Does it make sense to run partial least squares (PLS) on a data set that has many more columns (variables) than it has rows (data points)? I am using plsr in R.
I remember hearing this rule applied to principal component analysis (PCA). I'm not sure why though, and given that PLS is like PCA with Y response, I was wondering if this situation should be avoided here too.
I'd really like an explanation of this, if it is true for PLS or PCA.
You can do PLS in your situation. As usual just cross-validate the number of components you need. Also be careful with very high $R^2$ values (say higher than 0.9), they might indicate that you are overfitting. It is in some sense "easier" to capture most (co)variation with less components if there are few samples. You can try resampling methods to see if you are overfitting.
Yes.
One of the advantages of PLSR is its potential to overcome multicollinearity problem. Both in PCR and PLSR the projection provides projection of original data to smaller dimensions (If no variable is correlated originally, things get slightly complicated). Basically, the projected data, which have number of "new" variables equal to number of latent variable or principle components, is then used for regression.
