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{{AcademicPaperThis paper is published as: |Has title=Urban Start-up Agglomeration and Venture Capital Investment[[Delineating Spatial Agglomerations|Has author=Ed Egan,Jim Edward J. and James A. Brander|Has RAs=Peter Jalbert(2022), Jake Silberman"New Method for Identifying and Delineating Spatial Agglomerations with Application to Clusters of Venture-Backed Startups.", Journal of Economic Geography, Christy WardenManuscript: JOEG-2020-449.R2, Jeemin Simforthcoming.]] {{AcademicPaper|Has paper statustitle=Working paper}}Urban Start-up Agglomeration and Venture Capital Investment|Has author=Ed Egan,Jim Brander|Has RAs=Working PaperPeter Jalbert, Jake Silberman, Christy Warden, Jeemin Sim|Has paper status=Published}} The last =New Submission= A revised version of the paper, now co-authored with [[Jim Brander]] and based on the Houston narrative as version 3 rebuild, was submitted to the motivationJournal of Economic Geography. This is solely a methods paper, and is available from SSRNtitled: https://papers'''A New Method for Identifying and Delineating Spatial Agglomerations with Application to Clusters of Venture-Backed Startups'''.ssrnThe policy application would need to be written up as a separate paper.com/abstract ==Acceptance==3537162 The Management Science submission version has a more conventional front end and is as followsOn July 5th 2022, the paper was accepted to the Journal of Economic Geography: <pdf>File:AgglomerationV8* Manuscript ID JOEG-2020-Reduced449.pdf</pdf>R2* Title: A new version, written by Jim, is in the works!New Method for Identifying and Delineating Spatial Agglomerations with Application to Clusters of Venture-Backed Startups* Author(s): Edward J. Egan and James A. Brander.=New Work=* Editor: Bill Kerr, HBS: wkerr@hbs.edu* Abstract: ===Another list of items=== Jim asked for the following This paper advances a new approach using hierarchical cluster analysis (in order of delivery schedule, not importanceHCA):for identifying and#A dataset and STATA do file delineating spatial agglomerations and applies it to implement table 5, complete with an exploration of which regressors to includeventure-backed startups. HCA identifies nested clusters at varying aggregation levels. We describe two methods for selecting a#An implementation of particular aggregation level and the 'real elbow associated agglomerations. The “elbow method” relies entirely on geographic information. Our preferred method', then integration with (1)the “regression method”, also uses venture capital investment data and identifies finer agglomerations, often the size of a small neighborhood.We use heat maps to illustrate how agglomerations evolve and indicate how our#A (set of) comparison(s) between the max R2 method and the elbow methodscan aid in evaluating agglomeration support policies.#A new heatmap or two, based on a different location* Permanent link for code/data: https://www.edegan.com/wiki/Delineating_Spatial_Agglomerations ====Implementing The paper is now in production. I will build a wiki page called [[Delineating_Spatial_Agglomerations]] that structures the documentation of the '''Real Elbow Method'''build process and shares code and some data or artifacts. Currently, that page redirects here. ==R&R == I calculated the between and withinFiles:*Pdf: [[File:Egan Brander (2020) -cluster variances, as described below, using the Euclidean distance by using the ST_Distance function on PostGIS geographies A New Method for Identifying and Delineating Spatial Agglomerations (i.e., accounting for an ellipsoid earth using reference system WGS1984Submitted to JEG). pdf]]* In E:\projects\agglomerationThe output of the python HCL clustering script has around 40m observations (place** Last document was Agglomeration Dec 15.docx** Build is Version 3-6-2-statecode2. ** SQL file is: AgglomerationVcdb4.sql After some inquiries, yearwe heard from Bill Kerr, layerthe associate editor, clusterthat the paper had new reviews on Aug 11th. On Aug 23rd, startup), and some we recieved an email titled "Journal of the intermediate tables took several minutes to buildEconomic Geography - Decision on Manuscript ID JOEG-2020-449" giving us an R&R. As Overall, the process should