Steve asks: << Here is a question that's bugged me for a while: Is there a standard or obviously-right way of defining the median for a bag (like a set but with possibly repeated members) of points in 2D or higher space? One way of closing in on it is to ask what properties I want it to have. I can think of a few: o The 1-D case should be the 1-D median. o Rotating, translating or scaling the points should do the same to the result. o Embedding in a higher-dimensional space shouldn't matter.
The method I've heard about is calld "convex peeling". This means take the convex hull of the data points and remove those lying on the boundary, and repeat. When eventually all remaining data point(s) lie on the most recent convex hull's boundary, take their average and that's the median. --Dan P.S. An interesting phenomenon that occurs at least in 2D: If the data points are a very large random sample from the square, then with high probability the successive convex hull boundaries seem to approach a circular shape (if say they're normalized to have constant diameter).
On 11/15/07, Dan Asimov <dasimov@earthlink.net> wrote:
Steve asks:
<< Here is a question that's bugged me for a while: Is there a standard or obviously-right way of defining the median for a bag (like a set but with possibly repeated members) of points in 2D or higher space?
One way of closing in on it is to ask what properties I want it to have. I can think of a few: o The 1-D case should be the 1-D median. o Rotating, translating or scaling the points should do the same to the result. o Embedding in a higher-dimensional space shouldn't matter.
Using the term "median" suggests to me that you intend the result to be a member of the original point set. Is that what you meant?
The method I've heard about is calld "convex peeling".
This means take the convex hull of the data points and remove those lying on the boundary, and repeat.
When eventually all remaining data point(s) lie on the most recent convex hull's boundary, take their average and that's the median.
--Dan
P.S. An interesting phenomenon that occurs at least in 2D: If the data points are a very large random sample from the square, then with high probability the successive convex hull boundaries seem to approach a circular shape (if say they're normalized to have constant diameter).
I haven't experimented, but it seems likely that this phenomenon will occur with any initially regular polygon, or even any properly convex polygon; at points on the boundary where curvature is high, abrasion by the above process will proceed more rapidly. [Obviously, there are a number of assumptions involved in this picture ....] WFL
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