Ren é Reitsma & Stanislav Trubin Accounting, Finance & Information Management

1. René Reitsma & Stanislav Trubin Accounting, Finance & Information Management Electrical Engineering & Computer Science Oregon State University Weight-proportional Information Space Partitioning Using Adaptive Multiplicatively-Weighted Voronoi Diagrams

2. 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. Weight-proportional Voronoi Information Spaces

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

4. 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.

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

6. 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

7. Information Space – Definitions • Objective function: • EChen et al. = .825 • Constraints: • inclusiveness: giє ri • exclusiveness:∑Ai = S • locality:Δgi = 0

8. 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).

9. 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.

10. Adaptive Multiplicatively Weighted Voronoi Diagram wi+1,j = wi,j + k(Aj – ai,j) ki = ki-1 × .95 Resolution effect.

11. Adaptive Multiplicatively Weighted Voronoi Diagram

12. Adaptive Multiplicatively Weighted Voronoi Diagram

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

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

15. 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. Human Subjects Testing - Hypotheses

16. 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. Human Subjects Testing - Experiment

17. 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.

18. 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 vs. AMWVD: χ2=69.30; D.F.=1; p<0.1. • Standard VD vs. AMWVD: χ2=133.44; D.F.=1; p<0.1.

19. Human Subjects Testing - results

20. 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

21. Voronoi Information Spaces - Conclusion • Adaptive Multiplicatively Weighted Voronoi Diagram solves weight-proportional partitioning subject to: • inclusiveness: giє 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.

22. Voronoi Information Space - Solutions

23. Voronoi Information Space - Solutions