Changes

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), and used <math>f''(x)</math> to find the layer index <math>i</math> at which <math>varexp_i = min(varexp)</math> for some city-year. I then marked <math>i+1</math> as the elbow layer for that city-year, as we are using forward differences, not central differences.

Note that the lens found by this elbow method is only slightly bigger than the lenses found using the other heuristic method and the maximum R2 method (and those two lenses are near identical!). It's easy to look at the differences in the medians or means (etc.) and see differences, but it's important to remember just how big those differences could be!