Determining Elliptical-ness of Data: A Statistical Approach?

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Aaron
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"Elliptical-ness" of Data

I have sets of data that ideally should be constrained to elliptical bounds. I am looking for a method to see how elliptical a set of data is. My inital approach involved mean-centering and normalizing my data, calculating the angle of the point relative to the origin, finding the frequency distribution of this data and comparing it to a standard distribution. Graphically this works well for clearly elliptical data, but as the data approaches a circular bound (which is also valid) the distribution becomes flat and the comparison poor.

Is there a good statistical way to determine how well a set of data is fit by a elliptical bound? What methods are available for n-dimensional ellipsoids?

Please let me know if you need any more details and thanks for the help.
 
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I think this is not an easy task, especially because the alternative hypothesis can be varied -- as opposed to "ellipse vs. sphere" (which is what your test sounds well-suited for) or even "(sphere or ellipse) ['spherellipse'?] vs. uniform."

In the case of the alternative hypothesis "spherellipse vs. uniform," you can compare your statistic against the uniform distribution; but that would not be a useful test when the data are neither spherelliptical nor uniform.

An approach that might be helpful is described here: http://ciks.cbt.nist.gov/~garbocz/paper134/mono134.html

I may be thinking of the problem as more complicated that it actually is; if so, please let me know.

Another potential approach is: http://www.nlreg.com/ellipse.htm Suppose you "peel" the outermost "crust" of your data, then apply this procedure to these boundary points. The R^2 statistic of the elliptical regression would be a test of how well an ellipse fits to the outer boundary of the data.
 
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In what I'm doing, after applying a transformation I would like to see the data in some sort of ellipse (circle valid of course). Often when the transformation is bad, I get plots representing Xs or ellipses with Xs through them. I was hoping for some statistically valid method to do this, but I like the idea of regressing the outer "crust" of points, that should really be sufficient for what I'm doing. I'll give that a shot and see if I like it. That first paper you linked to seems a bit more involved, but I'll take a look at that if the other method doesn't work.

Thanks for the help!