Three-way preprocessing

Multi-way Analysis With Applications in the Chemical Sciences 

Regardless of the structure of the offsets, the basic principle of centering is that the data must be preprocessed, so that they are projected onto the nullspace of vectors of ones in a 

All elements with same j have the same offset 

As explained for the two-way case, scaling is a transformation of a particular variable (or object) space. Instead of fitting the model to the original data, the model is fitted to the data transformed by a (usually) diagonal scaling matrix in the mode whose variables are to be scaled. This means that whole matrices instead of columns have to be scaled by the same value in three-way analysis. For a four-way array, three-way slabs would have to be scaled by the same scalar. Mathematically, scaling within the first mode can be described as 

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