Say we have $n$ points; $(x_1,y_1),(x_2,y_2),(x_3,y_3),$ and $\dots,(x_n,y_n)$.
Based on least squares, we need to find the best fitting curve that is of the form $$y=a-be^{cx^d}$$
I want to evaluate those parameters in terms of the known $x$'s and $y$'s
For example, linear relation that have the form $y=Sx+I$, we can find $S$ and $I$ as follows:
$$S=\frac{n\sum xy-\sum x \sum y}{n\sum x^2-(\sum x)^2}$$
$$I=\frac{\sum y-S\sum x}{n}$$
So in the given case (i.e. $y=a-be^{cx^d}$), how to obtain the values of the parameters $a,b,c,$ and $d$ in terms of known $x$'s and $y$'s?
This is probably known as Hockett-Sherby curve.
I found a website that calculates these parameters, but I really need to know the theory behind that parameters calculations.
Any help would be really appreciated. THANKS!
You want to minimize the sum $$ \mathcal{S}=\sum_i^n\Big(y_i-f(x_i;a,b,c,d)\Big)^2 $$ with respect to the four parameters, i.e calculate $$ \nabla \mathcal{S}(a,b,c,d)=\vec{0}. $$ This will give you four equations to be solved in terms of your four parameters.
For example the first equation you will find is $$ -2\sum_i^n\Big(y_i-f(x_i;a,b,c,d)\Big)\partial_af=0. $$