MATLABregressionstatistics
I have a noisy data set (the grey line in the graph below) that corresponds roughly to $y=m(1-2^{-x/k})$ where m and k are unknown constants.
How can I determine the best-fit value of m and k?
I can get an approximate value for k by guessing m and then doing linear regression on $-\log_2(1-y/m)$… by this I estimate m=0.96 and k=1000 (see red and blue dotted lines above), but is there a more systematic way?
Thanks in advance.
I am not sure to well understand exactly what you are looking for. I suppose that you want to find the parameters $A,B,C,D$ without using a software for nonlinear regression. Nevertheless I suppose that it is allowed to use a PC for drawing the points and curve. I suppose also that "manually" means that a PC is allowed for elementary numerical calculus.
The first step in order to simplify the problem is to find the symetry center of the tangent curve. This can be roughly done graphically "by eye" (Next figure).
Still roughtly but slightly better consists in drawing the graph rotated of 180° and making fit at the best to the original graph by trial and error (next figure, the original in red and the rotated in blue).
This gives a rough estimate of the coordinates of the symetry center : $$\begin{cases} x_s\simeq 2.6\\ y_s\simeq 1.2 \end{cases}$$
Then we translate the axes with new coordinates : $$\begin{cases} X=x-x_s\\ Y=y-y_s \end{cases}$$ This transforms the original equation : $$y=A\tan(Bx+C)+D\quad\implies\quad Y+y_s=A\tan(B(X+x_s)+C+n\:\pi)+D$$ to the new equation : $$\boxed{Y=A\tan(BX)}$$ with $$D=y_s\simeq 1.2\tag 1$$ $$C=\pi-Bx_s\simeq 3.14-2.6B\tag 2$$
The problem is reduced to a simple tangent function fitting involving only two parameters.
If you don't want use a software for nonlinear regression you can proceed by trial and error. Of course this is tiresome and not accurate.
Best Answer
Why not do nonlinear least-squares via Levenberg-Marquardt instead of futzing with linearizations? There is the
lsqnonlin()function available in MATLAB via the Optimization Toolbox. You will need to figure out good starting values for $m$ and $k$, though that LM can polish to a (hopefully) adequate answer.