Abstract
This paper addresses the theme of spatial autocorrelation impacting spatial equilibria, and hence an understanding of economic network resilience and vulnerability. It exploits the notion that spatial autocorrelation in the geographic distribution of origin and destination attributes and network autocorrelation in the flows between origins and destinations constitute two spatial autocorrelation components contained in spatial interaction data. It illustrates that a spatial interaction model specification needs to incorporate both components in order to furnish sound implications about associated economic network resilience and vulnerability. Such models also need to undergo sensitivity analyses in terms of changes in geographic scale and resolution. And, it furnishes a novel 3-D visualization of geographic flows, such as journey-to-work trips, in order to achieve a better comprehension of economic network resilience and vulnerability.
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Notes
A location-allocation problem is a spatial optimization problem in which m places need to be determined in such a way that they minimize an aggregate objective function associated with the allocation of n nearby points to them. The simplest is the classical Weberian location problem, which also is the spatial median problem, which minimizes the Euclidean distance separating the n points from the optimally located places to which they respectively are allocated.
The map mean was set to 5. An ESF was constructed that has extremely high positive spatial autocorrelation; its map pattern appears in Fig. 1. For each location i (one of the 729 locations on a 27-by-27 grid of points), the mean was calculated as (5 + w × ESFi)—where w = 0, 5, 10, and 20 for an increasingly prominent spatial autocorrelation component—and a drawing was made from a Poisson pseudo-random number generator with this mean. Next, 1 was added to the result so that no weight was 0.
The list of victims is available at http://911research.wtc7.net/cache/sept11/victims/victims_list.htm. Geocoding their community locations was done with http://www.census.gov/geo/maps-data/data/gazetteer2010.html and http://www.lat-long.com.
Using matrix notation, for a given geographic weights matrix C, MC = \( \frac{\mathrm{n}}{{\mathbf{1}}^{\mathrm{T}}\mathbf{C}\mathbf{1}}\frac{{\mathbf{Y}}^{\mathrm{T}}\left(\mathbf{I}-\mathbf{1}{\mathbf{1}}^{\mathrm{T}}/\mathrm{n}\right)\mathbf{C}\left(\mathbf{I}-\mathbf{1}{\mathbf{1}}^{\mathrm{T}}/\mathrm{n}\right)\mathbf{Y}}{{\mathbf{Y}}^{\mathrm{T}}\left(\mathbf{I}-\mathbf{1}{\mathbf{1}}^{\mathrm{T}}/\mathrm{n}\right)\mathbf{Y}} \), where n is the number of areal units, 1 is an n-by-1 vector of ones, Y is an n-by-1 vector of variable values, I is an n-by-n identity matrix, and T denotes the matrix transpose operation. The matrix (I-11 T/n) is the standard projection matrix that centers a variable. This MC has an expected value of −1/(n-1) for no spatial autocorrelation, and its maximum and minimum values are defined by the extreme eigenvalues of matrix (I − 11 T/n)C(I − 11 T/n). Because it is a cross-products index, it is similar to a Pearson product moment correlation coefficient.
1/Ai indexes the accessibility of all destinations vis-à-vis origin i, whereas 1/Bj indexes the accessibility of all origins vis-à-vis destination j. The summation terms in \( {\mathrm{A}}_{\mathrm{i}}=1/{\displaystyle \sum_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{B}}_{\mathrm{j}}{\mathrm{D}}_{\mathrm{j}}{\mathrm{e}}^{{\upgamma \mathrm{d}}_{\mathrm{i}\mathrm{j}}}} \) and \( {\mathrm{B}}_{\mathrm{j}}=1/{\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{A}}_{\mathrm{i}}{\mathrm{O}}_{\mathrm{i}}{\mathrm{e}}^{{\upgamma \mathrm{d}}_{\mathrm{i}\mathrm{j}}}} \) capture spatial autocorrelation in the distribution of origin and destination characteristics.
These data can be downloaded from www.census.gov/population/www/cen2000/commuting/mcdworkerflow.html.
This 3-D visualization is prepared with ArcGIS version 10.1 of ESRI Inc. First, the 3-D flow lines are created with a customized ArcMap Add-in function that is programmed with ArcObjects in the Visual Basic .NET 2010 environment. One part of this program to create 3-D lines is presented in Appendix A, and its full code is available at http://www.utdallas.edu/~ywchun/Tools/Gen3Dflows.zip. Next, detail mapping with the n lines and the polygon layers is conducted in ArcScene. A detailed procedure is explained in Appendix B. Appendix C furnishes additional examples of implementation for US Federal Reserve Bank money flows, and both Chcago and Dallas-Ft. Worth journey-to-work flows.
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Appendices
Appendix A
1.1 Program code to create 3-D lines with ArcObjects 10.1 in Visual Basic .NET 2010
Appendix B
A procedure to create a 3-D visualization with the ArcMap Add-In tool.
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1.
Download the Add-In file and install it on a local computer housing ArcGIS 10.1.
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2.
Open ArcMap and then choose Customize → Customize Mode… The Customize window will appear.
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3.
Choose the Commands tab, and then Add-In controls under Categories.
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4.
Drag and drop the Generating 3-D flows tool on a toolbar.
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5.
Open a polygon shapefile for spatial units (i.e., origins and destinations). Next, click the button
to execute the tool. The Generating 3-D flow lines window will pop-up (Figure 10).
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6.
In the window, choose a polygon layer and an ID field, and set an output filename.
