Reasoning behind variable importance in projection (VIP score)? I have been using partial least squares regression to do latent variable modeling of a few data sets. Now I am wanting to look at variable importance but am getting stumped on the reasoning behind the construction of the VIP score. I have tried to find some sources on the on this, but every source I have found simply states what it is, defines the different variables, and moves on. Also, unless I am missing something, the definitions seem to be slightly different in different sources (see [1] eq. 1 and [2] eq. 12.21). Why is the VIP score constructed the way it is, and why does that give insight into variable importance?
 A: Given a set of predictors $X\in\mathbb{R}^{N\times K}$ and a set of depenent variables $Y\in\mathbb{R}^{N\times M}$, the PLSR model tries to find a solution in the form $Y=X\tilde{B}+\tilde{B}_0$.
The VIP was first defined by Wold, Johansson and Cocchi in this book (p.541) as

A measure of the relative importance of the $X$-variables for all $Y$s
and all model dimensions [...] VIP is derived from the PLS weights $w_a,\quad a=1,2,...,A$ weighted by how much of $Y$ is explained in each model dimension. VIP is normalized so values larger than 1.0 indicate important variables. The value of 0.8 is often used as a limit below which the variables are considered unimportant.

It is formulated as
$$VIP_k=\sqrt{\frac{K\cdot\sum_{a=1}^{A}{w^2_{ak}SSY_a}}{A\cdot SSY_{total}}}$$
$K$ is the number of covariates in $X$; $A$ is the number of PLS components; $w_{ak}$ indicates the weight of $X_k$ in the $a^{th}$ component; component and $SSY_a$ is the sum of squares of explained variance for the $a^{th}$ component; $SSY_{total}$ is the total sum of squares explained of $Y$ in all components.
The rationale, as far as I can get it, is this: with $w_{ak},SSY_a$ and $SSY_{total}$ defined as above, the term $\frac{SSY_a}{SSY_{total}}$ is the relative explained variance, and $\sum_{a=1}^{A}{w^2_{ak}\frac{SSY_a}{SSY_{total}}}$ is a weighted sum of relative explained variances. Finally, multiplying by $\frac{K}{A}$ accounts for normalization.
