Circle fitting validation

Question

I tried to find out circle center for noisy data. For computation sake I implemented this algorithm https://dtcenter.org/met/users/docs/write_ups/circle_fit.pdf It works fine but sometimes data is more similar to the line rather than circle. I need some parameter which would show me that data is more likely an arc of circle. example

Answer 1

Perhaps read up on model selection. A circle has more parameters than a line (3 vs 2), and indeed a circle can, locally, approximate a line arbitrarily well. You should thus expect that a circle always gives a better fit than a line (i.e. the fitting error is always lower), even if the data really is a noisy line.

In order to choose between models with varying numbers of parameters, you need to penalize the models with more parameters.