Changes

[[File:Portland3HullsOvertimePortland3HullsHighest.png|right|300px]] Using a maximum R-squared approach to find the 'best layer' for a city is inherently problematic. A city might have 5 layers in 1980 and 80 layers in 2017, and so using layer 40, say, irrespective of year is somewhat meaningless. There are several alternative that make more sense. One is to use the fraction unclustered, much like with the elbow approach. The other is to find the layer with a certain hull count (or as close to it as possible). Hulls might tend to be somewhat stable over time, so three hulls in Portland in 2017 will be centered in more or less the same place as three hulls in Portland in 2003. This turns out to be somewhat true, as seen in the image on the right, which uses the last time (highest level) that there are three hulls, or two for 1998 and 1993 (one of which is out of frame).One issue with this approach is that the highest level with a certain hull count is that hulls almost always contain just three points. [[File:Portland3HullsLowest.png|right|300px]] An obvious alternative approach is to use the first time (lowest level) that there are three hulls. There is a big difference in the layer numbers for this. See the queries below. Essentially, the HCA algorithm often takes an original hull apart and then takes the resulting hulls apart, giving a quadratic for hull count again layer. But, as is apparent in the image, this leads to big areas that would only be good for overlap analysis and not for identifying individual clusters.  --Three hulls (or fewer) first and last time (i.e., lowest and highest layer) SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='2015' ORDER BY year,layer; --3, 41 SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='2010' ORDER BY year,layer; --3,17 SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='2005' ORDER BY year,layer; --7,23 SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='2000' ORDER BY year,layer; --3,17 SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='1995' ORDER BY year,layer; --1(1),8(1) SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='1990' ORDER BY year,layer; --2(2),5(2) SELECT * FROM DisplayHullsFull WHERE place='Portland' and statecode='OR' AND year='1985' ORDER BY year,layer; --1(1), 1(1)