What is the function of a curve like a "」"?

Question

I'm doing curve fitting and the curve look like this.

I've tried polynomial or exponetial functions, but none of them fit the curve very well.

Can someone give some advice? Thanks in advance!

Answer 1

To me, these curves exhibit two linear regimes with a smooth transition. Denoting by $a_1x+b_1$ the first (left) linear form and by $a_2x+b_2$ the second (right) linear form and by $t$ the point, where the two linear forms meet, one could mimick such a behavior by the function $$ f(x) = \frac{a_1x+b_1}{1+c^{x-t}} + \frac{a_2x+b_2}{1+c^{t-x}}, $$ where the exponential numerators serve as smooth step functions. The larger $c$, the sharper the transition. Here is a demonstration of how such a function looks like. For the dark blue curve, I heuristically get $a_1 \approx0.2, b_1\approx100,a_2 \approx100,b_2\approx-8.9\cdot10^4, t\approx900,c\approx1.7$.

In any case, for a rigorous fitting, it would be best to understand the origin of these curves.

Answer 2

Without information on the underlying phenomenon, it is indeed difficult to make precise guesses. As Andreas says, there seems to be a transition between two linear regimes.

A possible model is

$$y=\log(e^{ax}+e^b)+cx+d.$$

It realizes a smooth transition between the slopes $c$ and $a+c$.