Determining Elliptical-ness of Data: A Statistical Approach?

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In summary, a statistical method to determine how well a set of data is fit by an elliptical bound is possible, but may require more complex analysis than is needed for simple cases like spherelliptical or uniform data.
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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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  • #3
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!
 

1. What is the concept of "Elliptical-ness" in data?

The concept of "Elliptical-ness" in data refers to the shape of the data distribution, which can be described as an ellipse or an oval. It is a measure of how clustered or spread out the data points are around the center of the distribution.

2. How is "Elliptical-ness" different from normal distribution?

"Elliptical-ness" and normal distribution are related but not the same. Normal distribution is a specific type of elliptical distribution where the data is symmetrical and bell-shaped. However, "Elliptical-ness" can refer to any type of elliptical distribution, including skewed or multi-modal distributions.

3. How is "Elliptical-ness" measured in data?

"Elliptical-ness" is typically measured using statistical methods such as measures of central tendency (e.g. mean, median) and measures of dispersion (e.g. range, standard deviation). These measures can help determine the shape and spread of the data distribution.

4. What are the implications of high or low "Elliptical-ness" in data?

High "Elliptical-ness" in data indicates a more clustered and symmetrical distribution, which can be easier to analyze and make predictions from. Low "Elliptical-ness" may indicate a more complex or irregular distribution, which may require more advanced statistical techniques for analysis.

5. How can "Elliptical-ness" affect data analysis and interpretation?

The level of "Elliptical-ness" in data can impact the choice of statistical methods and models used for analysis. It can also affect the accuracy and reliability of results, as non-elliptical data may require more robust techniques to account for any outliers or irregularities in the distribution.

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