Improving a linear regression: Add predictors or change model? 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:


*

*Change the model of regression: add some polynomials or other depedencies?

*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.
 A: In my field (social science using cross sectional surveys), an adjusted R squared of .87 would be much too large. That would be a sure sign that you have done somehting meaningless like predict something with a second measure of itself. So whether or not you need to improve your model depends on the context, which you did not give us.
If you are looking for alternative transformations of your explanatory/right-hand-side/predictor-variables you could consider fractional polynomials:
Royston P, Altman DG. (1994): Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling (with discussion). Applied Statistics, 43:429-467.
Royston P, Ambler G, Sauerbrei W. (1999): The use of fractional polynomials to model continuous risk variables in epidemiology. International Journal of Epidemiology, 28:964-974.
Royston P, Sauerbrei W. (2004): A new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials. Statistics in Medicine, 23:2509-2525.
Royston P, Sauerbrei W. (2007): Improving the robustness of fractional polynomial models by preliminary covariate transformation: a pragmatic approach.  Computational Statistics and Data Analysis, 51:4240-4253.
Royston P, Sauerbrei W (2008): Multivariable Model-Building - A pragmatic approach to regression analysis based on fractional polynomials for continuous variables. Wiley.
Sauerbrei W. (1999): The use of resampling methods to simplify regression models in medical statistics. Applied Statistics, 48, 313-329.
Sauerbrei W, Meier-Hirmer C, Benner A, Royston P. (2006): Multivariable regression model building by using fractional polynomials: Description of SAS, STATA and R programs. Computational Statistics & Data Analysis, 50:3464-3485.
Sauerbrei W, Royston P. (1999): Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society A, 162:71-94.
Sauerbrei W, Royston P, Binder H (2007): Selection of important variables and determination of functional form for continuous predictors in multivariable model building. Statistics in Medicine, 26:5512-28.
A: It's very hard to tell. If your criterion is $R^2$ (even adjusted) it is usually possible to "improve" it in the ways you mention. Indeed a completely different non-linear specification may be better. But the risk is that your extra terms will not be important and/or just complicate the model. If you are already using a mix of functions, that could be a great idea or just bizarre adhockery. 
How you can try to improve it is a matter of scientific as well as statistical reasoning. There should be a scientific story for each change of model, but you can learn a lot by looking at the residuals, e.g. on judging whether a quadratic term is needed as well as a linear. (For "scientific" read "economic" or something else if you don't think you are doing science too.) 
There is also an art and a craft to modelling. Trying very hard to catch all the structure in a particular set of data can result in a model specification that works very badly with new data. This is part of what is warned about under headings like "overfitting". 
$R^2$ may sound a universal criterion, but what it means depends very much on how much variation there is that for practical reasons if no other will remain inexplicable. There are fields where $0.1$ is a tremendous achievement and fields where anything less than $0.99$ is abject failure.  
A: I think instead of totally relying on R^2, you should try cross validating techniques.. there are chances that the model is fine and you are trying to improve it further which may not be possible.
A: *

*What you could also do is test your model for excess parameters (I don't think you
need both sine and cosine functions). You could do an SVD of the structure matrix you
use for fitting and make sure none of the singular values are zero (or too small).

*You could also improve your model by cleaning your data sample with filters
prior to fitting (Wiener filter for example is pretty good
against white noise). If you want to use filters you will have to make
a discrete Fourier transformation of the data, apply Filter in frequency
space and then make an inverse discrete Fourier transformation.

*If your data is periodic (or if the disturbances are periodic) you could perform
autocorrelation of your data sample, and any periodicity will be enhanced and the
rest will be suppressed. Then you can check the power spectrum of the autocorrelation,
to see which frequencies you have which you can then in turn either keep, or
cut out with a filter.

*You could also use a non-linear regression model and try to find a different minimum.
It could be, that the $R^2$ minimum you found is only local, whereas you want to have
a global one. If the $R^2$ doesn't change too wildly (is smooth enough), you could
try to fit your data using a simplex amoeba method.

*If you insist on using a linear model, why not $\chi^2$?

