Need an Introduction to Generalized Non Linear Multiple Regression

I have been searching the internet for a generalized method for doing regression analysis on non linear data. My model can be represented as

$$Y = \beta_0f(X_0) + \beta_1g(X_1) + ... + \beta_nz(X_n) + \varepsilon$$

where I don't have any idea what $f() g() z()$ are. But I can constrict myself to a domain saying that

$$f(), g(), z(), \varepsilon \in [\sin(), \log(), x^2, x^3, 1/x, e^x, x]$$

Please forgive me for any terminology mistake, I mean $f(), g(), h()$ can be one of the functions given in that set.

I've researched that once we know the equation, in certain cases we can linearize it so the form becomes linear regression. Is there no way to do a regression analysis for this form then? Without knowing the equation itself?

I'm a better programmer than a statistician and so I'm not averse to taking an iterative approach substituting the functions in each stage as long as someone can please guide me through the iterative process.

Further, isn't this model more frequently encountered in real life? I haven't seen any examples of this at all on the web.

• Did you mean to include the error term ($\varepsilon$) as something that could take one of the functions? – gung - Reinstate Monica Nov 9 '14 at 16:32
• What is the nature of your response variable, Y? – gung - Reinstate Monica Nov 9 '14 at 17:12
• As a matter of terminology--which can help both in searching for information and interpreting it correctly once you find it--you situation is neither "generalized" nor "nonlinear." $Y$ is explicitly a linear function of the parameters $\beta_i$; this is what makes it a linear model. A "generalized" model would make specific assumptions about the distributional family of $\varepsilon$; these are usually called GLMs. Your problem seeks re-expressions of the independent variables in order to create a linear relationship. – whuber Nov 9 '14 at 17:44
• On this site there are loads of questions about this, with examples and much discussion of when to use alternative approaches. Try searching our site: stats.stackexchange.com/search?q=transform+independent+variable. – whuber Nov 9 '14 at 17:46
• you might want to investigate additive models. – Glen_b -Reinstate Monica Nov 9 '14 at 23:35

• @gung Thanks a lot for your answer. I did research in MARS and also played around the earth package in R. Can you just explain / point me to some simple theoretical foundation as to why Combinatorial will lead to over fitting? – Gaurav Ramanan Dec 6 '14 at 9:19