Prediction of independent data with PLS In Matlab's plsregress function and in many other statistic toolboxes, there is a BETA vector returned that simplyfies the regression problem to(excluding the intercept term in BETA):
Y=X*BETA

In almost all documentations, this BETA vector is used to predict the original data and to calculate residuals from there. This is for the data being used in regression. However, I couldn't find any documentation or method to predict unknowns and using solely the BETA vector for this purpose feels wrong. In other words if I were to calculate Y from a different X what steps should I follow? Is there a clear guide somewhere?
What is the case in cross validation?
Edit: Also in this article http://www.sciencedirect.com/science/article/pii/0003267086800289 the use of BETA is not mentioned for the PLS part.
Months later edit: Now I understand that the source of my confusion is differences between two main algorithms: NIPALS and SIMPLS.
 A: I'm mostly using the papers 


*

*Paul Geladi and Bruce R. Kowalski: Partial least-squares regression: a tutorial, Analytica Chimica Acta, 185, 1-17 (1986). DOI: 10.1016/0003-2670(86)80028-9 and 

*Mevik, B.-H. & Wehrens, R.: The pls Package: Principal Component and Partial Least Squares Regression in R, Journal of Statistical Software, 18, 1 - 24 (2007). DOI:     10.18637/jss.v018.i02 papers 


to extend @amoeba's comment into an answer here:
Let's start with the PLS X
$\mathbf X = \mathbf T \mathbf P' + \mathbf E$ and
$\mathbf T = \mathbf X \mathbf W'$   
and the Y matrices
$\mathbf Y = \mathbf U \mathbf Q' + \mathbf F$ 
(outer relations)
(take care to construct the weights $\mathbf W'$ and $\mathbf Q'$ so they refer directly to $\mathbf X$ and  $\mathbf Y$, not to deflated matrices!) 
Regression can then take place on the X and Y scores: $\hat u = t b$ (inner relation),
thus
$\mathbf Y = \mathbf T \mathbf B \mathbf Q' + \mathbf E$ 
$\mathbf{\hat Y} = \mathbf X \mathbf W' \mathbf B \mathbf Q'$ 
Now, the last three matrices ($\mathbf W' \mathbf B \mathbf Q'$) are all part of the PLS model parameters. We can therefore introduce one matrix $\mathbf B' = \mathbf W' \mathbf B \mathbf Q'$ which gives PLS coefficients in analogy to the usual MLR coefficients and write
$\mathbf{\hat Y} = \mathbf X \mathbf B'$
which is the usual form of a linear regression model. 
Your prediction can either use these "shortcut" coefficients, or the 3 steps of calculating


*

*X scores $\mathbf{\hat T} = \mathbf X \mathbf W'$, then 

*Y scores $\mathbf{\hat U} = \mathbf{\hat T} \mathbf B$, and finally

*$\mathbf{\hat Y} = \mathbf{\hat U} \mathbf Q'$


update: this procedure modeling both $\mathbf X$ and $\mathbf Y$ with latent variables and scores is known as PLS2. In contrast, PLS1 models only one dependent variable $\mathbf y$ (or $\mathbf Y^{(n \times 1)}$ at a time so that no Y-scores are obtained. Multiple dependent variates can be modeled by separate PLS1 models -- one per variate.
Whether multiple PLS1 or a single PLS2 model are better depends on the application, e.g. on whether the dependent variates are correlated and whether an underlying structure with few(er) latent variables is expected. 

In practice, you also need to take care of centering (standard practice) and possible scaling (less standard practice) of $\mathbf X$ and $\mathbf Y$.

For cross validation, the prediction works exactly the same way as for unknown cases: you fit the model on your training cases and then predict the left out cases like any other unknown case.
(Assuming this is not asking whether shortcut solutions exist to update a PLS model for exchanging one case during leave-one-out cross validation)
