7,612
edits
Changes
Jump to navigation
Jump to search
Urban Start-up Agglomeration and Venture Capital Investment (view source)
Revision as of 20:23, 1 December 2020
, 20:23, 1 December 2020→Jim's notes on the curvature
So it seems pretty clear that we want to use the second first difference at the base. More precisely, to get the concavity for layer 3 we want to divide the middle second difference by the forward first difference. (It would probably also be okay to use the middle second difference divided by the middle first difference, but I have not checked that out).
=====Formalizing Jim's Notes=====
Jim calculates the following (examples using layer 2):
*The '''first-order backward difference''' in column '''D1''': <math>f(x)-f(x-1) = 40-0=40</math>
*The '''second-order central difference''' in column '''D2''': <math>f(x+1)-2f(x)+f(x-1) = 75-2x40+0 = -5</math>
*'''Concavity''' (in col5) as -D2_l/D1_l, or -1 times the backward first over the central second: <math> --5/40 = 0.125 \approx 0.13</math>
*'''Concavity''' (in col6) as -D2_l/D1_{l+1}, or -1 times the central first over the central second: <math> --5/35 = 0.43 \approx 0.14</math>
The concavity measure in col6 is therefore the -1 times central first difference divided by the central second difference, but the central first isn't computable for a step of 1 (and gives a weird answer anyway, as it straddles the observation in question). The central second difference isn't defined for either the first or last layer, and the backward first difference isn't defined for the first layer. It seems likely that we don't want the last layer and might get it because D1 is small and drives the ratio. So, we could instead use the forward first difference - this isn't available for the last observation (for which we can't compute a second central anyway) but is available for the first observation - and increment the answer, much as Jim proposes decrementing it when using the backward layer.
===Version 3.5 build notes===