be O(n), this process could accommodate data that R&R is perhaps 100x to 1000x biggervery positive. Bill's comments:* Referees aligned on central issue of Census places* Too short: Wants application and suggests ("not contractual requirements"):** Diversity within and between in terms of types of VC investment (e.g., assuming a patient researcherBiotech vs. That would put an upper-bound at around 40b observations, as the hardware/software that we are running this on is pretty close to the ICT in Waltham)** Patent citiations made between VC backed firms Reviewer 1's comments (currentexcluding minor things) frontier.:* Explain projection (should have said it was WGS1984)=====Fixing an issue=====* Starting units: Suggests MSA level. Suppose cities that are close... can we find cases?* Identify clusters that have grown over time* Maybe try a cluster-level analysisThe within* Is ruling out the first second-difference too limiting? Can a city be a cluster variance ? (Vegas, baby?, Or starting from CMSA, probably yes in some sense.)* Discuss cluster boundaries (they aren't hard and so F-stat and variance explained) revealed an issue with the data that had to be fixedfast: The Python HCA script forces "think of these clusters as the decomposition kernels or seeds of multitons into singletons at the end of its run! We want VC-backedstartup hotspots") Reviewer 2's comments (excluding minor things):* Starting Units. Suggests MSA. * Explain R2 method better. He didn't say try cluster-level but that might be helpful to him too.* Change language (back) to stop the HCA when we have every location microgeographies! (or startup neighborhoods). * Tighter connection to lit. He gives papers to start.* Discuss overlap of clusters (a la patent clustering). Check findings in a separate point, rather than artificially forcing startups with the same location into separate pointsKerr and Kominers!!!* Discuss counterfactuals/cause-and-effect/application etc. This issue likely doesnShow/discuss that we didn't affect the maximum R2 method, but does affect the heuristic just find office parks. <pdf>File:JOEG1RndReviews.pdf</pdf>  ===Notes for further improvement=== We might want to add some things in/back in. These include technical notes:*To do the HCA we used the AgglomerativeClustering methodfrom the sklearn.cluster library (sversion 0.20.1) that rely on layer in python 3.7.1, with Ward linkage and connectivity set to none. This method is documented here: https://scikit-learn.org/stable/modules/clustering.html. I checked some of the early results against an implementation of Ward's method using the agnes function, available through the cluster package, in R. https://www.rdocumentation.org/packages/cluster/versions/2.1.0/topics/agnes*The data was assembled and processed in a Postgresql (version 10) database using PostGIS (version 2.4). We used World Geodetic System revision 84, known as WGS1984 (see https://en.wikipedia.org/wiki/World_Geodetic_System), as a coordinate system with an ellipsoidal earth, to calculate distances and areas (see https://postgis.net/docs/manual-2.4/using_postgis_dbmanagement.html). Shapefiles for Census Places were retrieved from the U.S. Census TIGER (Topologically Integrated Geographic Encoding and Referencing) database (see https://www.census.gov/programs-surveys/geography.html).*The statistical analysis was done in STATA/MP version 15.*All maps were made using QGIS v3.8.3. The base map is from Google Maps. City areas are highlighted using U.S. Census TIGER/Line Shapefiles.  The methodology has other applications:*Food deserts - one could study the agglomerations of restaurants and other food providers in urban environments.*Airports, cement factories, banana plantations, police/fire stations, hospitals/drug stores, etc.*We could think about commercial applications. Perhaps locating plants/facilities that are/aren't in clusters with a view to buying or selling them? =SSRN version of the paper (uses v2 build)= There are two 'final' papers based on the version 2 build. The one with Houston narrative as the motivation is available from SSRN: https://papers.ssrn.com/abstract=3537162 The Management Science submission version has a more conventional front end and is as follows: <pdf>File:AgglomerationV8-Reduced.pdf</pdf> =Version 3 Rebuild= ===Another round of refinements=== #The elbow method has issues in its current form, so we are going to try using the elbow in the curvature (degree of concavity) instead. #We might also try using elasticities...