Looking for an approximating function?

Question

Mathematics

Hello! Faced the following problem: there is a certain dependence of two quantities (in my case, this is the dependence of the battery emf on the degree of charge, but it doesn’t really matter) and it is known that the dependence is nonlinear and its form is unknown. There are sets of experimentally obtained points for different conditions. Those. in principle, each one can be approximated by the least squares by an n-th order polynomial, but it is interesting how to find the type of function that is most suitable for this data set? For example, if we have an exponential dependence of the experimental data, we, of course, can approximate it with a polynomial, but the function was originally an exponent. In a word, do the reverse process.
Maybe some genetic algorithms or something like that?

Answer 1

What you want to do is called "system identification".
There are methods based on enumerating models with the search for the best approximation under the condition of a minimum entropy of the model parameters - this will just filter out your "n-order polynomial".
There are methods based on multiple dependency differentiation.
There are other approaches...

Answer 2

Splines can pretty well be approximated. On Wikipedia, it is well written about them in general here and there is an example of building a cubic spline here

Answer 3

Yes, I understand that there are methods for a good approximate approximation, but the question is to find the function that lies at the root of the physical process - i.e. if we built a model and it was described by the function y =3 + ln x + (sin x)^2 , then the spline will be able to smooth the function, but we will get a rougher approximation than if we found the desired function. But the question is that the model can not always be built, or if you find the right function, then it is easier to build a model

Answer 4

Can I see the x,y data set?

Answer 5

Well, my data is not much different from those given in the literature for other batteries, for example, one of the curves
www.hindawi.com/isrn/applied.mathematics/2013/9537...

Answer 6

As an option - the method of "dimensional analysis"
there is a simple example, starting with the words "Let's give an example. The car starts off ..."
In your particular case, everything, of course, will be more complicated.
There is also an interesting trick - converting the picture into logarithmic coordinates. Then, for example, the exponential dependence becomes an inclined segment, and the exponent is easily determined from the slope. The exponent becomes a line segment if the x-axis is left on the usual, non-logarithmic scale, and the logarithm becomes the opposite, if the y-axis is in the usual one, and the x-axis is in the logarithmic one.
But in general, for electro-chemical processes, it is pointless to look for a beautiful formula. She is not there.