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3. The Tucker3 model

3. The Tucker3 model. Quimiometria Teórica e Aplicada Instituto de Química - UNICAMP. PARAFAC & Tucker3. The PARAFAC model has a strict trilinear structure: x ijk =  a ir b jr c kr + e ijk

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3. The Tucker3 model

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  1. 3. The Tucker3 model Quimiometria Teórica e Aplicada Instituto de Química - UNICAMP

  2. 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. =

  3. C (K 4) B (J 2) core array G (3  2 4) A (I3) The Tucker3 model e.g. Mode I has chemical rank 3 Mode J has chemical rank 2 Mode K has chemical rank 4 CT + = G BT E X K I A J X (IJK) E (IJK)

  4. 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)

  5. g111=97, this triad is important 27 41 3 6 97 2.1 0 26 R3 R1 R2 g211=0, this interaction does not exist 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

  6. First mode Second mode Third mode XIJK XJKI XKIJ Eigenvalue vs. PC Number 4.5 4 3.5 3 2.5 Eigenvalue 2 1.5 1 ‘Knee’ here - select 4 PC’s for first mode 0.5 0 1 2 3 4 5 6 7 8 9 PC Number Eigenvalue vs. PC Number 4.5 4 Select 3 PC’s for third mode 3.5 3 2.5 Eigenvalue 2 1.5 1 0.5 0 1 2 3 4 5 6 7 8 9 PC Number Select 2 PC’s for second mode How many components to use in each mode?Unfold along each mode and look at the eigenvalues Try Tucker3(4,2,3)

  7. Try Tucker3 (4,2,3) 35 batches 5 samples 24 hours 61 excitation ’s 201 emission ’s 12 sensors Try 3-component PARAFAC or Tucker3 (3,3,3) When to use PARAFAC or Tucker3? Fluorescence data: X rank = 3 rank = 3 X rank = 3 rank = 4 rank = 3 rank = 2 Process data:

  8. C (KR) B (JR) core array I (RRR) A (IR) PARAFAC as a restricted Tucker model • PARAFAC is a type of Tucker model for which the core array is a superidentity, I. CT + = I BT E X K I A J X (IJK) E (IJK) 0 0 1 0 0 1 0 0 R3 R1 R2

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

  10. 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...).

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