Changes

This paper is published as:  [[Delineating Spatial Agglomerations|Egan, Edward J. and James A. Brander (2022), "New Method for Identifying and Delineating Spatial Agglomerations with Application to Clusters of Venture-Backed Startups.", Journal of Economic Geography, Manuscript: JOEG-2020-449.R2, forthcoming.]] {{AcademicPaper|Has title=Urban Start-up Agglomeration and Venture Capital Investment|Has author=Ed Egan,Jim Brander|Has RAs=Peter Jalbert, Jake Silberman, Christy Warden, Jeemin Sim|Has paper status=Working paperPublished}} =Working PaperNew Submission= The last 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 Method for Identifying and Delineating Spatial Agglomerations with Application to Clusters of Venture-Backed Startups* Author(s): Edward J. Egan and James A new version. Brander.* Editor: Bill Kerr, written by Jim, is in the works!HBS: wkerr@hbs.edu* Abstract: =New Work= This paper advances a new approach using hierarchical cluster analysis (HCA) for identifying and delineating spatial agglomerations and applies it to venture-backed startups. HCA identifies===To Do=== nested clusters at varying aggregation levels. We describe two methods for selecting a particular aggregation level and the associated agglomerations. The “elbow method” reliesItems:#Rerun the code with just entirely on geographic information. Our preferred method, the 8 geometry variables“regression method”, also uses venture capital investment data and three factorsidentifies finer agglomerations, in often the maximum R2 selectionsize of a small#Write a paragraph justifying the 'heuristic' method neighborhood.#Write a couple of sentences about Guzman We use heat maps to illustrate how agglomerations evolve and Stern#Explore if/indicate how we could implement the variance-based elbow methodour methods can aid in evaluating agglomeration support policies.I also wanted to fix confusion between CSAs (Combined Statistical Areas)[https* Permanent link for code/data: https://enwww.wikipediaedegan.orgcom/wiki/Combined_statistical_areaDelineating_Spatial_Agglomerations The paper is now in production. I will build a wiki page called [[Delineating_Spatial_Agglomerations]] that structures the documentation of the build process and CMSAs (Consolidated Metropolitan Statistical Areas)shares code and some data or artifacts. Currently, that page redirects here. == R&R == Files:*Pdf: [[httpsFile://www2.census.gov/geo/pdfs/reference/GARM/Ch13GARMEgan Brander (2020) - A New Method for Identifying and Delineating Spatial Agglomerations (Submitted to JEG).pdf]]* In E:\projects\agglomeration** Last document was Agglomeration Dec 15. CMSA redirects to CSA on Wikipedia. However, it docx** Build is actually not clear if these are the same thingsVersion 3-6-2-2. OMB ** SQL file is the originator of both terms[https://wwwAgglomerationVcdb4.census.gov/programssql After some inquiries, we heard from Bill Kerr, the associate editor, that the paper had new reviews on Aug 11th. On Aug 23rd, we recieved an email titled "Journal of Economic Geography - Decision on Manuscript ID JOEG-surveys/metro2020-micro/about/masrp449" giving us an R&R. Overall, the R&R is very positive.html 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., Biotech vs. ICT in Waltham)** Patent citiations made between VC backed firms Reviewer 1's comments (excluding minor things):* Explain projection (should have said it was WGS1984)* 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 analysis* Is ruling out the first second-difference too limiting? Can a city be a cluster? (Vegas, baby?, Or starting from CMSA, probably yes in some sense.)* Discuss cluster boundaries (they aren't hard and fast: "think of these clusters as the kernels or seeds of 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 microgeographies! (or startup neighborhoods). * Tighter connection to lit. He gives papers to start.* Discuss overlap of clusters (a la patent clustering). Check findings in Kerr and Kominers!!!* Discuss counterfactuals/cause-and-effect/application etc. Show/discuss that we didn't 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 method from the sklearn.cluster library (version 0.20.1) 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 indices and 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. '''I created a new build (version 3.3) of the dataset, do file and log file, which includes the population variance explained elbow method, as well as the elbowmax method. It's in the dropbox.'''