Evaluating/Fitting an ellipse from scattered points

Question

Here is the deal. I have multiple points (X,Y) that form an \'ellipse like\' shape.

I would like to evaluate/fit the \'best\' ellipse possible and get its properties

Answer 1

The problem is to define "best". What is best in your case? The ellipse with the smallest area which contains n% of pointS?

If you define "best" in terms of probability, you can simply use the covariance matrix of your points, and compute the error ellipse.

An error ellipse for this "multivariate Gaussian distribution" would then contain the points corresponding to whatever confidence interval you decide.

Many computing packages can compute the covariance, with its corresponding eigenvalues and eigenvectors. The angle of the ellipse is the angle between the x axis and the eigenvector corresponding to the largest eigenvalue. The semi-axes are the reciprocal of the eigenvalues.

If your routine returns everything normalized (which it should), then you can decide by what factor to multiply everything to obtain an alpha-confidence interval.