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Regionalization of Information Space with Adaptive Voronoi Diagrams

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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

- 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.

- 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.

Cox & Patterson (National Center for Supercomputing Applications - NCSA) (1991) Visualization of NSFNET traffic

Content

Usage

Card, Robertson & York (Xerox) (1996) WebBook

Content

Usage

WebMap Technologies

Content

Usage

SOM: Kohonen, Chen, et al.

Content

Usage

Inxight hyberbolic web site map viewer

Content

Usage

Chi (2002)

- Infer or resolve geometry (dimension & metric) from secondary data using ordination techniques:
- Factorial techniques.
- Vector space models.
- Multidimensional scaling.
- Spring models.

- Content.
- Relationships (structure).
- Navigational records.

- Distance is inversely proportional to traffic volume.
- Observed data are noisy manifestation of a stable process.

http://blt.colorado.edu

- Can this space be regionalized? If so, how?

- 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.

Okabe A., Boots, B., Sugihara, K., Chiu,S.N. (2000) Spatial Tesselations; Wiley Series in Probability and Statistics.

- Honeycombs are regionalizations.
- Regularly spaced 'generators.'
- Coverage is inclusive.
- Mimimum material, maximum area.
- Minimum generator distance.

- 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.

- 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.

- Assignment (static) view:
- 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.”

- 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.

- 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).

- 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.

- 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}

- Additively Weighted:

- 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.

- Voronoi Diagrams:
- Based on distance:
- Area = f(position, weight).
- Peripheral generators claim peripheral space.
- Landlocking.

- Area = f(position, weight).
- 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.

- Based on distance:

- 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.

- 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))

- 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.

- How to select k in wi,j+1 = wi,j (1 + k(Ai - ai,j))?

- Search-and-rescue?
- Crop dusting and harvesting?
- Others?