How do i fit my data with two functions and a knot between them

I am trying to fit the data in the graph below. The fit should be composed of two functions.

f(x) for x<k

g(x) for x>k

I do not know if it is needed but this are the two functions:

f(x) = 3/4*b*((a-x)/a-(a-x)^3/3a^3+2/3)+c

g(x) = d

The function g(x) is just a constant line and should fit the data approximately between 3.5 and 7. a,b and c are parameters of the fit.

Answer 1

[Sky Sartorius]

The tanh function can be handy for creating a one-liner piece-wise function for feeding into a curve fitting tool. If you don't have the curve fitting toolbox, check out https://www.mathworks.com/matlabcentral/fileexchange/10176. In the ezfit tool, I can ask it to fit my own equation of the form:

(0.5-tanh((x-k)/eps)/2)*(F1) + (0.5+tanh((x-k)/eps)/2)*(F2);

In your case, replace F1 and F2 with your fit functions:

(0.5-tanh((x-k)/eps)/2)*(3/4*b*((a-x)/a-(a-x)^3/3*a^3+2/3)+c) + (0.5+tanh((x-k)/eps)/2)*(d);

You can also get some a priori information from your data that will help constrain the problem and help it converge to a good fit, for example you can bound d to be between the min and max of your data y values, and similarly for k and the data x values.

[artu' breuer]

I forgot to ask, what is eps?