Is there a formal reason for identifying and using "knee/elbow points"? I am trying to understand the reasoning behind identifying knee/elbow points as suitable thresholds.
From what my own intuition tells me, the use of a knee/elbow point is akin to determining an optimal point A of production quantity to minimise cost. 

Here are some examples of knee/elbow points being selected as thresholds:


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*In this post, the OP asks how he can keep the data that represents the majority of the information while filtering out the noise.

*In this article, the writer describes an algorithm for identifying knee/elbow points in running a large MapReduce job where the knee/elbow point is chosen as a threshold for speculatively re-executing processes in hopes that the MapReduce job would complete in a shorter time.
So in other words – what is the scientific reasoning behind selecting/identifying knee/elbow points?
Thanks!
 A: Your intuition about optimization is right. The elbow/knee considered as the point where the independant variable(s) will either start to or cease having a larger effect on the dependant variable. In other words, when increasing the independant variable by one unit, the marginal increase/decrease on the dependant variable will suddenly become faster/slower. Why is that?
If you look at a typical case with logistic regression:

We are looking at the probability of success in a test based on the score in a pre-test. As you can see, it seems that somewhere between 80 and 90, the "return on investment" is not as large. The participants have a larger chance of success of course, but the increase in the probability when you increase the score by 1 point is not as large. We can assess that by looking at some pre-test scores and the associated probability of non failing the post-test:
Pre-test score    Prob (rounded)  Change (rounded)
60                0.257            --
61                0.281            0.024
62                0.306            0.025
63                0.332            0.026
...               ...              ...

I added a "Change" column that indicated the change in prob when score is increased by 1. As you can see the change is rather stable, which is coherent with what we see in the graph in that interval. But around 81, the change starts getting smaller then will stabilize at a lower value:
Pre-test score    Prob (rounded)  Change (rounded)
78                0.755            0.023
79                0.776            0.022
80                0.797            0.020
81                0.816            0.019
82                0.833            0.017
83                0.850            0.016
...               ...              ...

Now suppose that I'm a student preparing for the course, and let's suppose that the pre-test score is an indicator of the time spent studying. I should aim for about 81% if I want to min-max my chances of passing because after that the gains are lower and my time would be spent less efficiently.
Optional - finding the exact value of the elbow/knees using the second derivative of the function
In my example, the "knees" are at 61* and 81, as calculated using the second derivative method. Here I plotted the second derivative (plot the "change" column against the pre-test scores, then plot the derivative of that). The 1st knee will be the hight point, the 2nd knee will be the lowest point:

This means that the association of the pre-test score and post-test success probability becomes stronger somewhere around 61%, and becomes weaker at about 81%. One could imagine that if the pre-test grades were able to vary to infinity, the link between the two variables would tend to be non-existent (the line would be flat). There isn't really a specific spot where the association beween the variables became null. 
The 81 point is just the place where the  curvature is the most pronounced, so that if you really had to identify a spot where there was an decrease/increase in correlation, the elbow might be it.
*61 may not be the real, "ideal" lower elbow, it could be an artifact resulting from the fact that my data in truncated at 60.
