I have been trying to do classification of my hyperspectral data. The variable selection were done with variable importance in projection in pls-da. The selected variables were then used for classification, but in pls-da it did not show any clear separation. However when the selected variables were then analysed with lda it showed clear separation. Now I am not sure is it statistically correct to use the variables selected with pls-da and the classification done in lda.I have read that variable selection done with pls (not with pls-da) and then classification with other methods.
PLS and PLS-DA are very similar methods. In particular PLS-DA is simply PLS with dependent variables being categorical. Thus, selecting variables with PLS or PLS-DA for classification is equivalent.
I do not expect a huge difference in model performances among PLS-DA and post-LDA.
In the case of multicollinearity or if the presence of unrelated variables are suspected, one may want to select the best variables for LDA. It is not incorrect and may work very well in some cases.
However, I personally find variable selection methods based on regression coefficients, VIP scores, loadings unreliable to cope with multicollinearity problem, especially if your aim is improving model accuracy rather than simplification of model. Even if multicollinarity is not the issue, I would give PLS-LDA a try before variable selection. (see this link). Since all the variable selection methods can be somewhat dangerous and may lead to overfitting, testing the model against a independent validation set is a must.