#Rerun the distance calculations -- avghulldisthm and avgdisthm are only computed for layers that we select with some method (like max r2). However, this table hadn't been updated for the elbow method, perhaps as well as some other methods, so some distances would have been missing (and replaced with zeros in the STATA script).#Create and run the new max R2 layer. In this variant, we'll use "the first layer a cluster number is reached as the representative layer for that cluster number"  I built two new curvature based elbow methods and so variables: curvaturelayer and curvaturelayerrestricted. They use the method described below and are identical except that curvaturelayerrestricted can't select layer 2 (both can't select the first and last layers as they use central second differences). For the example cities we have:{| class="wikitable" style="vertical-align:bottom;"|-! place! statecode! year! numstartups! elbowlayer! finallayer! curvaturelayer|-| Menlo Park| CA| 2,006| 68| 4| 51| 4|-| San Diego| CA| 2,006| 220| 3| 184| 181|-| Campbell| CA| 2,006| 38| 3| 26| 8|-| Charlotte| NC| 2,006| 30| 3| 30| 28|-| Waltham| MA| 2,006| 106| 3| 58| 55|} For these city-years, the curvaturelayer is the same as the curvaturelayerrestricted. As you can see, it is all over the place! I really don't think we can say that this method 'works' for any real value of 'works'. There's a sheet (Curvature Raw Data Examples) in ResultsV3-6.xlsx, and there's graphs for the selected cities on sheet "Elbow Curvature Selected Cities". ====New MaxR2 Layer==== I noticed a copy and paste error in the do file and I re-ran the existing max R2 method too, just to be sure. My process for the new method uses the code for the old chosenhullflayer variable. Key variables are:*firstlayer the layer at which numclusters first achieves that value*regfirst an indicator to select the right set of layers to run the max r2 estimation on*chosenhullflayer - the variable that records the layer number selected using firstlayer and the max r2 method*besthullflayer - the equivalent to besthulllayer but with the first layers instead of the lowest-highest ones*targetnumclustersf, besthullflayerisadded, maxr2flayerflag, etc.*'''regmaxr2f''' and '''regbestf''' - these are the dataset constraints to use. Everything is pushed through the database and back to generate them. The results for our sample cities are as follows:{| class="wikitable" style="vertical-align:bottom;"|-! place! statecode! year! finallayer! chosenhulllayer! style="font-weight:bold;" | chosenhullflayer! elbowlayer|-| Campbell| CA| 2,006| 26| 15| style="font-weight:bold;" | 3| style="font-weight:bold;" | 3|-| Charlotte| NC| 2,006| 30| 14| style="font-weight:bold;" | 3| style="font-weight:bold;" | 3|-| Menlo Park| CA| 2,006| 51| 33| style="font-weight:bold;" | 21| style="font-weight:bold;" | 4|-| San Diego| CA| 2,006| 184| 141| style="font-weight:bold;" | 12| style="font-weight:bold;" | 3|-| Waltham| MA| 2,006| 58| 31| style="font-weight:bold;" | 3| style="font-weight:bold;" | 3|} I build the max R2 graphs in the sheet '''New MaxR2''' in ResultsV3-6.xlsx ====Jim's notes on the curvature==== Suppose we have a function f. Then what I have been calling the curvature is -f’’/f’. If f is a utility function this is the coefficient of absolute risk aversion and it has quite often been called curvature in that context. However, in differential geometry curvature is described differently, although it is quite similar. Mas-Collel and others have suggested calling -f’’/f’ the “degree of concavity” instead. I came across this definition on the internet: :“The degree of concavity is measured by the proportionate rate of decrease of the slope, that is, the rate at which the slope decreases divided by the slope itself.” The general rationale for using this measure is that it is invariant to scale, whereas the straight second derivative, f’’, is not invariant. The same applies to the second difference of course." So our measure is the second difference divided by the first difference. However, it is not clear whether we should divide by the initial first difference or the second first difference or the average. I initially assumed that we should use the initial first difference. I now think that is wrong as it can produce anomalies. I think we should use the second (or “current”) first difference as the base.  