1. AMCIS 2006 Weight-proportional Information Space Partitioning Using Adaptive Multiplicatively-Weighted Voronoi Diagrams René Reitsma & Stanislav Trubin Accounting, Finance & Information Management Electrical Engineering & Computer Science Oregon State University

2. 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.

3. AMCIS 2006 Information Space – Problem Problem: Maps of Information Space: –Good correspondence. –Usability. Geometry: –Metric / distance. –Placement. –Partitioning.

4. AMCIS 2006 Information Space – Examples www.smartmoney.com (squarified) treemap. Two-dimensional, Euclidian. Partitioning is area-weight proportional: A i /A j = W i /W j However: placement is 100% function of partitioning.

5. AMCIS 2006 Information Space – Examples Chen et al. (1998): ET-map. SOM. Placement ≈ similarity. Area ≈ magnitude. However: approximation only. Poor resolution.

6. AMCIS 2006 Information Space – Examples Andrews et al. (2002): InfoSky. (Power) Voronoi diagram. Two-dimensional, Euclidian. W i > W j  A i > A j However: A i /A j ≠ W i /W j Δg i ≠ 0

7. AMCIS 2006 Information Space – Definitions Objective function: –E Chen et al. =.825 Constraints: –inclusiveness: g i є r i –exclusiveness: ∑A i = S –locality: Δg i = 0

8. AMCIS 2006 Voronoi Information Space – Standard Model V i = { x | |x-x i | ≤ |x-x j | } Borders are straight and orthogonally bisect Delaunay triangulations. Regions are contiguous. All space is allocated. However: Area = f(location).

9. AMCIS 2006 Voronoi Information Space – Multipl. Weighted Model V i = { x | |x-x i |/w i ≤ |x-x j |/w j } Borders are arcs of Appolonius circles. Regions can surround other regions. All space is allocated. Area = f(location, weights).  Solve for w i, minimizing Regions may be noncontiguous.

10. AMCIS 2006 Adaptive Multiplicatively Weighted Voronoi Diagram w i+1,j = w i,j + k(A j – a i,j ) k i = k i-1 ×.95 Resolution effect.

11. AMCIS 2006 Adaptive Multiplicatively Weighted Voronoi Diagram

12. AMCIS 2006 Adaptive Multiplicatively Weighted Voronoi Diagram

13. AMCIS 2006 E Chen et al.(20×10) =.825 E AMWVD(1200×1200) = 0.002 Adaptive Multiplicatively Weighted Voronoi Diagram

14. 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.

15. 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.

16. AMCIS 2006 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.

17. 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. RectangularStandard VoronoiAMWVD Area estimation error μ:.202μ:.407μ:.268 Rectangularμ/μ:.51 t: -8.98; DF: 575; p<.01 μ/μ:.75 t: -3.42; DF: 565; p<.01 Standard Voronoiμ/μ: 1.46 t: 6.54; DF: 535; p<.01

18. 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) IncorrectCorrectTotal Rectangular 6285291 Standard Voronoi 22177298 AMWVD 79219298 Total 306581887 –Rectangular vs. AMWVD: χ 2 =69.30; D.F.=1; p<0.1. –Standard VD vs. AMWVD: χ 2 =133.44; D.F.=1; p<0.1.

19. AMCIS 2006 Human Subjects Testing - results RectangularStandard VoronoiAMWVD log(time (ms)) to select largest region) μ : 3.648 μ : 3.632 μ : 3.667 Rectangular μ / μ : 1.004 t:.571; DF: 586; p:.57 μ / μ :.995 t: -2.245; DF: 583; p:.03 Standard Voronoi μ / μ :.9904 t: -2.862; DF: 592; p<.01 RectangularStandard VoronoiAMWVD Time (ms) used to numerically estimate the size relationship μ : 10,004 μ : 7,001 μ : 8,452 Rectangular μ / μ : 1.429 t: 5.576; DF: 432; p < 0.1 μ / μ : 1.184 t: 2.772; DF: 477; p <.01 Standard Voronoi μ / μ :.828 t: -3.830; DF: 578; p <.01

20. 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 E AMWVD continuous (n=181) =.270 –μ E AMWVD discontinuous (n=117) =.266

21. AMCIS 2006 Voronoi Information Spaces - Conclusion Adaptive Multiplicatively Weighted Voronoi Diagram solves weight- proportional partitioning subject to: –inclusiveness: g i є r i –exclusiveness: ∑A i = S –locality: Δg i = 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. AMCIS 2006 Voronoi Information Space - Solutions

23. AMCIS 2006 Voronoi Information Space - Solutions