Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the variables. The correlation effect, if exists, is not instant and it might take some time for the effect of a variable to show up in the observables. Also multiple variables can affect an observable.
From my understanding, if the correlation was instant, there are various methods that I could use, from simple regression and covariance matrix to PCA analysis. Since there might be a time lag, I think convolution/cross-correlation is more suitable. However, is there a systematic method/way for analyzing such problems? I can imagine such problems would arise in many different fields.
I mostly have a bio background and learned post-undergrad math on my own (not in a systematic way) and I would appreciate any hints on this problem. Also if you know a good approachable book on this topic (I guess multivariate data analysis?) I would appreciate if you let me know.