THREE-WAY
COMPONENT MODELS
880305- pages 66-76
By: Maryam Khoshkam
1
Tucker component models
Ledyard Tucker was one of the pioneers in
multi-way analysis.
He proposed a series of models nowadays called
N-mode PCA or Tucker models
[Tucker 1964- 1966]
2
TUCKER3 MODELS
: nonzero off-diagonal elements in its core.
3
In Kronecker product notation the Tucker3 model
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PROPERTIES OF THE TUCKER3 MODEL
Tucker3 model has rotational freedom.
TA : arbitrary nonsingular matrix
Such a transformation of the loading matrix A
can be defined similarly for B and C, using TB and
TC, respectively
5
Tucker3 model has rotational freedom,
But: it is not possible to rotate Tucker3
core-array to a superdiagonal form (and to
obtain a PARAFAC model.!
The Tucker3 model : not give unique component
matrices
 it has rotational freedom.
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rotational freedom
Orthogonal component matrices
(at no cost in fit by defining proper matrices TA, TB
and TC)
convenient : to make the component
matrices orthogonal
easy interpretation of the elements of the corearray and of the loadings by the loading plots
7
SS of elements of core-array
amount of variation explained by combination of factors in
different modes.
variation in X:
unexplained and
explained by model
Using a proper rotation all the variance of explained part can be
gathered in core.
8
The rotational freedom of Tucker3 models can also be used to
rotate the core-array to a simple structure as is also common
in two-way analysis (will be explained).
9
Imposing the restrictions A’A = B’B = C’C = I
: not sufficient for obtaining a unique solution
To obtain uniqe estimates of parameters,
1. loading matrices should be orthogonal,
2. A should also contain eigenvectors of X(CC’ ⊗ BB’)X’
corresp. to decreasing eigenvalues of that same matrix;
similar restrictions should be put on B and C
[De Lathauwer 1997, Kroonenberg et al. 1989].
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Unique Tucker
Simulated data:
Two components,
PARAFAC model
1
0.5
0
0
5
10
15
20
1
0.5
0
5
10
15
1
0.5
0
2
11
4
6
8
0.5
0.5
0.5
-15
1
Unique
Tucker3
component
model
1
1
1.5
1 1.5
5
0.8
4
0
1.5
-5
2
2
2
3
0.6
2.5
2.5
-10
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P=Q=R=3
x 10
6
3.5
2.5
3
1
2
-153.5
3
2
0.4
0.2
1
2
3
3
0 3.5
1
1
2
3
0
1
0.5
0
-0.5
0
2
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12
14
16
18
20
1
0.5
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-0.5
0
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4
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8
10
12
14
16
1
0
12
-1
1
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7
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Only two
significant
elements in
core
0.5
0.5
1
0.5
1
1.5
1
0
0
1.5
1.5
-0.2
2
-0.4 2
2.5
2.5
-0.62.5
3
-10 3
2
-5
-15
x 10
0
-5
-10
3.5
1
2
-0.8 3
3.5
-15
3
1
2
3
8
10
12
-1 3.5
1
2
3
0.5
0
-0.5
-1
0
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14
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18
20
1
0.5
0
-0.5
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4
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14
16
1
0
13
-1
1
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7
8
-15
0.5
0.5
0.5
0.5
1
1
1
1
1.5
1.5
1.5
1.5
2
2
2
2
2.5
0.5
1.5
2
2.5
2.5
0.5
1
1.5
2
2.5
2.5
0.5
1
0.5
0
0
-1
-0.5
0
5
10
15
20
0.5
0.5
0
0
-0.5
-0.5
0
2
4
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10
12
14
16
1
1
0
0
-1
14
1
-1
1
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8
0
0
1
1
1.5
2
2.5
2
4
6
8
2
4
2
1
10
6
3
2.5
0.5
12
8
4
1.5
14
10
5
2.5
16
12
6
2
18
14
7
20
16
8
In tucker 3
all three modes
are reduced
models where only two of the three modes are reduced,
:Tucker2 models.
a Tucker3 model is made for X (I × J × K)
C is chosen to be the identity matrix I, of size K × K.
no reduction sought in the third mode (basis is not changed.
↘Tucker2 model :
15
Tucker2 has rotational freedom:
G : postmultiplied by U⊗V
(B⊗A) : premultiplied by (U⊗V)−1
=>(B(U’)−1 ⊗A(V’)−1) without changing the fit.
 component matrices A and B can be made orthogonal
16
without loss of fit. (using othog U and V)
Tucker1 models : reduce only one of the modes.
+ X (and accordingly G) are matricized :
17
different models [Kiers 1991, Smilde 1997].
Threeway component models for X (I × J × K),
A : the (I × P) component matrix (of first (reduced) mode,
X(I×JK) : matricized X; A,B,C : component matrices;
G : different matricized core-arrays ;
I :superdiagonal array (ones on superdiagonal.
(compon matrices, core-arrays and residual error arrays :
differ for each model
=>
18 PARAFAC model is a special case of Tucker3 model.