Here is some data I sent before: {| class="wikitable" style="text-align:right;"|- style="background-color:#FFF; color:#222;"| Layer| SSR| D1| D2| Concavity| Concavity|- style="text-align:left; background-color:#FFF; color:#222;"| style="text-align:right;" | 1| style="text-align:right;" | 0| style="vertical-align:bottom;" | | style="vertical-align:bottom;" | | style="vertical-align:bottom;" | | style="vertical-align:bottom;" | |- style="background-color:#FFF; color:#222;"| 2| 40| 40| -5| 0.13| 0.14|- style="background-color:#FFF; color:#222;"| 3| 75| 35| style="background-color:#FF0;" | -20| 0.57| 1.33|- style="background-color:#FFF; color:#222;"| 4| 90| 15| -12| style="background-color:#FF0;" | 0.80| style="background-color:#FF0;" | 4|- style="background-color:#FFF; color:#222;"| 5| 93| 3| -1| 0.33| 0.5|- style="background-color:#FFF; color:#222;"| 6| 95| 2| -1| 0.50| 1|-| style="background-color:#FFF; color:#222;" | 7| style="background-color:#FFF; color:#222;" | 96| style="background-color:#FFF; color:#222;" | 1| style="background-color:#FFF; color:#222;" | -1| style="background-color:#FFF; color:#222;" | 1.00| style="text-align:left;" | |} The column at the far right uses the second first difference as the base, which I now think is correct. The column second from the right uses the first first difference at the base. Just to be clear, for layer 2 the first difference is 40 – 0 = 40. For layer 3 the first difference is 75 – 40 = 35. Therefore, for layer 2, the second difference is 35 – 40 = -5. I think this is what you would call the “middle second difference”. It tells how sharply the slope falls after the current layer, which is what we want. To correct for scaling, we need to divide by a first difference. In the first concavity column, for layer 2 I use 5/40 = 0.125. For the last column for layer 2 I use 5/35 = 0.143. Both approaches have a local max at layer 4, which is what we want. However, the second column from the right has a global max at the last layer, which is certainly not what we want. But is can happen at the end where the increments are very small. 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.  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. But seeing as we can't use the first observation we've gained nothing anyway! So we'll do Jim's method verbatim, and declare the result null if it comes out as either the first or last layer. ====Curvature==== {{Colored box|title=Specification|content=For layer <math>l</math>, I compute the curvature as -1 times the backward first difference in the variance explained ratio from layer <math>l+1</math> divided by the central second difference in the variance explained ratio from <math>l</math>. The first and last layers are forbidden results.}} The curvature results seem somewhat better than the elbow results but are still far from ideal. Here are some things I look for and/or don't like in a layer selection method:*Interior solutions are good, collapsing to the bounds, especially the lower bound is bad*Stable interior solutions are better - when the results approximate a quadratic so that margins generally decrease and then increase around a maximum, the interior results are stable and that's very desirable*Consistent solutions are good within cities - it's nice when adjacent years in the same city have more or less the same layer selected*Consistent solutions across cities are also good - When the method picks roughly similar layer indicies (i.e., % unclustered) across cities, particularly conceptually similar cities, that's a plus*From other analysis, I know that the equilibrium of agglomeration forces occurs when agglomerations have fairly small average hull sizes, perhaps on the order of 10hm2. ===Version 3.5 build notes=== In the process of building version 3.5, I noticed a discrepancy between tothulldensity and avghulldensity. This turned out to be correct. Both are measured in startups/hm2. Tothulldensity of the sum of the number of startups in hulls divided by the total hull area, whereas the avghulldensity is the average of the hull densities (computed as the number of startups in the hull divided by the hull area).  