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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 $x=x_i$? – Listing May 7 '11 at 20:56
@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. – isomorphismes May 7 '11 at 21:34
Hm, the polynomial representation does not respect ordering of the roots, so cannot accurately describe points in $N$-dimensional space. – Willie Wong May 8 '11 at 3:17
@Willie Wong You're right, I am not thinking about vectors of observations but contrasting sets of observations. Set1 versus set2 versus set3. – isomorphismes May 8 '11 at 17:46
@WillieWong In time series analysis observations are considered ordered, but in other statistics (which is much of the time) observations are considered unordered. – isomorphismes May 20 '14 at 20:54

Looks like the Andrews Curve was thought of for this purpose. Eg,

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