How to find the equation of this curve from the data?

Question

I have the following data:

x        y
-------------
60      47.78
30      45.35
20      42.98
15      40.66
12      38.4
10      36.19
6       27.93
5       24.13
4       18.85
3       11.17

How can I find the equation of the line? I have plotted the graph and it looks like this:

I'm not sure what degree or type of curve this is. I have tried using Logger Pro to automatically find the best fit line but it doesn't get it. The closest I could get to is a Natural Exponent curve. This is what it is:

How can I find the equation of this curve?

Answer 1

Unless you have a function with as much parameters as point, you won't find a perfect fit. Look for the best one into your Logger pro, it seems to do the job of parametrizing for you (I'd say you have some kind of log or $x^{\alpha}$ with $\alpha<1$).

If you don't want any of these models, choose a function $f(x,a_1,...,a_1)$ where $a_i$ are your parameters, and with your actual values of x, minimize

$ \min_{a_i} \sum_{\text{all }x}(f(x,a_1,..,a_n)-y)^2$

Answer 2

You may try to optimize these test functions using different types of sigmoids :

1. $\displaystyle\quad A\arctan\left(\frac{x-B}C\right)$
   For $A=31.5,B=1.55,C=3,77\;$ I obtained :
2. $\displaystyle\quad \frac {A\;x}{\sqrt{1+(x/B)^2}}$
   For $A=5.4,B=8.8\;$ I obtained :

and others...

Alpha's bestfit didn't help much here...