There are several methods for decomposing the three-way array that are more general than MPCA . These methods include parallel factor analysis (PARAFAC) [553, 555] and Tucker models [179, 597],
The Tucker Model. The Tucker model decomposes the 3-dimensional data as l m n
¿=1 m= 1 n=l where an is an element of the (I x L) loading matrix of component i, bjm is an element of the (J x M) loading matrix of the second component j, and Ckn is an element of the (K x N) loading matrix of the third component k. zimn denotes an element of the three-way core matrix Z and el3k is an element of the three-way residual matrix E. The core matrix Z represents the magnitude and interactions between the variables , A,B,C are column-wise orthonormal, and A,B,C and Z are chosen to minimize the sum of the squared residuals. In the Tucker model, each mode (/, J, K) may have a different number of PCs, which is defined -not chosen- by the least squares algorithm.
The PARAFAC Model. The PARAFAC model yields a trilinear model of X
x^k and eijk are the same as in Eq. 4.157. alg, bjg, and Ckg are the elements of the loading matrices A (I x G), B (J x G), and C (K x G). A, B, and C are chosen to minimize the sum of squared residuals. Under certain conditions, the solution does not converge due to the degeneracy of the problem [552, 555]. Degeneracy refers to the fact that the loadings A, B, and C are rotation-dependent i.e., there can be no multiple solutions for the calculated set of loadings. In terms of data manipulations, the PARAFAC model is a simplification of the Tucker model in two ways :
1. The number of components in all three modes (I, J, K) are equal, and
2. There is no interaction between latent variables of different modes.
The use of MPCA and three-way techniques have been reported for SPM in [641, 646].
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