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THREE-WAY COMPONENT MODELS

THREE-WAY COMPONENT MODELS. 880305- pages 66-76 By: Maryam Khoshkam. 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]. TUCKER3 MODELS.

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THREE-WAY COMPONENT MODELS

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  1. THREE-WAY COMPONENT MODELS 880305- pages 66-76 By: MaryamKhoshkam

  2. 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]

  3. TUCKER3 MODELS :nonzero off-diagonalelements in its core.

  4. In Kronecker product notation the Tucker3 model

  5. 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

  6. 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.

  7. 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 core-array and of the loadings by the loading plots

  8. 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.

  9. The rotational freedom of Tucker3 models can also be used to rotate the core-array to a simple structureas is also common in two-way analysis (will be explained).

  10. 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].

  11. Unique Tucker Simulated data: Two components, PARAFAC model

  12. Unique Tucker3 component model P=Q=R=3 Only two significant elements in core

  13. 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 :

  14. 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 • without loss of fit. (using othog U and V)

  15. Tucker1 models : reduce only one of the modes. + X (and accordingly G) are matricized :

  16. 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) : matricizedX; 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 => PARAFAC model is a special case of Tucker3 model.

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