The revised script and dataset is v3-5. ResultsV3-5.xlsx has all of the old redundant results removed and has new sheets for Descriptives (copied over with renamed column names from Descriptives.xlx, which is generated by the .do file), as well as for the new scatterplot. Its Bar and Whisker is also stripped down to the bare essentials.  ===Heuristic Layer=== [[File:AgglomerationInflectionScatterPlotAllDataCircles.png|500px|right]] I had previously calculated the heuristic layer by calculating the mean fracinhull (i.e., % of startups in economic clusters) for each percentage of the layer index (i.e., for 101 observations) and then fitting a cubic to it. I did this because excel can't handle fitting a cubic to the full data (i.e., all 148,556 city-year-layers). However, it is incorrect because of orthogonality issues in calculating mean square distances (I'm also unsure that the mean would be the best measure of central tendency). So I redid the plot using all the data, and calculated the cubic in STATA instead. See: '''inflection.do''' and '''inflection.log'''. The old result is in [https://www.edegan.com/wiki/Urban_Start-up_Agglomeration_and_Venture_Capital_Investment#Fixing_an_issue Fixing an issue] below, and is x≈0.483879. The corrected result is x≈0.487717 (note that R2 has dropped to 92.43%): :2.737323 x^3 - 4.005114 x^2 + 0.3262405 x + 0.9575088≈0.481497 at x≈0.487717 [https://www.wolframalpha.com/input/?i=inflection+points+2.737323+x3+-+4.005114x2++%2B+0.3262405x+%2B+0.9575088] I also calculated an '''inflectionlayer''' (as opposed to the heurflhlayer, where flh stands for fraction of locations in hulls, described above). This inflectionlayer is '''the first time''' that the second central difference in the '''share of startups in economic clusters''' switches sign. It is only possible to calculate this when there are at least 4 data points, as the central difference requires data from layer-1, layer and layer+1, and we need two central differences. The variable is included in dataset (and so do files, etc.) version 3-4 forwards. However, the inflectionlayer is really meaningless. The sign of the second central switches back and forward due to integer effects and I can't find a straight forward algorithm to pick the "correct" candidate from the set of results. Picking the '''first one''', which I currently pick, is completely arbitrary. There are a bunch of examples of the curves and the issue(s) in Results3-4.xlsx sheet 'Inflection'. I expect that if I put a bunch of time into this I could come up with some change thresholds to rule candidate answers in or out, but even then this isn't a good method.  Ultimately, the individual city-year inflection curves (i.e., across layers within a city-year) are just way too noisy. A variant of this noise problem is what makes the elbow method so problematic, but the noise is even worse with the inflection method. Using the heuristic result above (i.e., the one using all city-years) solves this noise problem by aggregating city-years together. One complaint made about the heuristic results is that it is near the middle (i.e., it's 48.7717%, which happens to be near 50%). Although the nature of any HCA on geographic coords implies that the result is unlikely to the close to the bounds (0 or 100%) and more likely to be near the middle (50%), it could be in an entirely different place. '''This result (i.e., the heuristic layer at 48.7717%) characterizes the agglomeration of venture-backed startup firms'''. You'd get a very different number if you studied gas stations, supermarkets, airports, or banana plantations! ====Comparing the Heuristic and R2 Layers==== {{Colored box|title=The Case for the Heuristic Method|content=The heuristic method (i.e., using the inflection in the plot from the population of city-year-layers) finds pretty much the same layer as the R2 method with almost no work, and it can be used in a within-city analysis without having to hold hull count constant.