Changes

:<math> f'(x) = f(x + 1) - f(x) </math>

:<math> f''(x) = f(x+2) - 2 f(x+1) + f(x)</math>

I required that a city-year had more than two layers, as it takes at least 3 layers to form an elbow. I then used <math>f'(x)</math> to determine the layer index from which the variance explained was monotonic (i.e., there was no change in sign in <math>f'(x)</math> in higher layer indices). This wasn't an issue when using the population variance explained. In an earlier version, when we used the sample variance explained, we had some non-monotonic sections of the curve resulting from integer division (<math>\frac{k-1}{n-k}</math>).