Given a large matrix with 10,000 rows (variables) and 20 columns (sampling timepoints), I am trying to build a linear model to find the sampling timepoint given some of the variables. So in essence, I want to find a small subset of variables that explains the data well enough that I can build a linear model like
timepoint ~ v1 + v2 + v3 + v4. I hope that this will allow me to only take measurements of the few variables that actually make a difference instead of measuring all 10,000 of them every time.
I have already tried using PCA for this, by doing a PCA on the whole matrix and then using the PCs as my variables. This works to some degree, but it does not actually solve my problem, since I would still have to measure all the variables for this to work. When I look at the loadings for the different PCs I usually get a large number of variables for each of them, so I cannot use this to determine the important variables either.
Is there a way I can do this?