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

ContentUsage

Information Space - Other A Priori Approaches

WebMap Technologies WebMap Technologies

ContentUsage

Information Space - Other A Priori Approaches

SOM: Kohonen, Chen, et al.

ContentUsage

Information Space - Other A Priori Approaches

Inxight hyberbolic web site map viewer

ContentUsage

Information Space - Other A Priori Approaches

Chi (2002)Chi (2002)

ContentUsage

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.

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

Case Study: Building as a Learning Tool (BLT)

http://blt.colorado.edu

Building as a Learning Tool (BLT)

Can this space be regionalized? If so, how?

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.

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).

(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.

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}

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.

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.

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.

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))

Adaptive MW Voronoi Diagram

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.

Some issues...

• How to select k in wi,j+1

= wi,j (1 + k(A

i - a

i,j))?

Any Applicability to the World?

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