I am trying to model a time series variable $Y\_{t}$ with $4$ physical predictor variables. I used the following linear regression:

$Y=\beta\_{0}+\beta\_{1}f\_{1}(X\_{1})+\beta\_{2}f\_{2}(X\_{2})+\beta\_{3}f\_{3}(X\_{3})+\beta\_{4}f\_{4}(X\_{4})+\epsilon$ where $f\_{i} \in \{\log, \tan, \sin, \cos,1/.,Id\}$. 

My best model gives me an adjusted determination coefficient $R^2\_\rm{adjusted}$ of $0.87$.
But I know that this indicator is not completely reliable. **So I would like to know if I can improve my model.**

**If yes, how can I know if I have to:**

1. Change the model of regression: add some polynomials or other depedencies?
2. Look for more predictors $X\_{i}$ that I may have not discovered yet?

Any ideas would be nice!

***Edit: My true purpose***

$Y$ is a value which is measured each day, not at the same hour. What I really want to do is studying the variation of $Y$ with time and "cleaned" from the other external variables $X\_{i}$ which are "blurring" my data.

So thanks to my regression I want to find the part of $Y$ which depends on this external varying factors, and then remove this part. At the end, if my estimation $\hat{Y}$ is correct, $Y-\hat{Y}$ will correspond to my data "unblurred", and I will be free to sudy a "normalized" or "cleaned" version of $Y$ with time.

I tested my model for another set of data of $Y$ measured a year ago, and it also fits the data with a $R^2\_{adjusted}$ around $0.8$. And I can see graphically that the model is not completely insane, but not perfect either.