C-method

The C-method [1] fits a function \(f\left(x; c\right) = \frac{x\left(e^c + 1\right)}{x e^c + 1}\), where x is your input’s \(x\) coordinates, y is your input’s \(y\) coordinates, and \(c\) is the parameter which is tuned. The \(c\) parameter tuning is done with a custom implementation of a Newton-Raphson method.

After the optimal \(c\) is found, the theoretical knee \(x*\) is located at \(x* = \frac{\sqrt{e^c+1}-1}{e^c}\), as per [1]. The function returns the input \(x\) coordinate closest to the theoretical knee.

[1] (1,2)

The interested reader is invited to take a look at The C method (blog.msmetko.xyz) for the derivation of both of these equations.