in my schoolwork mathematical exploration, I am struggling to find ways in which I can model a function after a set of data. From a simulation software, I have gathered data that resembles an exponential equation. See image attached:
I am looking to find ways in which I could manually derive an equation that would fit this set of data. I am looking for a variety of suggestions as it would allow me to compare each method's effectiveness, however, I do not consider myself to be proficient in mathematics as I am only a 12-grade student. Any method or suggestion is welcome, thank you in advance.
If you want an empirical method, but not accurate, plot the points on semi-logarithmic scale . The linearization is not very good (Figure 1). This can be improved in substracting a constant $a$ to $y$ , empirically $a\simeq 9$ by trial and error (Figure 2).
Coming back to the linear scale, we obtain the Figure 3. The fitting is better for the low values of $(x,y)$ than for the high values. This is not surprising since the linearisation was on logaritmic scale instead of linear, which leads to better mean squares relative error, but worse mean squares absolute error.
The fitting can be improved with another parameter $b$ (Figure 4).
Of course, all above is rough empirical approach. In order to have a better fitting one have to use some regression method involving iterative numerical calculus.
Nevertheless a direct method ( without iterative calculus and without initial guessed values of parameters ) is explained in the paper : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
The application to the present data gives the results below with the details of numerical calculus :
RMSAE : Root Mean Square Absolute Error ; RMSRE : Root Mean Square Relative Error .
So, with this non-iterative method (linearisation thanks to integral equation) the result is :
$a\simeq 11.219\quad\;\quad b\simeq 0.4224 \quad\;\quad c\simeq 5.758 \quad\;\quad$ RMSAE$\simeq 1.980 \quad\;\quad$ RMSRE$\simeq 0.0646$
NOTE :
If one want more specific results with some specified criterion of fitting, one have to use iterative method of regression. For example in the present case :
If the criterion of fitting is the least mean square ABSOLUTE error the result is :
$a\simeq 12.344\quad\;\quad b\simeq 0.30407 \quad\;\quad c\simeq 6.091 \quad\;\quad$ RMSAE$\simeq 1.779 \quad\;\quad$ RMSRE$\simeq 0.0906$
If the criterion of fitting is the least mean square RELATIVE error the result is :
$a\simeq 10.331\quad\;\quad b\simeq 0.53465 \quad\;\quad c\simeq 5.520 \quad\;\quad$ RMSAE$\simeq 2.239 \quad\;\quad$ RMSRE$\simeq 0.0572$
One can see that the above non-iterative method gives an intermediary result between the iterative method with respectively "asolute" and the "relative" criteria.