1. Information Space Partitioning with Adaptive Multiplicatively-Weighted Voronoi Diagrams
René Reitsma1, Stanislav Trubin2
1OSU College of Business
2Amazon.com, Inc.

2. Information Space Partitioning
Information spaces: problem, geometry & examples.
squarified treemap, a SOM, a Voronoi space, a cartogram.
Weight-area proportionality problem.
Adaptive Multiplicatively-Weighted Voronoi Diagram
method & case testing.
Human subjects experiment.

3. Information Space – Problem
Problem: Maps of Information Space:
Good correspondence.
Usability.
Geometry:
Metric / distance.
Placement.
Regions / Partitioning.

4. Information Space – Examples
(squarified) treemap.
Two-dimensional, Euclidian.
Partitioning is area-weight proportional: Ai/Aj = Wi/Wj
Wi = market cap(italization)
Location/placement is arbitrary.

5. Information Space – Examples
Chen et al. (1998): ET-map.
SOM.
Placement ≈ similarity.
Area ≈ magnitude.
Poor resolution.

6. Information Space – Examples
Andrews et al. (2002): InfoSky.
(Power) Voronoi diagram.
Two-dimensional, Euclidian.
Wi > Wj  Ai > Aj
Ai/Aj ≠ Wi/Wj
Δgi ≠ 0

7. Geographic Space – Examples
Cartograms; e.g., Tobler (2004), Gastner et al. (2004)
Two-dimensional, orthogonal, Euclidian.
Partitioning is area-weight proportional: Ai/Aj = Wi/Wj
Location & shape compromise.

8. Information Space – Definitions
Objective function: Ai/Aj = Wi/Wj
EChen et al = .825
Constraints:
inclusiveness: gi є ri
exclusiveness: ∑Ai = S
locality: Δgi = 0

9. Voronoi Information Space – Standard Model
Vi = { x | |x-xi| ≤ |x-xj| }
Borders are straight and orthogonally bisect Delaunay triangulations.
Regions are contiguous.
All space is allocated.
However: Ai = f(location).

10. Voronoi Information Space – Multipl. Weighted Model
Vi = { x | |x-xi|/wi ≤ |x-xj|/wj }
Borders are arcs of Appolonius circles.
Regions can surround other regions.
All space is allocated.
Ai = f(location, weight).
 Solve for wi, minimizing
 Adaptive Multiplicatively Weighted Voronoi Diagram (AMWVD).
Regions may be noncontiguous.

11. AMWVD: Iterative algorithm
Notation:
i Iteration
j Region
wi,j Weight of region j at iteration i
ai,j Area of region j at iteration i
Aj Goal area of region j (= Wj)
k Empirical scaling factor
Update weights at each iteration: wi+1,j = wi,j + Δwj
wi+1,j = wi,j + k(Aj – ai,j)

12. AMWVD Algorithm: Which k?
wi+1,j = wi,j + k(Aj – ai,j)
Large k  large weight adjustments:
Quick reduction of error  few iterations.
Little fine tuning.
Algorithm might oscillate.
Small k  small weight adjustments:
Slow reduction of error  many iterations.
Lots of fine tuning.
k0 = 1.0
Iterate (keep adjusting weights) until E no longer decreases.
ki+1 = ki × .95

13. AMWVD Algorithm: Vector vs. Raster
Vector solution: infinite precision, difficult computing (Mu, 2004).
Raster solution: easy computing, limited precision
Resolution increases allow higher precision; computations increase quadratically.
Boundary-only, quadtree resolution increases.

14. Adaptive Multiplicatively Weighted Voronoi Diagram

15. Adaptive Multiplicatively Weighted Voronoi Diagram

16. Adaptive Multiplicatively Weighted Voronoi Diagram

17. Adaptive Multiplicatively Weighted Voronoi Diagram

18. Adaptive Multiplicatively Weighted Voronoi Diagram
EChen et al.(20×10) = .825
EAMWVD(1200×1200) = 0.002

19. AMWVD – Human Subjects Testing
Can people correctly resolve the area information from AMWVDs?
‘Graduated’ Symbols in Cartography studies:
Chang (1977),
Cox (1976),
Crawford (1971, 1973),
Flannery (1971),
Groop and Cole (1978),
Williams (1956).
‘Unusual’ shapes.
Discontinuities.
Gestalt issues.

20. Human Subjects Testing - Hypotheses
H-I: Estimation of size differences is more accurate in rectangular (squarified treemap) and standard Voronoi partitioning than in (A)MWVD partitioning.
H-II: Size difference estimation error involving discontinuous areas is larger than for those not involving discontinuous areas.

21. Human Subjects Testing - Experiment
Three types of partitionings:
Rectangular (squarified) treemap.
Standard Voronoi diagram.
(A)MWVD.
Task:
Select the largest of two regions (ordinal).
Estimate the larger area as a proportion of the smaller area (metric).
One partitioning scheme per subject.
Variables measured:
Accuracy of comparisons.
Time used to make the comparisons.
Subjects: 30 undergraduate MIS students
10 subjects per partitioning.
30 comparisons per subject.

22. Human Subjects Testing - Results
H-I: Estimation of size differences is more accurate in rectangular (squarified treemap) and standard Voronoi partitioning than in (A)MWVD partitioning.
Rectangular
Standard Voronoi
AMWVD
Area estimation error
μ: .202
μ: .407
μ: .268
μ/μ: .50
t: -8.98; DF: 575; p<.01
μ/μ: .75
t: -3.42; DF: 565; p<.01
μ/μ: 1.52
t: 6.54; DF: 535; p<.01

23. Human Subjects Testing - Results
H-I: Underestimation of size differences in rectangular (squarified treemap) and standard Voronoi partitioning is less than in (A)MWVD partitioning.
Selection of largest region (ordinal)
Incorrect
Correct
Total
Rectangular 6
285
291
Standard Voronoi
221 77
298
AMWVD 79
219
306
581
887
Rectangular vs. AMWVD: χ2=69.30; D.F.=1; p<0.1.
Standard VD vs. AMWVD: χ2=133.44; D.F.=1; p<0.1.

24. Human Subjects Testing - Results
Rectangular
Standard Voronoi
AMWVD
log(time (ms)) to select largest region)
μ: 3.648
μ: 3.632
μ: 3.667
μ/μ: 1.004
t: .571; DF: 586; p: .57
μ/μ: .995
t: ; DF: 583; p: .03
μ/μ: .9904
t: ; DF: 592; p<.01
Rectangular
Standard Voronoi
AMWVD
Time (ms) used to numerically estimate the size relationship
μ: 10,004
μ: 7,001
μ: 8,452
μ/μ: 1.429
t: 5.576; DF: 432; p < 0.1
μ/μ: 1.184
t: 2.772; DF: 477; p < .01
μ/μ: .828
t: ; DF: 578; p < .01

25. Human Subjects Testing - Results
H-II: Size estimation error involving discontinuous areas is larger than for those not involving discontinuous areas.
μ EAMWVD continuous (n=181) = .270
μ EAMWVD discontinuous (n=117) = .266

26. Voronoi Information Spaces - Conclusion
Adaptive Multiplicatively Weighted Voronoi Diagram (AMWVD) solves weight-proportional partitioning subject to:
inclusiveness: gi є ri
exclusiveness: ∑Ai = S
locality: Δgi = 0
Squarified treemaps cannot do this.
Standard and additively weighted Voronoi diagrams cannot do this.
Adaptive multiplicatively weighted Voronoi diagrams perform well in human subject area comparisons:
Perform not as well as squarified treemaps (-25%).
Significantly outperform standard (and additively weighted) Voronoi diagrams.