I am looking for the name of a fitting method that will work even if points from multiple dataseries are meshed together.
As far as I understand there are two major methods, least squares and least absolute error; the method I am looking for would be the next step in the direction from LS to LAE:
- Least Squares fitting: Minimize the square of the distances of all data points to a function. The further a point is, the stronger it's deviation is weighted. May be strongly influenced by outliers. Will always converge to one globally unique solution (at least for simple curves), easy to find with gradient descent methods.
- Least absolute deviations: Minimize the sum of the distances of all datapoints. All deviations are weighted equal. More robust to outliers, may converge to potentially infinite solutions that all have the same total deviation
(generated by hand; images may not be accurate)
- "Least Square Root" ??? (Google does not really have many hits for this name). I am looking for a function that may converge towards multiple local minima, and weights deviations stronger the closer a point is. Which solution it would converge would depend on the starting parameters. I think a least-square-root of all deviations would do this, but I cannot find any reference to this sort of fit.
Is there an official name for a method like this, and what would it be called? Or is this sort of fit fundamentally not possible with curve fitting methods?
I am not looking for a method that would find all local minima, one local minimum depending on starting parameters would suffice.
(note that in the example pictures the functions were all lines, but ideally I am looking for a method that is able to iteratively fit points towards arbitrary n-dimensional functions)