In an actual Project, I have to find a function that best-fits given data. The Parameter-estimation of such a function should be hopefully not the Problem, but now I am struggling with which type of function I should choose for the Regression.
I added a Picture, where I plotted my gained data against the time, see this
It seems that there might be some function behind. Unfortunately I don’t know which type it could be.
Does anyone of you can give a hint or has an idea what type of function this could be?
Thank you in advance, questionmarkengineer
EDIT
Unfortunately, the path I tried to follow this weekend did not help me. So I add a few more details what I am trying to find out.
I have a function $$f_\text{EC}(t,t_i) = \phi\cdot\left(\frac{t-t_i}{\beta+t-t_i}\right)^{0.3},\ (t>t_i)$$ that delivers “measurement”-data over the time, starting at $t_i$ and $\phi$ and $\beta$ are constants.
This data should be approximated by the following function $$ f_\text{KV}(t, t_i) = \sum_{\mu=1}^N\frac{1}{E_\mu(t_i)}\cdot\left(1-\exp\left(\frac{-(t-t_i)}{\tau_\mu}\right)\right), $$ where $\tau_\mu$ are constants (ranging from 1 to 1000) and $E_\mu$ are parameters that can be found by Regression for each $t_i$. In my case, $N = 4$.
I now determinend these Parameters $E_\mu$ for several Points $t_i$ which are those points in time marked in the graph above. When I then plotted the values of $E_\mu$ against the corresponding $t_i$, I found out that there seems to be a functional connection behind ($E_1$, $E_2$ and $E_3$ changed, while $E_4$ remained nearly constant). And this connection is what I am trying to find, because I need the values of $E_\mu$ at nearly each Point in time, and a linear-Interpolation between my determined $t_i$ is not sufficient enough, especially between $t = 1, \ldots, 21$.
Maybe now someone has an idea what I can try? - It would also be helpful for me just to find a procedure for a (discrete) determination of the $E_\mu$ at each $t_i$ without using the (costly) regression.
I don't understand the problem that you're trying to model, but...
Two functions come to mind: $-c_1xe^{-c_2x}$, and $c_1e^{-c_2x} + c_3x$
Plot 1 - Plot 2