.  Note that the lens found by the population elbow method is slightly bigger than the lenses found using sample elbow method from before, but the lens found using the elbowmax method is about the same size as the sample elbow method, if not slightly smaller. I'm not sure about the justification of the elboxmax method though. ====Fixing the layer index==== I checked the implementation of the % layer index again, and fixed a mistake in it. In Heurcalc, I define fracunclusted as: CASE WHEN finallayer >1 THEN ((layer::real-1)/(finallayer::real-1)) ELSE NULL END AS fracunclustered, I then previously select the layer below or equal to % layer index 0.374833 in Heurflh: CASE WHEN floor((finallayer-1)*0.374833)=0 THEN 1::int ELSE floor((finallayer-1)*0.374833) END AS heurflhlayer This was incorrect. With some rearrangement, we can see:<math>index \le 0.374833 \implies \frac{layer-1}{final-1} \le 0.374833 \implies layer \le 0.374833(final-1) + 1 </math> I therefore fixed the query to: floor(((finallayer-1)*0.374833)+1) AS heurflhlayer I've updated the results in the New Do File section, and the two methods are now much closer to giving the same lens (see the descriptives!). ====New Do File==== There's a new do file, dataset and log file in the dropbox. The do file is reorganized and condensed into a single file. In order to select layers for specific purposes, like regressions, the do file uses flags (defined on lines 165-190). Do not try to run regressions using xt commands even with these flags, as the underlying layers used will be incorrect. Instead, put in the fixed effects yourself (using i.var) and put them at the front of the regression. STATA decides which variables to omit before doing the linear algebra, so you'll be able to get non-zero coefficients on omitted variables if you put them ahead of the fixed effects that would wipe them out. The actual regressions I chose are on lines 243-250. There have the log of next period growth VC per startup as the dependent variable. Each spec uses year and city fixed effects, and clusters the standard errors at the city level. However, only the second spec in each sequence uses explicit scale controls.  '''Here's extracts of the results for Max R2:''' reg growthinv18lfperstartup i.year i.placeid nohull tothullcountl tothullareal tothulldensityl avghulldisthml if regmaxr2==1, cluster(placeid) nohull | 0 (omitted) tothullcountl | -.0731116 .0170361 -4.29 0.000 -.1067484 -.0394748 tothullareal | .0166992 .0032354 5.16 0.000 .0103111 .0230873 tothulldensityl | -.0026603 .0034181 -0.78 0.438 -.0094092 .0040885 avghulldisthml | .0025771 .0060088 0.43 0.669 -.0092869 .0144411 reg growthinv18lfperstartup i.year i.placeid nohull tothullcountl tothullareal tothulldensityl avghulldisthml growthinv18l numstartupsl numdealsl if regmaxr2==1, cluster(placeid) nohull | .0045458 .0007755 5.86 0.000 .0030165 .0060752 tothullcountl | .0042399 .0087493 0.48 0.629 -.0130145 .0214943 tothullareal | .0045121 .0042544 1.06 0.290 -.0038778 .0129021 tothulldensityl | .0163778 .0048519 3.38 0.001 .0068095 .0259462 avghulldisthml | -.0024717 .0026637 -0.93 0.355 -.0077247 .0027813 growthinv18l | .0075835 .0025118 3.02 0.003 .0026301 .012537 numstartupsl | -.1492758 .016447 -9.08 0.000 -.1817106 -.1168411 numdealsl | .0121621 .005014 2.43 0.016 .002274 .0220501 '''Here's extracts of the results for the 1st Heuristic (Percentage of startups in clusters):''' reg growthinv18lfperstartup i.year i.placeid nohull tothullcountl tothullareal tothulldensityl avghulldisthml if regheur1==1, cluster(placeid) nohull | .0029323 .0007413 3.96 0.000 .0014704 .0043941 tothullcountl | -.0526231 .0113923 -4.62 0.000 -.0750896 -.0301567 tothullareal | .0155028 .0041201 3.76 0.000 .0073776 .023628 tothulldensityl | .0057162 .0040038 1.43 0.155 -.0021797 .0136121 avghulldisthml | -.0022889 .0029852 -0.77 0.444 -.0081759 .003598 reg growthinv18lfperstartup i.year i.placeid nohull tothullcountl tothullareal tothulldensityl avghulldisthml growthinv18l numstartupsl numdealsl if regheur1==1, cluster(placeid) nohull | .0047632 .0007554 6.31 0.000 .0032736 .0062528 tothullcountl | .0008837 .012177 0.07 0.942 -.0231303 .0248977 tothullareal | .0011863 .0042324 0.28 0.780 -.0071603 .0095329 tothulldensityl | .0067998 .0043843 1.55 0.123 -.0018464 .015446 avghulldisthml | -.0063252 .0028367 -2.23 0.027 -.0119194 -.0007311 growthinv18l | .0076041 .0024995 3.04 0.003 .0026749 .0125333 numstartupsl | -.1450849 .0158493 -9.15 0.000 -.176341 -.1138289 numdealsl | .0124877 .0050118 2.49 0.014 .002604 .0223714 Lines 252 to 255 of the do file also compare the lense given by the max r2 method and the 1st heuristic method. The results are below. Note that the second spec shows the 1st heuristic for all city-years and the third spec shows it for the city-years that have a max R2 layer, to make a fairer comparison.  . 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 3.862758 7.279538 2951 1 67 1 7 tothullcount | 8 19.18841 36.62401 2951 3 369 3 37 tothullarea | 15.2049 431.5689 2067.431 2951 4.04e-06 34780.04 .7658609 606.8908 tothullden~y | .8115587 258.0355 13679.93 2951 .0002282 743141.7 .0123656 12.06713 growthinv18 | 33.16967 141.5326 563.4799 2951 0 22282.6 1.412436 298.6927 numdeals | 3 6.630973 17.17611 2951 0 275 0 14 numstartups | 16 41.2528 90.67289 2951 6 1317 7 87 ---------------------------------------------------------------------------------------------- . 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 4.352647 8.107042 3797 0 108 1 9 tothullcount | 9 20.60706 40.26376 3797 0 566 4 42 tothullarea | 18.88252 110.757 348.9474 3797 0 6079.538 1.523386 236.2767 tothullden~y | .5860988 6.013232 43.74094 3797 0 1429.742 .0277004 7.329074 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 '''regheur1==1 & compok==1''' , stats(p50 mean sd N min max p10 p90) columns(statistics) variable | p50 mean sd N min max p10 p90 -------------+-------------------------------------------------------------------------------- nohull | 2 4.581159 8.548611 2951 0 108 1 9 tothullcount | 9 21.68519 42.78684 2951 0 566 4 42 tothullarea | 19.43593 114.4633 360.3934 2951 0 6079.538 1.86059 252.267 tothullden~y | .5797117 6.835335 49.19501 2951 0 1429.742 .0314084 7.848711 growthinv18 | 33.16967 141.5326 563.4799 2951 0 22282.6 1.412436 298.6927 numdeals | 3 6.630973 17.17611 2951 0 275 0 14 numstartups | 16 41.2528 90.67289 2951 6 1317 7 87 ---------------------------------------------------------------------------------------------- ===A list of items=== Items:#Rerun the code with just the 8 geometry variables, and three factors, in the maximum R2 selection#Fix the implementation of %unclustered (now called %complete)#Compare the changed definitions and try some test regressions#Write a paragraph justifying the 'heuristic' method.#Write a couple of sentences about Guzman and Stern#Explore if/how we could implement the variance-based elbow method I also wanted to fix confusion between CSAs (Combined Statistical Areas)[https://en.wikipedia.org/wiki/Combined_statistical_area] and CMSAs (Consolidated Metropolitan Statistical Areas)[https://www2.census.gov/geo/pdfs/reference/GARM/Ch13GARM.pdf]. CMSA redirects to CSA on Wikipedia. However, it is actually not clear if these are the same things. OMB is the originator of both terms[https://www.census.gov/programs-surveys/metro-micro/about/masrp.html]. ====New implementations==== We're making two small but important changes to the implementations of:#The maximum R2 method (now just 8 metadata vars)#%Complete - now i-1/I-1 =====Max R2===== I ran the code and pushed the results through the database. The newly selected layers are generally the same or one lower than the previously selected layers. I expect that this will be just fine. =====Fraction Complete===== Fraction complete isn't a variable in MasterLayersv3-0. It is used on the fly in queries to assess the elbow layer and then generated anew in AnalysisVcdb4.do (though it isn't really used there): gen fracunclustered=layer/finallayer I fixed the following, and reran the analysis using fracunclustered = (layer-1)/(finallayer-1):*elbowcalc->heurcalc*elbowdata->heurdata*Elbowflh-heurflh [[File:HeuristicLayerSelectionGraphv2.PNG|right|500px]] Plotting the data from heurdata.txt in excel gives: 2.5944 x^3 - 2.9174 x^2 - 0.6407 x + 1.0304Wolfram alpha[https://www.wolframalpha.com/input/?i=inflection+points+2.5944x3+-+2.9174x2++-+0.6407x+%2B+1.0304] says the inflection point is now: x≈0.374833, y≈0.516982 Note that this is now the fraction complete is relative to finallayer-1. For reference, the old value was 0.429388 relative to finallayer. ====Implementing The Elbow Method==== This section explores whether we could implement the '''actual''' elbow method (see https://en.wikipedia.org/wiki/Elbow_method_(clustering) ). The