Regionalization of Information Space with Adaptive Voronoi Diagrams

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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 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

• 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.

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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.

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Information Space - Analog Approaches
Cox Patterson (National Center for
Supercomputing Applications - NCSA) (1991)
Visualization of NSFNET traffic
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Information Space - Analog Approaches
Card, Robertson York (Xerox) (1996) WebBook
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Information Space - Other A Priori Approaches
WebMap Technologies
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Information Space - Other A Priori Approaches
SOM Kohonen, Chen, et al.
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Information Space - Other A Priori Approaches
Inxight hyberbolic web site map viewer
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Information Space - Other A Priori Approaches
Chi (2002)
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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.

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Buttenfield/Reitsma Proposal

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

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

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

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Criteria for Regionalization

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

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

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

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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.

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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.

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Weighted Voronoi Diagrams

• Multiplicatively Weighted Voronoi Diagram
• Vi x d(x, i)/wi ? d(x, j)/wj , i ??j
• wi wj gt 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.

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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).

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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 gt q -1.5
• As wj increases, p decreases, q increases gt
  hence, i's (circular) region gets smaller.

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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

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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.

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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.

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Inverse Voronoi Diagrams Cont.'d

• MWVD is a nice starting point
• Multiplicity reflects multiplicity in area.
• Distance friction is uniform in all directions
  gt concentric allocation.
• By increasing weights landlocked generators can
  'escape.'
• However
• Weights represent distance rather than area.
• Area proportionality requires bounding polygon.

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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,j1 wi,j ?w
• wi,j1 wi,j (1 k(Ai - ai,j))

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Adaptive MW Voronoi Diagram
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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 gt 2.

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Some issues...

• How to select k in wi,j1 wi,j (1 k(Ai -
  ai,j))?

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Any Applicability to the World?

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