I'm trying to fit data to the following formula:
$$y = a + b x + \frac{c}{\sqrt{x} + d}$$
$y=a + b x$ can be fitted easily with linear regression, but I'm lost when it comes to anything more complicated.
Can anyone explain to me how I can build regression formulas for more complicated equations?
The model being $$y=a+bx+\frac c {\sqrt x +d}$$ it is nonlinear with respect to the parameters (because of $d$) and adjusting coefficients $a,b,c,d$ will require nonlinear regression and this will also require some reasonables estimates of the parameters for starting it.
What you could notice is that, for a fixed value of $d$, the model is linear. For a given $d$, define $z_i=\frac 1 {\sqrt x_i +d}$ which makes the model to be $$y=a+bx+cz$$So, for this value of $d$, you can compute $a,b,c$ using multilinear regression. So, consider that $d$ is fixed at a given value and compute $$SSQ(d)=\sum_{i=1}^n\Big(a+bx_i+cz_i-y_i\Big)^2$$ Plot the function and locate more or less a minimum. At that point, you are ready for the nonlinear regression.
If you do not want or cannot use nonlinear regression, you could continue the above procedure refining more and more the location of the minimum (zooming more and more tha area of the minimum).