answer is that we might be able to, at least for some sub-sample of our data, but that it likely doesn't give us what we want. =====Background===== The elbow method plots the number of clusters (on x) against the percentage of variance explained (on y) and finds the elbow. The elbow is the point at which the "diminishing returns [in variance explained] are no longer worth the additional cost [of adding another cluster]'. For the variance explained there are two main options:  #Variance explained = between-group variance / total variance#Variance explained = between-group variance / within-group variance (Note that this is the ANOVA F-statistic). Using the [https://en.wikipedia.org/wiki/Law_of_total_variance Law of total variance], total variance = between-group variance + within-group variance. '''From Wikipedia:''' :[https://en.wikipedia.org/wiki/F-test The F-test] in one-way analysis of variance is used to assess whether the expected values of a quantitative variable within several pre-defined groups differ from each other. The "explained variance", or "between-group variability" is :<math>\sum_{i=1}^{K} n_i(\bar{Y}_{i\cdot} - \bar{Y})^2/(K-1)</math> where <math>\bar{Y}_{i\cdot}</math> denotes the sample mean in the ''i''-th group, <math>n_i</math> is the number of observations in the ''i''-th group,<math>\bar{Y}</math> denotes the overall mean of the data, and <math>K</math> denotes the number of groups. The "unexplained variance", or "within-group variability" is :<math>\sum_{i=1}^{K}\sum_{j=1}^{n_{i}} \left( Y_{ij}-\bar{Y}_{i\cdot} \right)^2/(N-K),</math> where <math>Y_{ij}</math> is the ''j''^th observation in the ''i''^th out of <math>K</math> groups and <math>N</math> is the overall sample size. This ''F''-statistic follows the ''F''-distribution with degrees of freedom <math>d_1=K-1</math> and <math>d_2=N-K</math> under the null hypothesis. The statistic will be large if the between-group variability is large relative to the within-group variability, which is unlikely to happen if the population means of the groups all have the same value. '''Also from Wikipedia:''' [https://en.wikipedia.org/wiki/Variance Variance] is the expectation of the squared deviation of a random variable from its mean. :<math> \operatorname{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 = \left( \frac{1}{n} \sum_{i=1}^n x_i^2 \right) - \mu^2, </math> where <math>\mu</math> is the average value. That is, :<math>\mu = \frac{1}{n}\sum_{i=1}^n x_i</math> =====Practical Consequences===== It is possible that any calculation of variance using the full sample of our data (layers x city-years) is computationally infeasible. It seems particularly unlikely that we are going to manage between-group variance. I had problems in the past calculating mean distances between all centroids for just hulls, let alone for all geometries! We could, however, do this for some meaningful sub-population. There is also the question as to whether this approach is sensible in our context. In its native form, we'd be selecting the number of statistical clusters. We could readily use it to select the number of hulls (economic clusters) instead. But, in either case, we'd have to be within-city for this to make sense.  We could normalize the number of clusters, dividing it by the maximum, to deal with the 'cities are different' problem. That is, we could put %unclustered (later called %complete) on the x-axis and %variance explained on the y-axis and fit a curve to a plot of city-year-layers. We could then pick a %unclustered value and apply it across cities. The difference between this and the 'heuristic method' is that we'd be choosing based on diminishing marginal returns in variance explained as opposed to in percentage locations in hulls. '''Addendum:''' #We could do the elbow method on a per city-year basis. The number of statistical clusters is equal to the number of layers, so we'd be indexing over layers, and selecting a layer, for each city-year. It might be worth trying this for some city-year, say Tulsa, 2003. The code would be reusable for a bigger sample. Estimate: 3hrs.