Two-way Analysis of Three-way Data

Two-way Analysis of Two-way Data

= XDY

D = X Y

23

Two-way Analysis of Two-way Data

=DY

D = X Q Y

zX

22

Three-way Data

D3

D1

D2

D

21

= X2D2Y2

D2 = X2 Q2 Y2

z2

= X1D1Y1

D1 = X1 Q1 Y1

z1

= X3D2Y3

D3 = X3 Q3 Y3

z3

Structure of three way data

20

Trilinearity

D3D1 D2

D3

D1

D2

Rank = r1

Rank = r2

19

= X2D2Y2

D2 = X2 Q2 Y2

z2

= X1D1Y1

D1 = X1 Q1 Y1

z1

= X3D2Y3

D3 = X3 Q3 Y3

z3

Similar X matrices

18

D3D1 D2 =

X1Y1z1

Y2z2

Y3z3

= X1Q1Y1 Q2Y2 Q3Y3

Row wise augmented MCR

17

Similar Y matrices

= X2D2Y2

D2 = X2 Q2 Y2

z2

= X1D1Y1

D1 = X1 Q1 Y1

z1

= X3D2Y3

D3 = X3 Q3 Y3

z3

16

Column wise augmented MCR

D3

D1

D2 = X2z2

X1z1

X3z3

Y1

= Q2X2

Q1X1

Q3X3

Y1

15

Similar X and Y matrices

= X2D2Y2

D2 = X2 Q2 Y2

z2

= X1D1Y1

D1 = X1 Q1 Y1

z1

= X3D2Y3

D3 = X3 Q3 Y3

z3

14

D3D1 D2 =

X1Q1Y1 Q2Y2 Q3Y3

Row wise or column wise augmented MCR

D3

D1

D2 = Q2X2

Q1X1

Q3X3

Y1

13

Trilinearity constraint

D3

D1

D2 = Q2X2

Q1X1

Q3X3

Y1

PCA

12

= X2D2Y2

D2 = X2 Q2 Y2

z2

= X1D1Y1

D1 = X1 Q1 Y1

z1

= X3D2Y3

D3 = X3 Q3 Y3

z3

Different X and Y matrices

11

D3

D1

D2

Y1

=Y2

Y3Q2X2

Q1X1

Q3X3

Column wise augmented MCR

10

D3D1 D2 =

Y1

Y2

Y3

Row wise augmented MCR

Q2X2Q1X1 Q3X3

9

PARAFAC model

=D X Y

Z

D3D1 D2 =

X M1 M2 M3

8

PARAFAC model

=D X Y

Z

D3

D1

D2 =

N1Y

N2

N3

7

PARAFAC model

=D X Y

Z

D3D1 D2

=

D4

P1 P2 P3 P4

Z

6

D

m

np

Reconstruction of kth slice of a three-way array

= XDkYzk

m

n

m

c

c

c

c

n

PARAFAC

5

D

m

np

Reconstruction of kth slice of a three-way array

= XkDkYzk

m

n

m

c

c

c

c

n

PARAFAC2

X1X1T = X2X2

T = … = XkXkT

4

D

m

np

Reconstruction of kth slice of a three-way array

= XDkY

m

n

m

r

r

c

c

n

TUCKER3

Mk

3

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

Comparison of three-way resolution methods for non-trilinear chemical data setsAnna de Juan, Roma TaulerJ. Chemometrics, 2001, 15, 749-772.

Comparison of different multiway methods for the analysis of geographical metal distributions in fish, sediments and river water ib CatoloniaEmma Pere-Trepat, Antonio Ginebreda, Roma Tauler.Chemom. Intel. Lab. Syst., 2007, 88, 69-83.

On rotational ambiguity in parallel factor analysisH. Abdollahi, S.M. SajjadiChemom. Intel. Lab. Syst., 2010, 103, 144-151.

Further studies:

1