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

The PARAFAC model has a strict trilinear structure: 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.

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

X = AG (CB)T + E The Tucker3 formula Loadings Core array 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)

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

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

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

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

Can have different number of components in each mode. Core array, G. 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.

The Tucker3 model is good for 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...).