Two-way centering

In order to understand when and how centering works, it is important first to consider the goals of centering and to realize how these goals are achieved in practice. These concepts are described in this section. 

As stated by Harshman and Lundy [1984], quite subjective and qualitative reasons are often given for performing centering. Sometimes, centering is used to estimate offsets, but this aspect of centering is not considered here. Rather, the focus is on removing offsets. 

It is possible to formulate rational reasons for centering on scientific grounds. Basically, centering should be performed only if there are common offsets in the data or if removing such offsets provides an approximately reasonable model. Centering should make a difference and this difference can manifest itself as follows  

If a component model of the raw data requires, say, R + 1 components to describe the data well, whereas a model of the centered data requires only R components, then centering is sensible because the model of the centered data only has R(I + J) + J parameters. The J parameters pertain to the calculated averages, assuming that centering is performed across the first mode. The alternative of fitting the (R + l)-component model to the raw data would lead to a model with (R + 1)(7 + J) parameters and thus would violate the parsimony principle [Judge et al. 1985, Seasholtz Kowalski 1993, Weinberg 1964], 

If the vector m holds in its jth element the average of the y th column of X, then m can be expressed as 

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