Adding nonlinear terms to multiple linear regression (MLR) I just need a hint.(still learning)
I'd like to create a simple model that uses MLR. Basically, I add some parameters (temperature, irradiation...) and predict a parameter. But I'd like to add also (temperature^2, irradiation^2... ) and so on. But is there a theory/(know how) on whether to use squared values or square roots of them? I know that I should use such parameters that have greatest Pearson correlation with my parameter. But how should I seek for the parameters? Should I just multiply them randomly between each other and then check or maybe there is some smarter way?
 A: One way to go about your problem would be to fit the MLR model with the predictors temperature, irradiation, etc., assumed to have linear, independent effects on the parameter of interest. 
You could then plot the residuals from this model against each of your predictors. If the resulting plots don't display any systematic pattern (i.e., the residuals in these plots have random, constant scatter about the horizontal line going through zero), then you would feel comfortable with the linearity assumption for the predictor effects and you wouldn't need to refine your model.
But if you see a systematic pattern in one of these plots (e.g., the residuals follow a quadratic trend as you move along the horizontal axis corresponding to the predictor), that would be a signal that the linearity assumption is not tenable for the effect of the predictor in question. The systematic pattern itself might give you valuable clues as to what transformation to consider. 
You could also examine the so-called component plus residual plots for each predictor in your initial model to understand if there is an indication of nonlinearity for some of the predictor effects and gain some insights into its form.
Another thing you may want to do is to look at the distribution of each of the predictors. Often people tend to apply certain transformations to the predictors (e.g., log, square root) to force the distribution of the transformed predictor to be approximately normal (even though the normality of each predictor is not a requirement for MLR, but the normality of the model errors is). 
The Box-Tidwell transformation can also be a useful tool for deciding whether you need to include a predictor X raised to the power alpha in your model, where the value of alpha is estimated from the data. For example, if alpha is estimated to be close to 0.5 (or 1/2), you may decide to include the predictor sqrt(X) in your model to ensure it has a linear effect on the outcome variable. 
The process of finding the appropriate transformation for each predictor variable falls under the general umbrella of model building and can take various forms, as suggested above. 
For example, if you suspect temp (i.e., temperature) may have a quadratic effect after examining the diagnostics for your initial model where both temp and irradiation have linear effects, you may fit the following models to your data: 
Model 1: outcome parameter ~ temp + irradiation 
Model 2: outcome parameter ~ temp + temp^2 + irradiation 
and compare their AIC values (in a predictive setting) or BIC values (in an explanatory setting). The model with a smaller AIC or BIC value would be preferred. 
I have a feeling the parameters you mentioned were collected over time, in which case temporal correlation for the outcome parameter values will be an issue and will need to be accounted for in your modelling.
