AMCIS 2006

Weight-proportional Information Space Partitioning Using

Adaptive Multiplicatively-Weighted Voronoi Diagrams

René Reitsma & Stanislav Trubin

Accounting, Finance & Information ManagementElectrical Engineering & Computer Science

Oregon State University

AMCIS 2006

Weight-proportional Voronoi Information Spaces

• Information space partitioning: problem, geometry & examples

– Squarified treemap, a SOM, and a Voronoi space.

• Weight-area proportionality problem.

• Adaptive Voronoi partitioning: method & case testing.

• Human subjects experiment.

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Information Space – Problem

• Problem: Maps of Information Space:

– Good correspondence.

– Usability.

• Geometry:

– Metric / distance.

– Placement.

– Partitioning.

AMCIS 2006

Information Space – Examples

• www.smartmoney.com

• (squarified) treemap.

• Two-dimensional, Euclidian.

• Partitioning is area-weight proportional: Ai/Aj = Wi/Wj

• However: placement is 100% function of partitioning.

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Information Space – Examples

• Chen et al. (1998): ET-map.

• SOM.

• Placement ≈ similarity.

• Area ≈ magnitude.

• However: approximation only.

• Poor resolution.

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Information Space – Examples

• Andrews et al. (2002): InfoSky.

• (Power) Voronoi diagram.

• Two-dimensional, Euclidian.

• Wi > Wj Ai > Aj

• However: Ai/Aj ≠ Wi/Wj

• Δgi ≠ 0

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Information Space – Definitions

• Objective function:

– EChen et al. = .825

• Constraints:

– inclusiveness: gi є ri

– exclusiveness: ∑Ai = S

– locality: Δgi = 0

n

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AW

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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: Area = f(location).

AMCIS 2006

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.

• Area = f(location, weights).

Solve for wi, minimizing

• Regions may be noncontiguous.

n

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AW

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AMCIS 2006

Adaptive Multiplicatively Weighted Voronoi Diagram

wi+1,j = wi,j + k(Aj – ai,j)

ki = ki-1 × .95

Resolution effect.

AMCIS 2006

Adaptive Multiplicatively Weighted Voronoi Diagram

AMCIS 2006

Adaptive Multiplicatively Weighted Voronoi Diagram

AMCIS 2006

• EChen et al.(20×10) = .825

• EAMWVD(1200×1200) = 0.002

Adaptive Multiplicatively Weighted Voronoi Diagram

AMCIS 2006

AMWVD – Human Subjects Testing

• Can people correctly resolve the area information from AMWVDs?

• Cartography studies:

– Chang (1977), Cox (1976), Crawford (1971, 1973), Flannery (1971), Groop and Cole (1978), Williams (1956).

• ‘Unusual’ shapes.

• Discontinuities.

• Gestalt issues.

AMCIS 2006

Human Subjects Testing - Hypotheses

• H-I: Size differences (under) estimation will follow Steven’s Rule.

• H-II: Underestimation of size differences in rectangular (squarified treemap) and standard Voronoi partitioning is less than in (A)MWVD partitioning.

• H-III: Size comparisons involving overlapping circle patterns will show the same amount of error as those not involving such patterns.

• H-IV: Size estimation error involving discontinuous areas is larger than for those not involving discontinuous areas.

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Human Subjects Testing - Experiment

• Three types of partitionings:– Rectangular (squarified) treemap.

– Standard Voronoi diagram.

– Adaptive multipl. weighted Voronoi diagram.

• Task:– Select the largest of two regions.

– Estimate how much larger the selected region is.

– 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.

AMCIS 2006

Human Subjects Testing - results

• H-II: Underestimation of size differences in rectangular (squarified treemap) and standard Voronoi partitioning is less than in (A)MWVD partitioning.

•

Rectangular Standard Voronoi AMWVD

Area estimation error

μ: .202 μ: .407 μ: .268

Rectangular μ/μ: .51t: -8.98; DF: 575;

p<.01

μ/μ: .75t: -3.42; DF: 565;

p<.01

Standard Voronoi μ/μ: 1.46t: 6.54; DF: 535;

p<.01

ratio Actual

ratio Estimated ratio ActualError

AMCIS 2006

Human Subjects Testing - results

• H-II: 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 79 219 298

Total 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.

AMCIS 2006

Human Subjects Testing - results

Rectangular Standard Voronoi AMWVD

log(time (ms)) to select largest region)

μ: 3.648 μ: 3.632 μ: 3.667

Rectangular μ/μ: 1.004t: .571; DF: 586; p: .57

μ/μ: .995t: -2.245; DF: 583; p: .03

Standard Voronoi μ/μ: .9904t: -2.862; 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

Rectangular μ/μ: 1.429t: 5.576; DF: 432; p < 0.1

μ/μ: 1.184t: 2.772; DF: 477; p < .01

Standard Voronoi μ/μ: .828t: -3.830; DF: 578; p < .01

AMCIS 2006

Human Subjects Testing - results

• H-III: Size comparisons involving overlapping circle patterns will show the same amount of error as those not involving such patterns.

• H-IV: 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

AMCIS 2006

Voronoi Information Spaces - Conclusion

• Adaptive Multiplicatively Weighted Voronoi Diagram 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.

AMCIS 2006

Voronoi Information Space - Solutions

AMCIS 2006

Voronoi Information Space - Solutions