}}  . tabstat nohull tothullcount tothullarea tothulldensity growthinv18 numdeals numstartups if regmaxr2==1, stats(p50 > mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 3.531407 7.07922 2977 1 68 1 6 tothullcount | 8 17.4565 35.65118 2977 3 380 3 30 tothullarea | 14.76523 448.029 2063.824 2977 .0049029 34780.04 .5275311 732.4005 tothullden~y | .7640136 11.32988 63.62256 2977 .0002282 1425.338 .0115537 16.15439 growthinv18 | 33.53101 142.5 561.6696 2977 0 22282.6 1.53118 309.0208 numdeals | 3 6.71347 17.06682 2977 0 275 0 15 numstartups | 16 41.28955 89.98027 2977 6 1317 7 90 ---------------------------------------------------------------------------------------------- . tabstat nohull tothullcount tothullarea tothulldensity growthinv18 numdeals numstartups if regheur1==1, stats(p50 > mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 4.279958 8.433203 3797 0 119 1 9 tothullcount | 8 20.08954 42.99372 3797 0 673 3 43 tothullarea | 11.32983 49.42803 158.7375 3797 0 2569.169 1.660208 93.94627 tothullden~y | .946713 3.48483 10.93185 3797 0 212.8198 .06182 7.601018 growthinv18 | 31.8453 133.0608 508.1196 3797 0 22282.6 1.235763 292.4397 numdeals | 2 6.629181 16.46614 3797 0 275 0 15 numstartups | 15 38.74743 83.6814 3797 6 1317 7 83 ---------------------------------------------------------------------------------------------- Analyzing layers: Method Avg. Layer Index Std. Dev Layer Index Max R2 0.392473192 0.2380288695 Heuristic 0.43423652 0.0495630531  '''The Max R2 and Heuristic layers are identical in 12.6% of cases!''' Some of these cases are found in city-years with a large number of layers, for instance, there are 90 city-years that have more than 20 startups and identical heuristic and max r2 layers. The table below shows city-years with more than 50 startups and identical heuristic and max R2 layers: {| class="wikitable" |- style="font-weight:bold;"! place! statecode! year! numstartups! chosenhulllayer! heurflhlayer|-| San Francisco| CA| 2,009| 503| 175| 175|-| Los Angeles| CA| 2,012| 213| 93| 93|-| Redwood City| CA| 2,012| 151| 49| 49|-| Redwood City| CA| 2,013| 151| 49| 49|-| Seattle| WA| 2,000| 113| 48| 48|-| Houston| TX| 2,007| 92| 40| 40|-| Waltham| MA| 2,012| 73| 24| 24|-| Pittsburgh| PA| 2,008| 70| 25| 25|-| Bellevue| WA| 2,001| 64| 25| 25|-| Bellevue| WA| 2,003| 61| 23| 23|-| Pleasanton| CA| 2,004| 54| 20| 20|-| Menlo Park| CA| 2,004| 52| 22| 22|-| Durham| NC| 2,009| 50| 22| 22|} In fact, 84% of city-years (which have both heuristic and max R2 layers) have heuristic and max R2 layers that are separated by less than or equal to 5 layers, and 59% have them separated by less than or equal to 2 layers! '''More than a third (36.3%) of city-years have their heuristic and max R2 layers separated by less than or equal to 1 layer.''' ===Another list of items=== Jim asked for the following (in order of delivery schedule, not importance):#A dataset and STATA do file and to implement table 5, complete with an exploration of which regressors to include#An implementation of the 'real elbow method', then integration with (1).#A (set of) comparison(s) between the max R2 method and the elbow methods#A new heatmap or two, based on a different location. All done... see the sections below. ====Heatmaps==== I built '''unbuffered heatmaps using maximum R2 layers from 1995 to 2018''' for a set of "interesting" cities. I often built the same city at multiple scales. Only the zoomed-in maps are in the gallery below. I can now quite quickly build more cities if needed. It is worth noting the following:*Because we are using unbuffered hulls, heatmap components are angular and non-diffuse.*Agglomerations are smaller in cities with higher startup counts but are small everywhere. *Agglomerations don't come close to overlapping city boundaries. Agglomerations within Palo Alto don't overflow into Mountain View and it isn't meant meaningful to talk about Boston-Cambridge agglomerations, except as a broad set. An agglomeration is typically a few square blocks (we knew this from the mean and median hull sizes). *Some famous policy interventions appear to have no effect. There is no agglomeration, let alone a concentration of them, in Boston's North End, where hundreds of millions were plowed into a TIF (and MassChallenge). <gallery widths=300 