Two-way Analysis of Three-way Data

1. Two-way Analysis of Three-way Data

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

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

4. Three-way Data D3 D D2 D1 21

5. Structure of three way data Y1 z1 D1 X1 = D1 = X1 Q1 Y1 Y2 z2 D2 X2 = D2 = X2 Q2 Y2 Y3 z3 D2 X3 = D3 = X3 Q3 Y3 20

6. Trilinearity D2 D1 D3 Rank = r1 D1 D2 Rank = r2 D3 19

7. Similar X matrices Y1 z1 D1 X1 = D1 = X1 Q1 Y1 Y2 z2 D2 X2 = D2 = X2 Q2 Y2 Y3 z3 D2 X3 = D3 = X3 Q3 Y3 18

8. Row wise augmented MCR D2 D1 D3 = Y2 Y1 z2 z1 X1 Y3 z3 Q1Y1 Q2Y2 Q3Y3 X1 = 17

9. Similar Y matrices z1 Y1 X1 D1 = D1 = X1 Q1 Y1 z2 Y2 X2 D2 = D2 = X2 Q2 Y2 z3 Y3 X3 D2 = D3 = X3 Q3 Y3 16

10. Column wise augmented MCR Y1 z1 D1 X1 z2 D2 = X2 z3 D3 Y1 X3 Q1X1 = Q2X2 Q3X3 15

11. Similar X and Y matrices z1 Y1 X1 D1 = D1 = X1 Q1 Y1 z2 Y2 X2 D2 = D2 = X2 Q2 Y2 z3 Y3 X3 D2 = D3 = X3 Q3 Y3 14

12. Row wise or column wise augmented MCR D2 D1 D3 = Q1Y1 Q2Y2 Q3Y3 X1 Y1 D1 Q1X1 D2 = Q2X2 D3 Q3X3 13

13. Trilinearity constraint Y1 D1 Q1X1 D2 = Q2X2 D3 Q3X3 PCA 12

14. Different X and Y matrices Y1 z1 D1 X1 = D1 = X1 Q1 Y1 Y2 z2 D2 X2 = D2 = X2 Q2 Y2 Y3 z3 D2 X3 = D3 = X3 Q3 Y3 11

15. Column wise augmented MCR Y1 D1 Q1X1 Y2 D2 = Q2X2 Y3 D3 Q3X3 10

16. Row wise augmented MCR D2 D1 D3 = Q1X1 Q3X3 Q2X2 Y1 Y2 Y3 9

17. PARAFAC model Z Y X D = = D2 D1 D3 M1 M2 M3 X 8

18. PARAFAC model Z Y X D = Y D1 N1 N2 D2 = D3 N3 7

19. PARAFAC model Z Y X D = D1 D2 D3 D4 = Z P1 P2 P3 P4 6

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

21. Reconstruction of kth slice of a three-way array p n D m PARAFAC2 n c c n Y zk Dk Xk = c c m m X1X1T = X2X2T = … = XkXkT 4

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

23. Conclusions: 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. 2

24. Further studies: 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 ibCatolonia 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. 1