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How might I go about trying to detect the knee point for families of curves similar to this one?

Curve

I've tried fitting to exponentials, logs, rationals - but I'm still not sure where I would progress onwards from there. I've tried detecting inflection points with inflection-package in R but I don't think the curve actually has an inflection point.

Are there any ways of computing knee points?

Thanks in advance to you all.

Luke Chang
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  • Is there an actual mathematical definition of a knee point, or is it more of an informal concept? I thought it was the latter. – Hong Ooi Jul 10 '13 at 17:35
  • I believe it is the latter. No formal definition, I think. – Luke Chang Jul 10 '13 at 17:44
  • Well, then you're going to have trouble programming a search for it, no? If you want to do this anyway, the first thing would be to devise a suitable definition that you can use in your code. – Hong Ooi Jul 10 '13 at 17:46
  • I was actually thinking a bit about the curve again, and it seems the scaling of the y axis is confusing my thoughts a bit. If I instead think of it scaled 1:1 to the x-axis, perhaps I can find a threshold by computing finite differences and choosing a 1st derivative cutoff. Any suggestions? – Luke Chang Jul 10 '13 at 17:48
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    @LukeChang I think you are on the right track, but like Hong Ooi mentioned, you'll have to find some definition first, then the programming will be easier. Have a look at [this paper](http://www1.icsi.berkeley.edu/~barath/papers/kneedle-simplex11.pdf) which provides an interesting discussion of the concept and their own technique for detecting it. – nograpes Jul 10 '13 at 17:55

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