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I have a sorted list of integers. From this list, I would like to find intervals of numbers in which most of the numbers are concentrated.

I have used K-Means with R and played around with the k parameter to visually identify ranges, but I am not satisfied with the justification of this method as I'm not sure how to evaluate how good each clustering scheme is.

Would it be possible to use something like Kernel Density Estimation to get the ranges?

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    $\begingroup$ Try to be more precise about what is an "interval" for you, and what would be the best solution. One dimensional data is easy, and finding the optimum solution may be in O(n) or O(n^2) if you spell it out, rather than hoping k-means happens to gind qhat you are looking for (if you even know yet, what you really desire). $\endgroup$ Jan 15, 2016 at 9:38
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    $\begingroup$ See stats.stackexchange.com/questions/163778/… $\endgroup$
    – Tim
    Jan 15, 2016 at 14:23
  • $\begingroup$ An interval to me is a pair of minimum and maximum numbers defining a range of numbers. I would like these ranges to cover all of the integers in my sorted list. I would also like the ranges to separate groups of frequently occurring numbers. I thought k-means would make sense here because I would be finding the centers of the intervals I seek. $\endgroup$
    – Duck
    Jan 15, 2016 at 14:25
  • $\begingroup$ It's clear whether you're looking for an algorithm or method? $\endgroup$
    – Aksakal
    Jan 15, 2016 at 14:30
  • $\begingroup$ Either algorithm or method would be appreciated. $\endgroup$
    – Duck
    Jan 15, 2016 at 14:48

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Well, yes, indeed. You could estimate the point density via kernel estimation and what would be likely a smoother density function. Perhaps you could try out the Parzen window methodology.

You'll be facing a trade-off between bias and variance. When assessing the "quality" of your estimate, you'll likely have to decide where you'd like to stand wrt that trade-off (in the case, eg, of the Parzen window, by adjusting the width of the window)

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  • $\begingroup$ How would I be able to extract the number intervals after I do a kernel density estimation? $\endgroup$
    – Duck
    Jan 14, 2016 at 21:08
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    $\begingroup$ @Duck You might start with something like a hpd-type interval based off the KDE and then adapt your interval from there, perhaps by looking at where the actual data are near the endpoints $\endgroup$
    – Glen_b
    Jan 15, 2016 at 1:12

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