I'm trying to fit a set of data to a power curve of the form f(x)=a*(x+b)^c+d, is there a formula to determine the a/b/c/d components?
For example, if I have the following data:
| X | Y |
|---|---|
| 11 | 740 |
| 12 | 420 |
| 13 | 313.3333333 |
| 14 | 260 |
| 15 | 228 |
| 16 | 206.6666667 |
| 17 | 191.42857143 |
| 18 | 180 |
| 19 | 171.11111111 |
| 21 | 164 |
| 22 | 158.18181818 |
I'm expecting b = 10, and d = 100... is there a formula to derive a/b/c/d based on the data above, and the fact that I'm looking for a power curve?
The usual way is to use a specialised software for "non-linear regression". https://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html This involves numerical iterative calculus starting from "guessed" initial values of the parameters $a,b,c,d$.
A non-usual method (shown below) isn't iterative and doesn't requires "guessed" initial values. But the criteria of fitting isn't the classical LMSE. Since no criteria of fitting is specified in the question this non-usual method is convenient.
If a particular criteria of fitting (LMSE or LMSRE or LMAE or other) was specified a software for non-linear regression would be necessary as well as "guessed" initial values of $a,b,c,d$. This is not always easy and sometimes leads to not convergent iteration. Instead of, the values obtained above would be very good initial values instead of "guessed".
General principle of the non-iterative method : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales . The fitting isn't wrt the equation model but wrt a linear integral equation to which the equation model is solution. In the present case the linear integral equation is : $\int y\,dx = \frac{1}{c+1}x\,y+\frac{b}{c+1}y-\frac{c\,d}{c+1}x+$constant , with $S_k\simeq\int_{x_1}^{x_k} y(\chi)\,d\chi$ .