In principle components regression, we mean center(or standardize) X and using this X we decompose it so that:

X = T.P' + E

And regression to Y becomes(with using a certain portion of T):

Y = T.b where b is a vector without intercept term

Similarly in PLS, after mean centering both X and Y the equations(shortly) are X = T.P' + E and Y = U.Q' while the inner relation to be solved is U = T * b where b is, again, a vector having linear regression coefficents.

Does it make any sense to add a intercept term to the b vectors I have mentioned above? In other words, do PLS and/or PCA themselves make sure the scores(T and/or U) are bias free mathematically or due to initial mean centering of the data?

  • As you say, after mean-centering no further intercept term is needed, as the data is guaranteed to go through the origin. In that sense the center is the intercept term of the model and the standard PLS models do have $x$ and $y$ intercept terms.

  • That being said, there are reasons for centering the data on some point that is not the mean of $\mathbf X$ (and $\mathbf Y$, respectively).
    E.g. I often center on the weighted mean (mean of group means) of the data when using PLS-LDA so that both PLS and the subsequent LDA use the same origin. Sometimes it is also sensible to put the origin say at the mean of the control group or the mean of all blank samples.
    Still, all "sensible" choices for origin I've encountered mean that no further intercept term is needed. But keep in mind that you always need very good reasons to force your model through a specific point.

  • (Remember that you can always model an intercept in a bilinear model using a constant non-zero variate. I.e. using $\mathbf{X'} = [ \mathbf X ~ \mathbf 1 ]$.)


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