I want to do multiple linear regression and then to predict new values with little extrapolation. I have my response variable in the range from -2 to +7, and three predictors (the ranges about +10 - +200). The distribution is nearly normal. But the relationship between the response and the predictors is not linear, I see curves on the plots. For example like this: http://cs10418.userapi.com/u17020874/153949434/x_9898cf38.jpg
I would like to apply a transformation to achieve linearity. I tried to transform the response variable by checking different functions and looking at the resulting plots to see a linear relationship between the response and predictors. And I found that there are many functions which can give me visible linear relationship. For example, functions
$t_1=\log(y+2.5)$
$t_2=\frac{1}{\log(y+5)}$
$t_3=\frac{1}{y+5}$
$t_4=\frac{1}{(y+10)^3}$
$t_5=\frac{1}{(y+3)^\frac{1}{3}}$ etc. give the similar results: http://cs10418.userapi.com/u17020874/153949434/x_06f13dbf.jpg
After I am going to back-transform the predicted values (for $t=\frac{1}{(y+10)^3}$ as $y’=\frac{1}{t^\frac{1}{3}}-10$ and so on). The distributions are more or less similar to normal.
How can I choose the best transformation for my data? Is there a quantitative (and not very complicated) way to evaluate linearity? To prove that selected transformation is the best or to find it automatically if possible.
Or the only way is to do the non-linear multiple regression?
plot(lm(1/(y+5)~r))
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