Note: the height value for the origin points of 3-D lines to be created is set with the height of the polygon layer’s extent, by default.
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7.
Click on the OK button, which executes the program to crate a 3-D line shapefile.
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8.
In ArcScene, open the 3-D line shapefile.
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9.
Next, open the polygon layer twice: once for origins on the top layer, and once for destinations on the bottom layer.
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10.
Set the Base Height of the top layer (origins) with the height value used when the 3-D line shapefile is created (see Fig. 10). Also, change its transparency level to make any overlapped features of this layer visible.
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11.
Change symbols of the three layers, if necessary.
A graphical user interface to set up parameters to create 3-D flow lines
Note: the 3-D line shapefile contains a field named NetID whose values are concatenations of origin and destination IDs. Hence, a modeling result such as that from spatial interaction models can be joined with the field.
Appendix C
3.1 Additional geographic flows visualizations
This section presents a number of additional flows visualizations in order to furnish a more general sense of the technique’s applicability.
3.1.1 Money flows in the coterminous US Federal Reserve Bank districts
Figure 11 portrays the money flows map for spatial interaction among the 12 districts into which the conterminous US has been partitioned for central banking purposes (Tobler 1987). As noted previously, the top GIS layer polygons represent origins, whereas the bottom GIS polygons represent destinations. The portrayed geographic distributions are the spatial autocorrelation components for the origins (top) and destinations (bottom). In this case, these distributions are the same. The choropleth maps portray the geographic distributions of the origin and destination balancing factors. These two distributions have a MC value of −0.28166; in other words, they contain weak negative spatial autocorrelation. Meanwhile, the inflated flows (Figs. 11b–c) tend to concentrate in the eastern part of the country, whereas the less numerous deflated flows (Figs. 11d–e) tend to originate in New York City, or terminate in New York City or Philadelphia. A perturbation to the system would tend to disrupt inflated flows, with relatively small impacts upon spatial autocorrelation neutral and deflated flows. Not surprisingly, New York City appears to be the most vulnerable part of this banking system.
A 3-D visualization of monetary flows among the US Federal Reserve Bank districts. a: flows among the districts. b: spatial autocorrelation inflated flows in 2-D. c: spatial autocorrelation inflated flows in 3-D d: spatial autocorrelation deflated flows in 2-D. e: spatial autocorrelation deflated flows in 3-D. f: spatial autocorrelation neutral flows in 2-D. g: spatial autocorrelation neutral flows in 3-D
3.1.2 The commuting landscape of the Chicago MSA
Figure 12 portrays the journey-to-work flows map for spatial interaction among the 13 counties comprising the Chicago MSA, with the City of Chicago located in the center of the eastern part of this metropolitan region. As noted previously, the top GIS layer polygons represent origins, whereas the bottom GIS polygons represent destinations, and as choropleth maps, they portray the geographic distributions of the origin and destination balancing factors. These geographic distributions are the spatial autocorrelation components for the origins (top) and destinations (bottom), and have respective MC values of 0.27768 and 0.21943; in other words, they contain weak positive spatial autocorrelation. Meanwhile, both the inflated (Fig. 12b–c) and deflated (Fig. 12d–e) flows are spread across the MSA, whereas the less numerous deflated flows tend to cover shorter distances than the inflated flows. A perturbation to the system focused on the City of Chicago appears as though it would nudge the system from equilibrium, being most disruptive to the inflated flows.
A 3-D visualization of county-level Chicago MSA journey-to-work flows. a: flows among the counties. b: spatial autocorrelation inflated flows in 2-D. c: spatial autocorrelation inflated in 3-D. d: spatial autocorrelation deflated flows in 2-D. e: spatial autocorrelation deflated flows in 3-D. f: spatial autocorrelation neutral flows in 2-D. g: spatial autocorrelation neutral flows in 3-D
3.1.3 The commuting landscape of the Dallas MSA
Figure 13 portrays the journey-to-work flows among the 12 counties comprising the Dallas MSA, with Dallas-Ft. Worth located roughly in the center of this metroplex. As noted previously, the top GIS layer polygons represent origins, whereas the bottom GIS polygons represent destinations. The portrayed geographic distributions are the spatial autocorrelation components for the origin balancing factors (top) and the destination balancing factors (bottom). These two distributions have respective MC values of 0.21607 and 0.13202; in other words, they contain weak positive spatial autocorrelation. Meanwhile, all the spatial autocorrelation inflated (Figs. 13b–c) and deflated (Figs. 13d–e) flows are spread across the MSA, with the spatial autocorrelation deflated flows being far more numerous than the inflated flows. A perturbation to this system most likely would only nudge it from equilibrium.
A 3-D visualization of county-level Dallas-Ft. Worth MSA journey-to-work flows. a: flows among the counties. b: spatial autocorrelation inflated flows in 2-D. c spatial autocorrelation inflated flows in 3-D. d: spatial autocorrelation deflated flows in 2-D. e: spatial autocorrelation deflated flows in 3-D. f: spatial autocorrelation neutral flows in 2-D. g: spatial autocorrelation neutral flows in 3-D
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Griffith, D.A., Chun, Y. Spatial Autocorrelation in Spatial Interactions Models: Geographic Scale and Resolution Implications for Network Resilience and Vulnerability. Netw Spat Econ 15, 337–365 (2015). https://doi.org/10.1007/s11067-014-9256-4
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DOI: https://doi.org/10.1007/s11067-014-9256-4