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According to this question and answer, the sum of variances of all partial least squares (PLS) components is normally less than 100%: Why do all the PLS components together explain only a part of the variance of the original data?

Can somebody provide (further) evidence for this? When would this occur?

When running PLS in SAS using the SIMPLS method, I find that 100% of the variance is explained when I use all components.

I have confirmed that the weight vectors produced by PLS in my case are not orthogonal. However, according to The Elements of Statistical Learning the outputed components themselves are orthogonal:

partial least squares produces a sequence of derived, orthogonal inputs or directions $z_1, z_2, \ldots, z_M$.

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    $\begingroup$ Components are orthogonal (i.e. have correlation zero), that's right. This does not contradict my answer in the linked thread. What exactly do you mean, when you say that 100% variance is explained when you use all components? Are you sure you mean the sum of the variances of the projections on the unit weight vectors? $\endgroup$ – amoeba Jul 30 '15 at 20:34
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    $\begingroup$ Please give more detail of your circumstances where you get 100%. How many observations? How many components? etc. Do you have a small example (small enough to give the data) where this happens? $\endgroup$ – Glen_b Jul 30 '15 at 22:09
  • $\begingroup$ I have approximately 100 response and 100 predictor variables and approximately 2,000 observations. I am fairly confident that I mean what I say regarding the variances. Is it possible that the variance explained converges to 100% as the number of observations increases? I did some quick testing in R with the plsr function from the pls package and this seems to occur. $\endgroup$ – Iggy25 Jul 31 '15 at 23:35
  • $\begingroup$ Did you check whether this 100% corresponds to the matrix $X$ or $Y$? Because the explained variance for matrix $X$ has to be 100% but for matrix $Y$ can be less since you try to approximate $Y$ by a linear model. $\endgroup$ – Kirill Aug 3 '15 at 8:43
  • $\begingroup$ We are discussing the X Matrix. $\endgroup$ – Iggy25 Aug 3 '15 at 14:14

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