Regionalization of Information Space with Adaptive Voronoi Diagrams René F. Reitsma Dept. of Accounting, Finance & Inf. Mgt. Oregon State University Stanislaw Trubin Dept. of Electrical Engineering and Computer Science Oregon State University Saurabh Sethia Dept. of Electrical Engineering and Computer Science Oregon State University
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Regionalization of Information Space with Adaptive Voronoi Diagrams René F. Reitsma Dept. of Accounting, Finance & Inf. Mgt. Oregon State University Stanislaw.
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Regionalization of Information Space with Adaptive Voronoi
Diagrams
René F. ReitsmaDept. of Accounting, Finance & Inf. Mgt.
Oregon State University
Stanislaw TrubinDept. of Electrical Engineering and Computer Science
Oregon State University
Saurabh SethiaDept. of Electrical Engineering and Computer Science
Oregon State University
Regionalization of Information Space with Adaptive Voronoi Diagrams
Information space: contents & usage. Pick or infer a spatialization? Loglinear/multidimensional scaling approach. Regionalization based on distance: Voronoi Diagram. Regionalization based on area: Inverse/Adaptive Voronoi
Diagram. Conclusion and discussion.
Information Space
Dodge & Kitchin (2001) Mapping Cyberspace. Dodge & Kitchin (2001) Atlas of Cyberspace. Chen (1999) Information Visualization and Virtual
Environments. J. of the Am. Soc. for Inf. Sc. & Techn. (JASIST). ACM Transactions/Communications. Annals AAG: Couclelis, Buttenfield & Fabrikant, etc. IEEE INTERNET COMPUTING. INFOVIS Conferences.
Information Space - Analog Approaches
Cox & Patterson (National Center for Supercomputing Applications - Cox & Patterson (National Center for Supercomputing Applications - NCSA) (1991) Visualization of NSFNET trafficNCSA) (1991) Visualization of NSFNET traffic
Information Space - Analog Approaches
Card, Robertson & York (Xerox) (1996) WebBookCard, Robertson & York (Xerox) (1996) WebBook
Nongenerator points get allocated to the closest generator --> Voronoi Diagram.
Area point of view:
Generators have claims on the surrounding space --> Inverse Voronoi Diagram.
Voronoi Diagram Regionalization Based on Distance
Okabe A., Boots, B., Sugihara, K., Chiu,S.N. (2000) Spatial Tesselations; Wiley Series in Probability and Statistics.
Voronoi Diagrams
Honeycombs are regionalizations. Regularly spaced 'generators.' Coverage is inclusive. Mimimum material, maximum
area. Minimum generator distance.
Ordinary Voronoi Diagrams
Vi = {x | d(x, i) d(x, j) , i j}
Thiessen Polygons. Bisectors are lines of
equilibrium. Bisectors are straight lines. Bisectors are perpendicular to
the lines connecting the generators.
Bisectors intersect the lines connecting the generators exactly half-way.
Three bisectors meet in a point.
Exterior regions go to infinity.
Ordinary Voronoi Diagrams
Vi = {x | d(x, i) d(x, j) , i j} is a special case:
Assignment (static) view: Distance (friction) is uniform in all directions for all
generators.
Growth (dynamic) view: All generators grow their regions at the same rate. All generators start growing at the same time. Growth is uniform in all directions.
Boots (1980) Economic Geography: Weighted versions “produce patterns which are free of the
peculiar and, in an empirical sense, unrealistic characteristics of patterns created by the Thiessen polygon model.”
Weighted Voronoi Diagrams
Multiplicatively Weighted Voronoi Diagram:
Vi = {x | d(x, i)/wi d(x, j)/wj , i j}
wi = wj ==> Ordinary Voronoi Diagram.
wi wj:
Static View: distance friction i distance friction j.
Dynamic View: generators start growing at the same time, but grow at different rates.
WeightedVoronoi Diagrams Cont.'d
Multiplicatively Weighted Voronoi Diagram: Vi = {x | d(x, i)/wi d(x, j)/wj , i j}
Bisectors are lines of equilibrium.
Bisectors become curved when wi wj.
Bisectors divide the lines connecting generators i and j in portions wi/(wi + wj) and wj/(wi + wj).
Low weight regions get surrounded by high weight regions.
Highest weight region goes to infinity (surrounds all others).
Weighted Voronoi Diagrams Cont.'d
Bisectors are Appolonius Circles: “Set of all points whose distances from two fixed points are in a constant ratio” (Durell, 1928).