3. The Tucker3 model

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3. The Tucker3 model

• Quimiometria Teórica e Aplicada
• Instituto de Química - UNICAMP

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PARAFAC Tucker3

• The PARAFAC model has a strict trilinear
  structure
• xijk ?airbjrckr eijk
• Another generalization of PCA for multiway data
  is able to use a different number of components
  for each mode the Tucker3 model.

etc.

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The Tucker3 model
CT


G
BT
E
X
K
I
A
J
X (I ? J ? K)
E (I ? J ? K)

• e.g. Mode I has chemical rank 3
• Mode J has chemical rank 2
• Mode K has chemical rank 4

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The Tucker3 formula
X AG (C?B)T E

• Loadings
• A (I ? R1) describes variation in the first mode
• B (J ? R2) describes variation in the second mode
• C (K ? R3) describes variation in the third mode

• Core array
• G (R1 ? R2 ? R3) is matricized into GR1?R2R3 (R1
  ? R2R3)

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What does the core array mean?

• The core array describes the significance of the
  interactions between the different loadings, e.g.
  the Tucker3 (2,2,2) model can be written as
• X g111a1(c1?b1)T
• g112a1(c2?b1)T
• g121a1(c1?b2)T
• g122a1(c2?b2)T
• g211a2(c1?b1)T
• g212a2(c2?b1)T
• g221a2(c1?b2)T
• g222a2(c2?b2)T
• E

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How many components to use in each mode?Unfold
along each mode and look at the eigenvalues
Try Tucker3(4,2,3)
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When to use PARAFAC or Tucker3?
Fluorescence data
X
rank 3
rank 3
X
rank 3
rank 4
rank 3
rank 2
Process data
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PARAFAC as a restricted Tucker model
CT


I
BT
E
X
K
I
A
J
X (I ? J ? K)
E (I ? J ? K)
0
0

• PARAFAC is a type of Tucker model for which the
  core array is a superidentity, I.

1
0
0
1
0
0
R3
R1
R2
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PARAFAC vs Tucker3
PARAFAC
Tucker3
Can have different number of components in each
mode. Core array, G.
Same number of components in each mode.
Core array is superidentity, I.
Rotational freedom. More difficult to
interpret.
Solution is unique. Easy to interpret.
Multiway subspace model. Good for
exploratory analysis.
Strict, trilinear model. Good for some
types of data.
Algorithm sometimes slow and problematic.
Algorithm fast and robust.
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Conclusions

• The Tucker3 model is good for
• general exploratory analysis
• multiway data which have modes of different rank
• Like PARAFAC, the Tucker3 model is estimated
  using ALS, with an extra step for the estimation
  of G.
• Like PCA, Tucker loadings have rotational
  freedom, making model interpretation more
  difficult than for PARAFAC. The use of
  constraints can help.
• Restricted Tucker3 models have been used for
  chemical calibration (more about this later...).