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