Method for estimating (thermodynamic) scaling fit parameters Let's say I have some measurements, label them $N$, which depend on the variables $x$ and $y$.
Further, I hypothesize that there is a scaling relationship between $N$, $x$, and $y$ of the form $x^{\alpha}N = f(x^{\beta}(y-\gamma))$, where $\alpha$, $\beta$, and $\gamma$ are unknown parameters and $f$ is some unknown function.
If I have many values of $N$, recorded for many values of $x$ and $y$, is there a way to estimate the values of the parameters $\alpha$, $\beta$, and $\gamma$? I don't really care what the function $f$ is.
I'm sorry if this is unclear, or simplistic; I'm embarrassed to admit I know very little statistics.
It would be extremely helpful if you know of an existing computational library (e.g. an R package) that would be helpful here, too.
Thanks!
 A: First of all, I'd start by restating your hypothesized relationship in the more "conventional" format, i.e., with the dependent variable $N$ appearing on one side of the equation, and all of the independent variables and unknown parameters on the other side, like so: $$N = g(x,y;\alpha,\beta,\gamma) = \frac{f(x^{\beta} (y-\gamma))}{x^{\alpha}}$$
Restated in this fashion, you have a classic nonlinear regression problem (i.e., the mathematical model is "nonlinear" in the parameters $\alpha$, $\beta$, and $\gamma$, because some of these parameters appear as exponents within the model equation--not to mention the fact that the function $f$ itself may also be nonlinear as well).
There is a package in R called nls which is designed to solve problems of this type.  If you are actually working on some kind of experimental physics problem (which I surmise based on the appearance of the word "thermodynamics" in the question title), please be aware that the international academic physics community also has independently developed its own computation package, minuit, precisely for addressing nonlinear parameter estimation problems such as this.  I believe that MATLAB has tools for this class of problem as well, e.g., lsqnonlin or LFMsolve.
For a good and very approachable introduction to non-linear regression for the physical sciences, I recommend chapter 8 of "Data Reduction and Error Analysis for the Physical Sciences", by Philip R. Bevington and D. Keith Robinson.  However, the material is built extensively on ideas first introduced in previous chapters, so if you want to really "get it," you're probably going to have to read a sizeable preceding chunk of the book. 
