Finding multi dimensional, analytical relationship (curve fitting) for experimental data I have an unknown relationship like

$$
y = f(M, R, \sigma, P)
$$

I would like to find an analytical relationship that fits observed data the best.

As an example, if I keep $\sigma=0.3$ and $P=7$ constant, I get the following curves from the experiment:



I can fit each of these curves into equations of the form

$$
y = a (M + b)^n
$$

where $a$, $b$, $n$ are fitting parameters which I find via Least Squares.

However, how do I incorporate the relationship for $R$?

If I repeat the experiment but pick another $\sigma$ and $P$, the curves have the same shape (i.e., can  be described again by $y = a (M + b)^n$) but $a$, $b$, $n$ are all different. How can I incorporate $\sigma$ and $P$ to find the generic $f$ of a form $y = a (M + b)^n$?
There are two ways:


You know the science behind your experiment, so you can guess an analytic form. It looks like it's not the case for your problem.
You need to fit for many $R, \sigma, P$ parameters the curve and plot $a, b, n$ as a function of $R, \sigma, P$
There are two ways:


You know the science behind your experiment, so you can guess an analytic form. It looks like it's not the case for your problem.
You need to fit for many $R, \sigma, P$ parameters the curve and plot $a, b, n$ as a function of $R, \sigma, P$