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I was wondering if the meaning of an elbow applies to graphs like the following. Basically for each curve obtained, I am looking to find out if there is a way to determine a point such as the one marked by 0 in the graph. In layman terms, I want to find out the maximum trade-off point across all the curves. I am not sure about the exact terminology though. In other words, I am trying to find out the point on the curve where the curve undergoes the maximum change. In this graph however, every change looks almost identical but the amount by which the curve falls before changing differs.

I tried looking at the point at which we obtain the maximum second derivative using the second derivative test but am having a hard time to see if it actually applies to curves like this and if there are any references explaining this.

Can someone please suggest related techniques that can be used to find the elbow point on curves like this? I found the following method called the L-Method but wasn't sure if it is applicable in my case.

EDIT: An attempt to describe what I am trying to do:

I am aware that I will be having a hard time explaining what I am doing but I will try my best. I am trying to fix two thresholds: one that is used to obtain the line and the other that I use to filter noise. The '0' on the graph marks the thresholds. In this particular case, I am using Threshold 3 and filtering my population to contain only those people that have >= 2 friends. While this is working well for me, I am trying to determine if there is systematic procedure to explain why these thresholds are better than others. For instance, Threshold 2 is not such a good filter because it behaves like Threshold 1 in that it is not filtering any information from the population (which is around 1500). Utilizing Threshold 4 is not so useful because the returns in going from Threshold 2 to Threshold 3 is much higher than that going from Threshold 3 to Threshold 4.

enter image description here

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So, you're trying to find the jump discontinuities of some step function? –  J. M. Aug 1 '11 at 7:04
    
(meta: functions seemed best here; if the tags I removed are in fact justifiable, please add them back.) –  J. M. Aug 1 '11 at 7:06
    
Oh... so its called jump discontinuities then. Thank you for cleaning up the tags. I am no expert in this so I will trust your selection of tags. Basically, I am trying to figure out across all the curves, which one has the metric as the maximum. –  Legend Aug 1 '11 at 7:08
    
Yeah, it's the term of art, but I think I like "elbow points" more now. :) –  J. M. Aug 1 '11 at 7:10
    
Oh! :) I knew elbow points existed for curves like this: nda.ox.ac.uk/wfsa/html/u09/u09c_t01.gif but I wasn't so sure if it makes sense for a step function. The second derivative test gave me some acceptable solution but I was not sure if I was using the wrong technique to get the right answer. –  Legend Aug 1 '11 at 7:11

1 Answer 1

Assuming that I'm reading the graph correctly... The derivative (or local change) isn't define on elbows, and since each pair of two adjacent elbows are connected by a straight line, the derivative in each such region (and thus in it's entire area of definition) is constant. Thus, the maximal derivative occurs where the longest "altitude" is lost in the least "time". If we assume that the distance between two adjacent elbows which aren't of the same height is constant $\Delta$ we get that the greatest change occurs in the plunge around 5.

Now for a more constructive approach. This graph seems to represent a function of discrete values (i.e., the values it takes on represents people, which is always a natural number). A very numerically convenient analogue to the derivative in this case is simply $f(n+1)-f(n)$. The larger $f(n)$ will be in relation to $f(n+1)$ the larger the plummet is, in a way independent of the actual values of $f(n)$ and $f(n+1)$. That is, the value of $n$ where $f(n+1)-f(n)$ is maximal (in absolute value) is where the steepest plummet/rise (up to sign) occurred in the graph. I know it seems ridiculously trivial, but it works, and it's also very easy to take averages of and generate some more meaningful statistics.

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"since each pair of two adjacent elbows are connected by a straight line" - I couldn't tell if the horizontal sections are in fact joined by lines, or are just plotting artifacts (like what you see when you plot the floor or unit step functions on a "dumb" plotter)... –  J. M. Aug 1 '11 at 7:13
    
Neither could I :\ That's why I explicitly mentioned it was an assumption... –  user864940 Aug 1 '11 at 7:15
    
Actually, they are straight lines. This is a CCDF plot of some data that I obtained through a survey that I conducted. For example, the 0 on the graph says that there are at least 2 people with around 130 friends. What I am trying to find out is the right threshold to apply along with the elbow point. –  Legend Aug 1 '11 at 7:17
    
+1 for your time. I updated my question with some more details on what I am trying to achieve. Any comments would be greatly appreciated. –  Legend Aug 1 '11 at 7:24
    
@Legend: Okay, so they're piecewise linear curves and not step functions... –  J. M. Aug 1 '11 at 7:27

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