THREE-WAY COMPONENT MODELS pages 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 ] 2

3 TUCKER3 MODELS : nonzero off-diagonal elements in its core.

In Kronecker product notation the Tucker3 model 4

PROPERTIES OF THE TUCKER3 MODEL Tucker3 model has rotational freedom. T A : arbitrary nonsingular matrix Such a transformation of the loading matrix A can be defined similarly for B and C, using T B and T C, 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.! 6 The Tucker3 model : not give unique component matrices  it has rotational freedom.

rotational freedom Orthogonal component matrices (at no cost in fit by defining proper matrices T A, T B and T C ) convenient : to make the component matrices orthogonal  easy interpretation of the elements of the core- array and of the loadings by the loading plots 7

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 structure as is also common in two-way analysis (will be explained).

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]. 10

11 Unique Tucker Simulated data: Two components, PARAFAC model

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

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all three modes are reduced In tucker 3 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 without loss of fit. (using othog U and V) 16

Tucker1 models : reduce only one of the modes. + X (and accordingly G) are matricized : 17

18 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 => PARAFAC model is a special case of Tucker3 model.