# (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 --- EDIT --- 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 $f_\text{EC}(t,t_i) = \phi\cdot\Bigl(\frac{t-t_i}{\beta+t-t_i}\Bigr)^{0.3}, (t>t_i)$ 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 $$f_\text{KV}(t, t_i) = \sum_{\mu=1}^N\frac{1}{E_\mu(t_i)}\cdot\Bigl(1-\exp\Bigl(\frac{-(t-t_i)}{\tau_\mu}\Bigr)\Bigr),$$ 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 ($E_1$, $E_2$, $E_3$ 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 $t = 1 ... 21$.

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 '17 at 20:26
• What kind of system/value are you measuring? Aug 18 '17 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 '17 at 0:24
• ... but anyway, it would be of interest to me if there exist some (simple?) functions that show such behaviour ... Aug 19 '17 at 7:45

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