I'm considering circular regression and elliptic regression on a computational and conceptual basis. If we fit an ellipse to our data then we deal with the principal components as reference for the main and for the secondary axis; we set the lengthes of the axes according to the sqrt of the eigenvalues.
The idea behind this is, that in the first principal component we have the maximal variance and the second the minimal variance. Accordingly we construct the confidence-ellipse this way. Now I was thinking: assume we have a huge number of normal distributed data, slightly correlated such that we have a nice elliptic shape. If we rotate to the principal components we get the two values for the length of the axes by the variance in each direction. But if we would think of, say, 40 angular segments of the ellipse, consecutively ordered around the origin, then shouldn't the shape of the confidence-curve follow the variance in each direction/each of the 40 segments? What would such a curve in the limit look like? Suddenly, as I think about this I' getting doubt it would be an ellipse...
