Under what conditions can a PLS regression model be expressed by single linear equation? I am confused by two, yet inconsistent for me, facts: Since the PLS regression is expressed by  matrices of scores and loadings as
$$X=TP^T+E\\Y=UQ^T+F$$
how it can be translated into linear equation like $Y=a+b_1X_1+b_2X_2+...+b_nX_n$?
(I have found this in several papers).
 A: There is a good summary of PLS regression in

Rosipal, R and Krämer, N (2006). Overview and recent advances in
  Partial Least
  Squares.
  In Saunder, C, Grobelnik, M, Gunn, S, and Shawe-Taylors (Eds.),
  Subspace, Latent Structure and Feature Selection, pp. 34-51, Springer.

Using your notation, where $P$ and $Q$ are matrices of loadings, of dimensions $(N\times p)$ and $(M\times p)$, and $T$ and $U$ are $(n\times p)$ matrices of $p$ scores for $n$ individuals ($p\le\min(n,N)$), component scores are constructed as $t=Xw$ and $u=Yc$, and we find orthogonal weight vectors, $w$ and $c$, by maximizing covariance between $t$ and $u$, with appropriate constraints on $w$ and $c$ ($\|w\|=\|c\|=1$).
This is usually done iteratively using the popular NIPALS algorithm, but other methods have been proposed (SVD, SIMPLS), and different models of PLS lead to different deflation strategies. A good overview is available in 


*

*Wegelin, JA (200). A Survey of Partial Least Squares (PLS) Methods, with Emphasis on the Two-Block Case. Technical Report, Department of Statistics, Univeristy of Washington, Seattle.

*Höskuldsson, A (1988). PLS regression methods. Journal of Chemometrics, 2, 211-228.


(The chemometrics R vignette gives a short overview as well.)
In PLS regression (with a single or multiple outcomes, aka PLS-2), we further assume a linear (inner) relation between the components scores, $U=TD+H$, where $D$ is a $(p\times p)$ diagonal matrix of weights, and $H$ is the matrix of residuals. Therefore, following Rosipal and Krämer's notation, the $Y$ matrix can be decomposed as follows:
$$Y=TDQ^T+\underbrace{(HQ^T+F)}_{residuals}$$
which is roughly an OLS decomposition considering the orthogonal predictors $T$, and where $C^T=DQ^T$ is the $(p\times M)$ matrix of regression coefficients. Its estimate can be computed as $\hat C=(T^TT)^{-1}T^TY$. 
It is also possible to express the PLS regression model using the observed block $X$ directly, as $Y=XB+(HQ^T+F)$. Using the fact that $T=XW(P^TW)^{-1}$ ($T^TT=\mathbb I_p$), Rosipal and Krämer shows that the regression coefficients are then estimated as $\hat B=X^TU(T^TXX^TU)^{-1}T^TY$. This allows to express separate $\hat y$ (in case of PLS-2) as a linear combination of the $x_i$ ($i=1,\dots,N$).
