Regionalization of Information Space with Adaptive Voronoi Diagrams

1. 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 Regionalization of Information Space with Adaptive Voronoi Diagrams

2. 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.

3. 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.

4. Information Space - Analog Approaches Cox & Patterson (National Center for Supercomputing Applications - NCSA) (1991) Visualization of NSFNET traffic

5. Content Usage Information Space - Analog Approaches Card, Robertson & York (Xerox) (1996) WebBook

6. Content Usage Information Space - Other A Priori Approaches WebMap Technologies

7. Content Usage Information Space - Other A Priori Approaches SOM: Kohonen, Chen, et al.

8. Content Usage Information Space - Other A Priori Approaches Inxight hyberbolic web site map viewer

9. Content Usage Information Space - Other A Priori Approaches Chi (2002)

10. Information Space - A Posteriori Approaches • Infer or resolve geometry (dimension & metric) from secondary data using ordination techniques: • Factorial techniques. • Vector space models. • Multidimensional scaling. • Spring models. • Sources of secondary data: • Content. • Relationships (structure). • Navigational records.

11. Buttenfield/Reitsma Proposal • Distance is inversely proportional to traffic volume. • Observed data are noisy manifestation of a stable process.

12. Case Study: Building as a Learning Tool (BLT) http://blt.colorado.edu

13. Building as a Learning Tool (BLT) • Can this space be regionalized? If so, how?

14. Criteria for Regionalization • Define our points as 'generators.' • Distance point of view: • Nongenerator points get allocated to the closest generator --> Voronoi Diagram. • Area point of view: • Generators have claims on the surrounding space --> Inverse Voronoi Diagram.

15. Voronoi Diagram Regionalization Based on Distance Okabe A., Boots, B., Sugihara, K., Chiu,S.N. (2000) Spatial Tesselations; Wiley Series in Probability and Statistics.

16. Voronoi Diagrams • Honeycombs are regionalizations. • Regularly spaced 'generators.' • Coverage is inclusive. • Mimimum material, maximum area. • Minimum generator distance.

17. 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.

18. 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.”

19. 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.

20. 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).

21. 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). • (j – q) / (i – q) = (j – p) / (p - i) = wj / wi = 5 • q cannot be -p = -1 as (j – q) / (i – q) = (6 - -1) / (0 - -1) = 7  5 • (6 – q) / –q = 5 ==> q = -1.5 • As wj increases, p decreases, q increases ==> hence, i's (circular) region gets smaller.

22. Weighted Voronoi Diagrams Cont.'d • Other weighting schemes: • Additively Weighted: • Vi = {x | d(x, i) - wi d(x, j) - wj , i j} • Generators grow at identical rates but start growing at different times. • Bisectors are hyperboles. • Compoundly Weighted: • Vi = {x | d(x, i)/wi1 - wi2 d(x, j)/wj1 - wj2 , i j} • Power Diagram: • Vi = {x | d(x, i)p- wi d(x, j)p - wj , i j}

23. Weighted Voronoi Diagrams Cont.'d • Applications in Geography: • Huff, D. (1973) Delineation of a National System of Planning Regions on the Basis of Urban Spheres of Influence; Regional Studies; 7; 323-329.

24. Inverse Voronoi Diagrams • Voronoi Diagrams: • Based on distance: • Area = f(position, weight). • Peripheral generators claim peripheral space. • Landlocking. • Based on area: • Generator regions have areas proportional to a(ny) given variable. • Space is uniform; i.e., distance friction is uniform in all directions. • Weight = f(position, area). • Inverse Voronoi diagram.

25. Inverse Voronoi Diagrams Cont.'d • MWVD is a nice starting point: • Multiplicity reflects multiplicity in area. • Distance friction is uniform in all directions ==> concentric allocation. • By increasing weights landlocked generators can 'escape.' • However: • Weights represent distance rather than area. • Area proportionality requires bounding polygon.

26. Adaptive MW Voronoi Diagram • Weight = f(position, area) • Let Ai= target area of generator i (prop.). • Let ai,j= allocated area of generator i (prop.) after iteration j. • Objective function: minimize Ai - ai,j • Let wi,j = weight of generator i at iteration j. • wi,0 = Ai • wi,j+1 = wi,j + w • wi,j+1 = wi,j (1 + k(Ai - ai,j))

27. Adaptive MW Voronoi Diagram

28. Adaptive MW Voronoi Diagram • Summary: • Interest in information space visualization. • LLM/MDS method provides dimensionality, location and a measure of size or 'force' (od). • MW Voronoi diagrams provide a good 'multiplicative' starting point but area = f(position, distance). • AMW Voronoi diagrams can solve for weights = f(position, area). • Applies to dimensionalities > 2.

29. Some issues... • How to select k in wi,j+1 = wi,j (1 + k(Ai - ai,j))?

30. Any Applicability to the World? • Search-and-rescue? • Crop dusting and harvesting? • Others?