heights=300>File:Bellevue125000MaxR2UnbufferedHeatmap.png| Bellevue, WA, 1:125kFile:PaloAlto50000MaxR2UnbufferedHeatmap.png| Palo Alto, CA, 1:50kFile:Boulder50000MaxR2UnbufferedHeatmap.png| Boulder, CO, 1:50kFile:Waltham65000MaxR2UnbufferedHeatmap.png| Waltham, MA, 1:65kFile:Boston50000MaxR2UnbufferedHeatmap.png| Boston, MA, 1:50k</gallery> I also built three buffered heatmaps of Boston as a proof of concept. I used either the average distance between the points on the edge of the hull and the centroid, or half of it, as a buffering distance. I also varied the intensity of the shading (down to 10% per layer from 20% in the 1:70000 image). Boston should have 17 agglomerations according to the maximum R2 method, so the half distance buffer might be best for picking them out. <gallery widths=300 heights=300>File:Boston70000MaxR2buffered1xHeatmap.png| Boston, MA, 1:70k, 1x buffer, 10% opacityFile:Boston50000MaxR2bufferedHalfxHeatmap.png| Boston, MA, 1:50k, 0.5x buffer, 20% opacityFile:Boston50000MaxR2buffered1xHeatmap.png| Boston, MA, 1:50k, 1x buffer, 20% opacity</gallery> ====Comparing the Methods==== Summaries of the meta-data on geometries created by each lens is probably the best method of comparison. These are in the do file: . //Compare how their lenses look: . tabstat nohull tothullcount tothullarea tothulldensity growthinv18 numdeals numstartups if regmaxr2==1, stats(p50 > mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 3.531407 7.07922 2977 1 68 1 6 tothullcount | 8 17.4565 35.65118 2977 3 380 3 30 tothullarea | 14.76523 448.029 2063.824 2977 .0049029 34780.04 .5275311 732.4005 tothullden~y | .7640136 11.32988 63.62256 2977 .0002282 1425.338 .0115537 16.15439 growthinv18 | 33.53101 142.5 561.6696 2977 0 22282.6 1.53118 309.0208 numdeals | 3 6.71347 17.06682 2977 0 275 0 15 numstartups | 16 41.28955 89.98027 2977 6 1317 7 90 ---------------------------------------------------------------------------------------------- . tabstat nohull tothullcount tothullarea tothulldensity growthinv18 numdeals numstartups if regheur1==1, stats(p50 > mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 4.305768 8.498714 3797 0 119 1 9 tothullcount | 8 20.27153 43.51017 3797 0 675 3 43 tothullarea | 11.43336 49.81455 159.0983 3797 0 2569.169 1.661926 94.14735 tothullden~y | .9422739 3.452804 10.89704 3797 0 212.8198 .06182 7.47113 growthinv18 | 31.8453 133.0608 508.1196 3797 0 22282.6 1.235763 292.4397 numdeals | 2 6.629181 16.46614 3797 0 275 0 15 numstartups | 15 38.74743 83.6814 3797 6 1317 7 83 ---------------------------------------------------------------------------------------------- . tabstat nohull tothullcount tothullarea tothulldensity growthinv18 numdeals numstartups if regelbow==1, stats(p50 > mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 3.152638 3.973168 3374 0 48 0 7 tothullcount | 12 37.32721 87.6828 3374 0 1303 0 91 tothullarea | 55.78572 898.919 3849.938 3374 0 74067.25 0 1589.324 tothullden~y | .169715 31.79024 1726.042 3374 0 100257.3 0 1.841935 growthinv18 | 36.56511 146.5069 537.3532 3374 0 22282.6 1.816288 326.6357 numdeals | 3 7.232662 17.32508 3374 0 275 0 16 numstartups | 17 42.33225 88.08184 3374 6 1317 7 98 ---------------------------------------------------------------------------------------------- Another way to compare the methods is to look at the layers they select. This is visible in a box plot as well as summary statistics. The following is from ResultsV3.xlsx '''Separate Samples (All Available City-Years)'''{| class="wikitable" style="vertical-align:bottom;"|- style="font-weight:bold;"! Method! N! Avg Layer Index! Std. Dev Layer Index|-| Maximum R2| 3080| 39%| 18%|-| Startups in Clusters Inflection| 6,743| 35%| 16%|-| Variance Explained Elbow| 4,799| 43%| 30%|} '''Using Common City-Years'''{| class="wikitable" style="vertical-align:bottom;"|- style="font-weight:bold;"! Method! N! Avg Layer Index! Std. Dev Layer Index! L Index < Peak! L Index < Max R2! X < Max R2|-| Maximum R2| 2662| 40%| 24%| 0| 2662| |-| Startups in Clusters Inflection| 2662| 44%| 5%| 1102| 167| 6%|-| Variance Explained Elbow| 2662| 31%| 22%| 53| 297| 11%|} [[File:BoxPlot.PNG|frame|600px]] Finally, we can look at a city where different methods select different layers and look at those layers. Here's Cincinnati, Ohio, in 2018: <gallery widths=300 heights=300>File:Cincinnati2018_Level16of25(MaximumR2).png| Layer:16/25 MaxR2File:Cincinnati2018_Level12of25(FractionInHullsInflection).png| Layer:12/25 InflectionFile:Cincinnati2018_Level8of25(VarianceExplainedElbow).png| Layer:8/25 Elbow</gallery> ====Implementing the '''Real Elbow Method'''==== I calculated the between and within-cluster variances, as described below, using the Euclidean distance by using the ST_Distance function on PostGIS geographies (i.e., accounting for an ellipsoid earth using reference system WGS1984).  