#I've rechecked the code and I now think it is computationally feasible. What I was trying to do before was find the average distance between every set of coordinates, which is an order more complex than what we need to do to calculate even within-group variance (between-group variance is simpler). Think O(n) rather than O(n^2), and we have around ~20million statistical clusters spread over ~200k layers. Estimate, given (1) above: 2hrs. =====The "Right" Implementation===== I could implement the elbow method using scalar distances. Specifically. I could measure <math>\bar{Y}_{i\cdot} - \bar{Y}</math> as the distance between a cluster's centroid and the overall centroid, and measure <math>Y_{ij}-\bar{Y}_{i\cdot}</math> as the distance between a cluster's constituent location and its centroid. There's a sense in which this approach is "the right thing to do", and the distance measurements are pretty straight-forward in PostGIS (and would account for an elipsoid earth).  However, in actuality, we have vectors of locations <math>(a,b)</math>, and not scalar distances as fundamental inputs. This changes the math[https://online.stat.psu.edu/stat505/lesson/8/8.2], as well as the ultimate test statistic [https://online.stat.psu.edu/stat505/lesson/8/8.3] that we might use. Specifically, <math>Y_{ij}</math> is a vector: <math>\mathbf{Y_{ij}} = \begin{pmatrix} Y_{ija} \\ Y_{ijb}\end{pmatrix}</math>, as are the sample mean <math>\mathbf{Y_{i\cdot}} = \begin{pmatrix} \bar{Y}_{i\cdot a}\\ \bar{Y}_{i\cdot b} \end{pmatrix}</math> and the grand mean <math>\mathbf{Y} = \begin{pmatrix} \bar{Y}_{a}\\ \bar{Y}_{b} \end{pmatrix}</math>. So the between-variance, which is called the '''H'''ypothesis sum of squares and cross products, is a 2x2 matrix and has K-1 degrees of freedom: :<math>H=\sum_{i=1}^{K} n_i(\mathbf{\bar{Y}_{i\cdot}} - \mathbf{\bar{Y}})(\mathbf{\bar{Y}_{i\cdot}} - \mathbf{\bar{Y}})'</math> And the within-variacnce, called the '''E'''rror sum of squares and cross products, is a 2x2 matrix and has N-K degrees of freedom: :<math>E=\sum_{i=1}^{K}\sum_{j=1}^{n_{i}} ( \mathbf{Y_{ij}}-\mathbf{\bar{Y}_{i\cdot}} )( \mathbf{Y_{ij}}-\mathbf{\bar{Y}_{i\cdot}} )'</math> The '''T''' sum of squares and cross products is <math>\mathbf{T=H+E}</math>. This is all potentially problematic because relational databases don't naturally do linear algebra very well. And we likely don't want to do this in some other software for two reasons. First, we'd have to move the data out and back into the database, which is costly. And, second the other software isn't likely to handle an elipsoid earth and the coordinates are not actually on a flat plane. (Though, assuming a flat earth might be reasonable as most cities are sufficiently small that curvature won't materially affect results, no matter how conceptually offensive it is.)  We could calculate the elements of '''H''' and '''E''' separately in PostGIS, taking advantage of ST_Distance. That wouldn't be too bad as the matrices are only 2x2s. However, it's worth asking whether that would be "right". Let's look at how '''H''' captures information about 'distance' (ignoring flat earth issues). Denote <math>\mathbf{\bar{Y}_{i\cdot}} - \mathbf{\bar{Y}} = \begin{pmatrix} \Delta A \\ \Delta B\end{pmatrix}</math>. Then: :<math>\mathbf{H}=\sum_{i=1}^{K} n_i \begin{pmatrix} \Delta A^2 & \Delta A \Delta B \\ \Delta A \Delta B & \Delta B^2\end{pmatrix} </math> So, the determinant and trace of '''H''' are: :<math>\det(\mathbf{H}) = \Delta A^2 \Delta B^2 - (\Delta A \Delta B)^2 \quad \mbox{and} \quad Tr(\mathbf{H}) = \Delta A^2 + \Delta B^2</math> Thus the trace is the square of the distance between points (for H, between a point within a cluster and the cluster's centroid). The trace of a product is the product of the traces only under specific circumstances and not in general, though we likely meet those circumstances (no zero elements). =====Standard Test Statistics===== We'd like to use some measure of variance explained, but variance is now a matrix. The standard test statistics for MANOVA, which are the closest equivalents to the ANOVA F-statistic from earlier, are: Wilk's Lambda: <math>\Lambda^* = \frac{\det \mathbf{E}}{\det(\mathbf{H}+\mathbf{E})}</math> Hotelling-Lawley Trace: <math>T = Tr(HE^{-1})</math> Pillai Trace: <math>V = Tr(H(H+E)^{-1})</math> So... the two trace statistics are very close to what we would get if we used scalar distances and used either scalar definition of variance explained. The main difference is the lack of correction for degrees of freedom. =====An Opportunity?