Different kinds of PLS weights, loadings, and what to look at?
Sep 12, 2009
The age of Partial Least Squares (PLS) regression (as opposed to PLS path modeling) began with the SIAM publication of Svante Wold et. al. in 1984 . Many of us learned PLS regression from the rather more accessible paper by Geladi and Kowalski from 1986  which described the Nonlinear Iterative PArtial Least Squares (NIPALS) algorithm in detail.
For univariate y, the NIPALS algorithm is really more sequential than iterative. There is a specific sequence of fixed steps that are followed to find the weight vector w (generally normalized to unit length) for each PLS factor or Latent Variable (LV). Only in the case of multivariate Y is the algorithm really iterative, a sequence of steps is repeated until the solution for w for each LV converges.
Whether for univariate y or multivariate Y the calculations for each LV end with a deflation step. The X data is projected onto the weight vector w to get a score vector, t (t = Xw). X is then projected onto the score t to get a loading, p, (p = X’t/t’t). Finally, X is deflated by tp’ to form a new X, Xnew with which to start the procedure again, Xnew = X – tp’. A new weight vector w is calculated from the deflated Xnew and the calculations continue.
So the somewhat odd thing about the weight vectors w derived from NIPALS is that each one applies to a different X, i.e. tn+1 = Xnwn+1. This is in contrast to Sijmen de Jong’s SIMPLS algorithm introduced in 1993 . In SIMPLS a set of weights, sometimes referred to as R, is calculated, which operate on the original X data to calculate the scores. Thus, all the scores T can be calculated directly from X without deflation, T = XR. de Jong showed that it is easy to calculate the SIMPLS R from the NIPALS W and P, R = W(P’W)-1. (Unfortunately, I have, as yet, been unable to come up with a simple expression for calculating the NIPALS W from the SIMPLS model parameters.)
So the question here is, “If you want to look at weights, which weights should you look at, W or R?” I’d argue that R is somewhat more intuitive as it applies to the original X data. Beyond that, if you are trying to standardize outputs of different software routines (which is actually how I got started on all this), it is a simple matter to always provide R. Fortunately, R and W are typically not that different, and in fact, they start out the same, w1 = r1, and they span the same subspace. Our decision here, based on relevance and standardization, is to present R weights as the default in future versions of PLS_Toolbox and Solo, regardless of which PLS algorithm is selected (NIPALS, SIMPLS or the new Direct Scores PLS).
A better question might be, “When investigating a PLS model, should I look at weights, R or W, or loadings P?” If your perspective is that the scores T are measures of some underlying “latent” phenomena, then you would choose to look at P, the degree to which these latent variables contribute to X. The weights W or R are merely regression coefficients that you use to estimate the scores T. From this viewpoint the real model is X = TP’ + E and y = Tb+ f.
If, on the other hand, you see PLS as simply a method for identifying a subspace within which to restrict, and therefore stabilize, the regression vector, then you would choose to look at the weights W or R. From this viewpoint the real model is Y = Xb + e, with b = W(P’W)-1(T’T)-1T’y = R(T’T)-1T’y via the NIPALS and SIMPLS formulations respectively. The regression vector is a linear combination of the weights W or R and from this perspective the loadings P are really a red herring. In the NIPALS formulation they are a patch left over from the way the weights were derived from deflated X. And the loadings P aren’t even in the SIMPLS formulation.
So the bottom line here is that you’d look at loadings P if you are a fan of the latent variable perspective. If you’re a fan of the regression subspace perspective, then you’d look at weights, W or preferably R. I’m in the former camp, (for more reasons than just philosophical agreement with the LV model), as evidenced by my participation in S. Wold et. al., “The PLS model space revisited,” . Your choice of perspective also impacts what residuals to monitor, etc., but I’ll save that for a later time.
 S. Wold, A. Ruhe, H. Wold, and W.J. Dunn III, “The Collinearity Problem in Linear
Regression. The Partial Least Square Approach to Generalized Inverses”, SIAM J. Sci.
Stat. Comput., Vol. 5, 735-743, 1984.
 P. Geladi and B.R. Kowalski, “PLS Tutorial,” Anal. Chim. Acta., 185(1), 1986.
 S. de Jong, “SIMPLS: an alternative approach to partial least squares regression,” Chemo. and Intell. Lab. Sys., Vol. 18, 251-263, 1993.
 S. Wold, M. Høy, H. Martens, J. Trygg, F. Westad, J. MacGregor and B.M. Wise, “The PLS model space revisited,” J. Chemometrics, pps 67-68, Vol. 23, No. 2, 2009.