Question

I have a simple loglog curve as above. Is there some function in Matlab which can fit this curve by segmented lines and show the starting and end points of these line segments ? I have checked the curve fitting toolbox in matlab. They seems to do curve fitting by either one line or some functions. I do not want to curve fitting by one line only.

If there is no direct function, any alternative to achieve the same goal is fine with me. My goal is to fit the curve by segmented lines and get locations of the end points of these segments .

Answer 1

First of all, your problem is not called curve fitting. Curve fitting is when you have data, and you find the best function that describes it, in some sense. You, on the other hand, want to create a piecewise linear approximation of your function.

I suggest the following strategy:

1. Split manually into sections. The section size should depend on the derivative, large derivative -> small section
2. Sample the function at the nodes between the sections
3. Find a linear interpolation that passes through the points mentioned above.

Here is an example of a code that does that. You can see that the red line (interpolation) is very close to the original function, despite the small amount of sections. This happens due to the adaptive section size.

function fitLogLog()
   x = 2:1000;
   y = log(log(x));

   %# Find section sizes, by using an inverse of the approximation of the derivative
   numOfSections = 20;
   indexes = round(linspace(1,numel(y),numOfSections));
   derivativeApprox = diff(y(indexes));
   inverseDerivative = 1./derivativeApprox;
   weightOfSection =  inverseDerivative/sum(inverseDerivative);   
   totalRange = max(x(:))-min(x(:));   
   sectionSize = weightOfSection.* totalRange;

   %# The relevant nodes
   xNodes = x(1) + [ 0 cumsum(sectionSize)];
   yNodes = log(log(xNodes));

   figure;plot(x,y);   
   hold on;
   plot (xNodes,yNodes,'r');
   scatter (xNodes,yNodes,'r');
   legend('log(log(x))','adaptive linear interpolation');
end