===== We can't find a decent, let alone seminal, reference for using the elbow method to select the number of clusters. Our problem, which uses geographic coordinates, is also a special case anyway. So, we could implement a method using scalar distance and put a description of it, and its relationship to other measures, in the appendix. It might be a good value-added for the paper. One final thought: We could weight the distance between locations and the mean(s) by the fraction of startups in the location. ====The Heuristic Method Justification==== An attempt at a paragraph justifying the 'heuristic' method::Our heuristic method provides an objective technique for picking a city-year layer, which identifies and maps its clusters. It uses two measures. First, we are interested in clusters rather than lines or points, so we measure the percentage of locations in clusters. Second, we want to view each city-year through the same lens. As layer indices are not comparably across city-years, we use the HCA's 'fraction complete' to measure a layer's lens.:As an HCA progresses towards completion, it takes locations out of clusters at an increasing and then decreasing rate. Accordingly, a city-year plot, with the fraction complete on the x-axis and the percentage of locations in clusters on the y-axis, gives an S-curve. The inflection point of this curve marks a conceptual transition between refining clusters and dismantling them.  '''Note:''' For each layer of the HCA from i=1 to i=I, we define the HCA's fraction complete as (i-1)/(I-1). The HCA's fraction complete is then zero for the first layer when all locations are in a single hull, and one for the last layer when it has decomposed every cluster into separate locations. From version 8::My ‘elbow method’ fits a cubic function to the relationship between the percentage unclustered and the percentage of locations that are in hulls and determines the inflection point. The inflection point finds the layer beyond which further unclustering moves locations out of hulls at a decreasing rate.:As a rough guide, divisive clustering of geographic data of things like startups in cities moves through three stages. In the first stage, the algorithm identifies outliers as points, lines, or small-population hulls. Highest-first layers with low hull counts occur in this stage. In the second stage, the algorithm breaks apart core areas until it achieves the maximum number of hulls. Then, in the third stage, these hulls are refined, providing a progressively tighter lens on the core groupings, and dismantled, until all that remains are points. The elbow method identifies layers in this third stage at the tipping point between refining hulls and dismantling them. ====Guzman and Stern===='''The objective:''':"I think less is more. We just want to try to immunize ourselves against a referee who thinks we might be unaware of that data. We need a sentence saying it exists and another sentence saying why we don’t use it. And citing one source is probably enough." '''A hard-and-fast two-sentence version, with a one-sentence pillow:''':Guzman and Stern (2015) use business registration to find non-venture-backed startups. We do not use this data as it does not have a performance measure to demonstrate the representative layer's selection. However, understanding the relationships between clusters of different types of startups is an exciting topic for future research. '''A paragraph that covers the reviewer's comments too:''':We use data on venture capital-backed startups as their venture capital investment provides a direct performance measure to identify their clusters. However, other entrepreneurial firms may have clusters that influence clusters of venture-backed firms. In particular, pre-venture capital startups, firms that have experienced an acquisition or an initial public offering, and high-growth startups that do not raise venture capital (see Guzman and Stern, 2015, and related papers) may all affect the agglomeration of venture-backed startups. We leave it to future research to examine the relationships between clusters of different types of startups. '''A two-sentence version:''':We use data on venture capital-backed startups as their venture capital investment provides a direct performance measure to identify their clusters. However, we expect that other entrepreneurial firms, such as those identified from business registration data in Guzman and Stern (2015), may have a second-order effect on the clustering of venture-backed firms. '''We should cite their Science paper:''':Guzman, Jorge and Stern, Scott (2015), "Where is Silicon Valley?", Science, Vol. 347, No.6222, pp. 606-609, American Association for the Advancement of Science Bibtex: @article {Guzman606, author = {Guzman, Jorge and Stern, Scott}, title = {Where is Silicon Valley?