1D1 Y1Y1 D 1 = X 1 Q 1 Y 1 z1z1 = X3X3 D2D2 Y3Y3 D 3 = X 3 Q 3 Y 3 z3z3 Different X and Y matrices 11

15. D3D3 D1D1 D2D2 Y1Y1 = Y2Y2 Y3Y3 Q2X2Q2X2 Q1X1Q1X1 Q3X3Q3X3 Column wise augmented MCR 10

16. D3D3 D1D1 D2D2 = Y1Y1 Y2Y2 Y3Y3 Row wise augmented MCR Q2X2Q2X2 Q1X1Q1X1 Q3X3Q3X3 9

17. PARAFAC model =D X Y Z D3D3 D1D1 D2D2 = X M1M2M3 8

18. PARAFAC model =D X Y Z D3D3 D1D1 D2D2 = N1 Y N2 N3 7

19. PARAFAC model =D X Y Z D3D3 D1D1 D2D2 = D4D4 P1P2P3P4 Z 6

20. D m n p Reconstruction of kth slice of a three-way array = XDkDk Y zkzk m n m c c c c n PARAFAC 5

21. D m n p Reconstruction of kth slice of a three-way array = XkXk DkDk Y zkzk m n m c c c c n PARAFAC2 X 1 X 1 T = X 2 X 2 T = … = X k X k T 4

22. D m n p Reconstruction of kth slice of a three-way array = X DkDk Y m n m r r c c n TUCKER3 MkMk 3

23. MCR-ALS is a quite adaptable method for different kinds of non-trilinear data sets. MCR-ALS with trilinearity constraint is equivalent to PARAFAC. MCR-ALS is conceptually simple, can constrain all modes and works satisfactorily in a large variety of situations. When a data set presents a PARAFAC2 structure, this method can provide unique solutions. MCR-ALS is the preferred option to deal with non-trilinear data sets. Conclusions: 2

24. Comparison of three-way resolution methods for non-trilinear chemical data sets Anna de Juan, Roma Tauler J. Chemometrics, 2001, 15, 749-772. Comparison of different multiway methods for the analysis of geographical metal distributions in fish, sediments and river water ib Catolonia Emma Pere-Trepat, Antonio Ginebreda, Roma Tauler. Chemom. Intel. Lab. Syst., 2007, 88, 69-83. On rotational ambiguity in parallel factor analysis H. Abdollahi, S.M. Sajjadi Chemom. Intel. Lab. Syst., 2010, 103, 144-151. Further studies: 1