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One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$.

Thinking about data visualisation, one can portray a set of $N$ observations as a curve in 2-D. Imagine you have several sets of observations and want to eyeball the difference between them, other than with a histogram.

Since data often comes as sets of real numbers (or "factors", or "levels"), rather than as complex numbers, polynomial projection via real roots seems to be a less-than-ideal solution. Worse, important statistical differences (mean, modes, moments) don't jump out much more than unimportant differences.

Is there a better way to project these sets onto curves? tangent of a polynomial with simple integer roots

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    Why not use the curve that has small peaks where $x=x_i$?2011-05-07
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    @user3123 If you extended that idea to include multiplicities you'd have the density estimate of the histogram. I guess I was wondering about less literal projections.2011-05-07
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    Hm, the polynomial representation does not respect ordering of the roots, so cannot accurately describe points in $N$-dimensional space.2011-05-08
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    @Willie Wong You're right, I am not thinking about vectors of observations but contrasting sets of observations. Set1 versus set2 versus set3.2011-05-08
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    @WillieWong In time series analysis observations are considered ordered, but in other statistics (which is much of the time) observations are considered unordered.2014-05-20
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    @isomorphismes: now that we have the site stats.stackexchange.com, I think perhaps this question would be a better fit there. Unfortunately, this question is waaaaaay too old to be migrated `:-)`, so you will have to ask it there yourself.2014-05-21
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    @WillieWong Not a bad idea.2014-05-21

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Looks like the Andrews Curve was thought of for this purpose. Eg, http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/mvahtmlnode9.html