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 f'(x) to determine the layer index from which the variance explained was monotonic (i.e., there was no change in sign in f'(x) in higher layer indices), found and used f''(x) to find the layer index <math>i</math> at which <math>varexp_i = min(varexp)</math> for some city-year, and . I then marked <math>i+1</math> as the elbow layer for that city-year, as we are using forward differences, not central differences.