# When do all PLS components together explain only part of the variance of the original data?

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 $z_1, z_2, \ldots, z_M$.

• 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? – amoeba Jul 30 '15 at 20:34
• 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? – Glen_b Jul 30 '15 at 22:09
• 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. – Iggy25 Jul 31 '15 at 23:35
• 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. – Kirill Aug 3 '15 at 8:43
• We are discussing the X Matrix. – Iggy25 Aug 3 '15 at 14:14