Compute the 'elbow' for a curve automatically and mathematically

Question

One example for curve is shown as below. The elbow point might be x=3 or 4. How to compute the elbow for a curve automatically and mathematically?

Answer 1

What you really want is the point with maximum curvature. When the slope is much smaller than 1, this can be approximated by the second derivative (as @ebo points out), but this is not always the case.

Answer 2

Functions like this one are usually called L-curves for their shapes. They appear when solving ill-posed problems through regularization.

The 'elbow'-point is the point on the curve with the maximum absolute second derivative.

Answer 3

You might want to look for the point with the maximum absolute second derivative which, for a set of discrete points x[i] as you have there, can be approximated with a central difference:

secondDerivative[i] = x[i+1] + x[i-1] - 2 * x[i]

As noted above, what you really want is the point with maximum curvature, but the second derivative will do, and this central difference is a good proxy for the second derivative.

Answer 4

I created a Python package that attempts to implement the Kneedle algorithm.

To recreate the function above and detect the point of maximum curvature:

x = range(1,21)
y = [0.065, 0.039, 0.030, 0.024, 0.023, 0.022, 0.019, 0.0185, 0.0187,
     0.016, 0.015, 0.016, 0.0135, 0.0130, 0.0125, 0.0120, 0.0117, 0.0115, 0.0112, 0.013]

kn = KneeLocator(
    x,
    y,
    curve='convex',
    direction='decreasing',
    interp_method='interp1d',
)

print(kn.knee)
7

import matplotlib.pyplot as plt
plt.xlabel('x')
plt.ylabel('f(x)')
plt.xticks(range(1,21))
plt.plot(x, y, 'bx-')
plt.vlines(kn.knee, plt.ylim()[0], plt.ylim()[1], linestyles='dashed')

update
Kneed has an improved spline fitting method for handling local minima, use interp_method='polynomial'.

kn = KneeLocator(
    x,
    y,
    curve='convex',
    direction='decreasing',
    interp_method='polynomial',
)

print(kn.knee)
4

And the new plot:

plt.xlabel('x')
plt.ylabel('f(x)')
plt.xticks(range(1,21))
plt.plot(x, y, 'bx-')
plt.vlines(kn.knee, plt.ylim()[0], plt.ylim()[1], linestyles='dashed')