The output of the python HCL clustering script has around 40m observations (place-statecode, year, layer, cluster, startup), and some of the intermediate tables took several minutes to build. As the process should be O(n), this process could accommodate input data that is perhaps 100x bigger, assuming a patient researcher, which would imply source data perhaps 10x bigger. Note that the hardware/software that we are running this on is pretty close to the (current) frontier. =====Fixing an issue===== [[File:HeuristicLayerSelectionGraphv3-1.PNG|400px|right]]The within-cluster variance (and so F-stat and variance explained) revealed an issue with the data that had to be fixed: The Python HCA script forces the decomposition of multitons into singletons at the end of its run! We want to stop the HCA when we have every location in a separate point, rather than artificially forcing startups with the same location into separate points. This issue likely directly affects the heuristic method(s) that rely on layer indicesand indirectly (by changing observation counts) affects the maximum r2 layer choice. I pushed through the change and reran everything. It is build '''version 3.1''', and includes a new .do file, new .txt data files, and a new .log file.  The new elbow layer is: 2.5795 x^3 - 3.7445 x^2 + 0.1989 x + 0.9808≈0.492554 at x≈0.483879 [https://www.wolframalpha.com/input/?i=inflection+points+2.5795x3+-+3.7445x2++%2B+0.1989x+%2B+0.9808].  {{Colored box|title=NOTICE|content= The results in the section below are outdated! The updated results are similar but not the same. Do not rely on the results on the wiki page. '''Always check the log file in the dropbox (or in E:\projects\agglomeration) for the latest results!'''}} =====Trying to find the elbow===== The objective is to apply the [https://en.wikipedia.org/wiki/Elbow_method_(clustering) Elbow Method], which involves finding the [https://en.wikipedia.org/wiki/Knee_of_a_curve Knee of the curve] of either the F-statistic or variance explained. I used distances calculated by ST_Distance and calculated the '''variance explained''' using the following equations: :<math>SS_{exp}=\sum_{i=1}^{K} n_i(\bar{Y}_{i\cdot} - \bar{Y})^2</math>:<math>SS_{unexp}=\sum_{i=1}^{K}\sum_{j=1}^{n_{i}} \left( Y_{ij}-\bar{Y}_{i\cdot} \right)^2</math>:<math>R^2 = \frac{SS_{exp}}{SS_{exp}+SS_{unexp}}</math> I then calculated forward differences, and added one to the answer, as using central differences left truncates the data. (An inspection of the data revealed that it is vastly more likely that the 'correct' answer is found at the left end of the data than the right. Also central first difference bridge the observation, which can lead to misidentification of monotonicity.) Specifically, I used: :<math> f'(x) = f(x + 1) - f(x) </math>:<math> f''(x) = f(x+2) - 2 f(x+1) + f(x)</math> See https://en.wikipedia.org/wiki/Finite_difference 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>). I used <math>f''(x)</math> to find the layer index <math>i</math> at which <math>varexp_i = min(varexp)</math> (for elbowlayer) or for which <math>varexp_i = max(varexp)</math> (for elbowmaxlayer), for some city-year. I then marked <math>i+1</math> as the elbow (or elbowmax) layer for that city-year, as we are using forward differences, not central differences. Note that the biggest change in slope could be found using max(abs(f''(x))) but this is essentially always min(f''(x)), i.e., the elbow layer, as the change in slopes are mostly negative. However, the changes in slopes do often go positive, and the elbowmax layer captures the biggest positive change in slope.

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