The most accurate curve fitting function for the attached plot (It is a thermodynamic properties behavior)

Question

I have a diagram that has been attached

Which combination of functions can fit my function in an accurate way?

If a math professional offer me the general form of the fit function I can find the constant of the generalized form.

The attached plot

Answer 1

They are an infinity of functions suitable.

It is impossible to answer correctly without data. The data obtained from a graph isn't accurate enough to distinguish if the fitting to a function is better or not than the fitting to another function.

Moreover the information about what you are looking for is not sufficient to chose the kind of function which is likely to satisfy.

For example the function below is well fitting to the curve of your graph (from numerical scanning of the image) :

In blue : Your curve (digitized from the scan of your gragh).

In red : The curve drawn from this function : $$y\simeq 100(1-e^{-0.0228\,x^{0.63}+0.000465x})$$

Nevertheless I bet that this function will not satisfy you because they are probably other criteria of fitting and property of the function which don't agree with some physical and/or mathematical requirements not clearly specified in the wording of the question.

Answer 2

After the comments from MEng and the data table given later, it appears that the wanted fitting is not for the first curve posted, but for its inverse ($x$ and $y$ inverted).

This is a different problem because the empirical function first proposed is not analytically invertible.

Since the problem was changed, I give a different answer, but there is no raison to delete the first answer which is correct in the original context.

Nevertheless in order to not waste more time, the new search will be less advanced and the new function proposed will be more simplistic.

The next figure is drawn from MEng's data :

We observe that $\ln(y)$ drawn as function of $x$ is not far from linear. So, a simplistic method consists in approximating $\ln(y)$ by a polynomial with a number of terms such as the fitting be accurate enough. $$\ln(y)\simeq c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5$$ $$y(x)\simeq e^{c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5} $$ In the wording of the question the expected accuracy is missing. So one cannot define the number of terms of the series. For example, six terms were used without justification.

Also in the wording of the question the criteria for fitting is missing. For example least mean square relative errors was used as a criteria. Note that if the criteria was least mean square errors or least mean absolute errors the results would be different from the results below.

Result of regression : $\quad\begin{cases} c_5= 4.423019\:10^{-9}\\ c_4= -1.03187 \:10^{-6}\\ c_3= 9.466449\:10^{-5}\\ c_2= -0.004239\\ c_1= 0.143353\\ c_0= 2.282453\\ \end{cases}$

The points from the given data are drawn in blue.

The curve drawn in red of equation $y(x)\simeq e^{c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5} $ is quite confused with the blue curve.

The mean relative error is about 1.3% . Of course this is a mean and the relative error is higher in some points. For more accuracy one have to take more terms in the series.

Of course they are a lot of other kind of smarter functions which are likely to fit the data with less parameters to optimise. But this need advanced search.