# (nonlinear) Regression, but: What type of function is this?

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?

## 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.

• If this is real data from a real system, then you should try using a mathematical model to derive a model curve to fit. Have you done that ? Aug 18, 2017 at 20:26
• What kind of system/value are you measuring? Aug 18, 2017 at 22:16
• It is the elastic modulus in a spring-dashpot-system (4-Kelvin-Chain-Model) that is fitted to resulting creep-curves starting from each time-step marked in the graphs above. The shown graphs are the elastic modules of the springs that lead to the considered creep curve (one of these springs has a constant module). I just realized that Bazant/Xi did something like that and presented a dirichlet-series-approximation when the given creep-curve(s) is differentiable. I hope that I can work it out by this weekend. If one is interested, I can post the solution or the way I used this Bazant-algorithm. Aug 19, 2017 at 0:24
• ... but anyway, it would be of interest to me if there exist some (simple?) functions that show such behaviour ... Aug 19, 2017 at 7:45

Two functions come to mind: $-c_1xe^{-c_2x}$, and $c_1e^{-c_2x} + c_3x$