}, volume = {347}, number = {6222}, pages = {606--609}, year = {2015}, doi = {10.1126/science.aaa0201}, publisher = {American Association for the Advancement of Science}, issn = {0036-8075}, URL = {https://science.sciencemag.org/content/347/6222/606}, eprint = {https://science.sciencemag.org/content/347/6222/606.full.pdf}, journal = {Science} } =====Previous attempts to include Guzman and Stern===== '''From version 8:'''Finally, Guzman and Stern (2015a,b) and related papers use business registration data to suggest that non-venture high-growth, high-technology startups outnumber their venturebacked counterparts by around three or four to one. These papers map and analyze nonventure startup activity within states, cities, zip codes, and for an individual street. A natural extension to this paper would consider agglomeration economies of non-venture startups, as well as their effects on the agglomeration of venture-backed firms. '''From version 4.1:'''Catalini et al. (2019) suggest that there are high-growth, high-technology startups that are not supported by venture capital. These firms may have been turned down by venture capitalists, may never approach venture capitalists, or may not have had their venture investment recorded at the time the authors conducted their study. Regardless, according to Catalini et al. (2019), non-venture HGHT startups outnumber venture-backed firms by around three or four to one. One obvious question is then whether they enhance or detract from agglomeration economies among venture-backed startups.  A natural extension of this paper would consider the impact of these firms. Catalini et al. (2019) derives its data from the processing of business registration data, which is explored for Massachusetts in Guzman & Stern (2015a), California in Guzman & Stern (2015b), firms leaving Delaware in Guzman (2017), and 15 U.S. states in Guzman & Stern (2016). This data provides full addresses, and so readily lends itself to Geographic Information System mapping, as well as a hierarchical cluster analysis decomposition to create microgeographies. =====Comments from the Associate Editor and two reviewers at Management Science===== '''Associate Editor:'''Multiple referees recommend supplementing your analyses with additional datasets. An expensive but informative path would be the National Establishment Time Series, which has street addresses for every business that reports data to Dun & Bradstreet; this would solve your longitudinal address problem. Alternatively, at no cost you can take advantage of Guzman & Stern’s Startup Cartography project data, which provides counts of the high-potential startups you are probably trying to capture with VX (and also see high-potential startups that don’t raise VC). '''Reviewer 2:'''First, the analysis only considers startup ventures which have already received venture funding—an important component of the ecosystem, for sure, but not necessarily the most important for agglomeration purposes. This results in an incomplete and possibly distorted picture of what agglomeration actually means. if a district has 50 startups that have not yet received VC funding and one that has, would that qualify as agglomeration? And what about companies that have been acquired or have gone public? At a minimum, a more accurate count of active ventures would be required to draw any legitimate conclusions about the effects of agglomeration, along the lines of what Andrews, Fazio, Guzman, Liu, & Stern (2019) have done with the Startup Cartography Project, especially if it is true that “non-venture high-growth, high-technology startups outnumber their venture-backed counterparts by around three or four to one” (p. 5). '''Reviewer 4:'''The authors note the challenges of using VC database addresses. This needs to be more closely considered given the persistence/agglomeration application currently contemplated. If ala Guzman’s work, you have a venture being ported to Redwood City for a big financing round and you also assign all prior rounds to Redwood City, you create quite a biased sample to extra growth around winner places. ===Back and Forth=== Big things:*Q: What is this technique for? A: Endogenous selection!

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