# How to compute the confidence intervals on regression coefficients in PLS?

The underlying model of PLS is that a given $n \times m$ matrix $X$ and $n$ vector $y$ are related by $$X = T P' + E,$$ $$y = T q' + f,$$ where $T$ is a latent $n \times k$ matrix, and $E, f$ are noise terms (sssuming $X, y$ are centered).

PLS produces estimates of $T, P, q$, and a 'shortcut' vector of regression coefficients, $\hat{\beta}$ such that $y \sim X \hat{\beta}$. I would like to find the distribution of $\hat{\beta}$ under some simplifying assumptions, which should probably include the following:

1. The model is correct, i.e. $X = T P' + E,y = T q' + f$ for unknown $T, P, q$;
2. The number of latent factors, $k$, is known, and used in the PLS algorithm;
3. The actual error terms are i.i.d. zero-mean normal with known variances;

This question is somewhat underdefined because there are scores of variants of 'the' PLS algorithm, but I would accept results for any of them. I would also accept guidance on how to estimate the distribution of $\hat{\beta}$ via e.g. a bootstrap, but perhaps that is a separate question.

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Do you know this article: PLS-regression: a basic tool of chemometrics? Deriving SE and CI for the PLS parameters is described in §3.11.

I generally rely on Bootstrap for computing CIs, as suggested in e.g., Abdi, H. Partial least squares regression and projection on latent structure regression (PLS Regression). I seem to remember there are theoretical solutions discussed in Tenenhaus M. (1998) La régression PLS: Théorie et pratique (Technip), but I cannot check for now as I don't have the book. For now, there are some useful R packages, like plsRglm.

P.S. I just discovered Nicole Krämer's article, in reference to the plsdof R package.

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