Tucker models core array

Three-way PCA allows a much easier interpretation of the information contained in the data set, since it directly takes into account its three-way structure. If the TuckerS model is applied, the hnal result is given by three sets of loadings together with a core array describing the relationship among them. If the number of components is the same for each way, the core array is a cube. Each of the three sets of loadings can be displayed and interpreted in the same way as a score plot of standard PCA. 

Tucker Models In Af-way PC A or Tucker models, the number of components for the different modes can vary, in contrast to PARAFAC models. Tucker models can be derived for a different number of modes and are called Tuckerl, Tucker2, and so on. Consider a three-way array X. Then the corresponding TuckerS model is given for an element, x f, of this matrix by the loadings a, b, and c and the element g of a core matrix by... 

Id,Ie and Ip being the identity matrices of dimensions D, E and F, respectively, as this significantly improves interpretability both of the loadings and of the core array [50,54]. Under these assumptions, it is possible to demrai-strate that the sum of the squared elements of the core array corresponds to the amount of variation explained by the TuckerS model. Indeed, if the TuckerS approximation of the original data array is defined as X, then it can be shown that Equation (14) leads to... 

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