A Visualization Model Based on Adjacency Data

Transcription

1 A Visualization Model Based on Adjacency Data by Edward Condon Bruce Golden S. Lele S. aghavan Edward Wasil Presented at Miami Beach INFOMS Conference Nov.

2 Focus of Paper The focus of this paper will be on a visualization project based on adjacency data (Fiske data) The paper illustrates the power of visualization Visualization generates insights and impact

3 Motivation Typically, data are provided in multidimensional format A large table where the rows represent countries and the columns represent socio-economic variables Alternatively, data may be provided in adjacency format Consumers who buy item a are likely to buy or consider buying items b, c, and d also Students who apply to college a are likely to apply to colleges b, c, and d also

4 Motivation -- continued More on adjacency If the purchase of item i results in the recommendation of item j, then item j is adjacent to item i Adjacency data for n alternatives can be summarized in an n x n adjacency matrix, A = (a ij ), where a ij = if item j is adjacent to item i, and otherwise Adjacency is not necessarily symmetric

5 Motivation -- continued Adjacency indicates a notion of similarity Given adjacency data w.r.t. n items or alternatives, can we display the items in a two-dimensional map? Traditional tools such as multidimensional scaling and Sammon maps work well with data in multidimensional format Can these tools work well with adjacency data? 4

6 Sammon Map of World Poverty Data Set (World Bank, 99) 5

7 Obtaining Distances from Adjacency Data How can we use linkage information to determine distances? c a b items adjacent to a items adjacent to b d items adjacent to c e items adjacent to d 6

8 Obtaining Distances from Adjacency Data -- continued. Start with the n x n - asymmetric adjacency matrix. Convert the adjacency matrix to a directed graph Create a node for each item (n nodes) Create a directed arc from node i to node j if a ij =. Compute distance measures Each arc has a length of Compute the all-pairs shortest path distance matrix D The distance from node i to node j is d ij 7

9 4. Modify the distance matrix D, to obtain a final distance matrix X Symmetry Disconnected components Example Obtaining Distances from Adjacency Data -- continued A = 5 6 4

10 Find shortest paths between all pairs of nodes to obtain D Average d ij and d ji to arrive at a symmetric distance matrix X Example -- continued = D = X

11 Example A and B are strongly connected components The graph below is weakly connected There are paths from A to B, but none from B to A MDS and Sammon maps require that distances be finite A B

12 Ensuring Finite and Symmetric Basic idea: simply replace all infinite distances with a large finite value, say If is too large The points within each strongly connected component will be pushed together in the map Within-component relationships will be difficult to see If is too small Distances Distinct components (e.g., A and B) may blend together in the map

13 Ensuring Finite and Symmetric Distances -- continued must be chosen carefully (see Technical eport) This leads to a finite distance matrix D Next, we obtain the final distance matrix X where x = x = + ij ji ( d d )/ X becomes input to a Sammon map or MDS procedure ij ji

14 Application: College Selection Data source: The Fiske Guide to Colleges, edition Contains information on colleges Approx. 75 pages Loaded with statistics and ratings For each school, its biggest overlaps are listed Overlaps: the colleges and universities to which its applicants are also applying in greatest numbers and which thus represent its major competitors

15 Overlaps and the Adjacency Matrix Penn s overlaps are Harvard, Princeton, Yale, Cornell, and Brown Harvard s overlaps are Princeton, Yale, Stanford, M.I.T., and Brown Note the lack of symmetry Harvard is adjacent to Penn, but not vice versa 4

16 Proof of Concept Start with colleges and the associated adjacency matrix From the directed graph, several strongly connected components emerge We focus on the four largest to test the concept ( schools) Component A has 74 schools Component B has southern colleges Component C has 8 mainly Ivy League colleges Component D has 7 California universities 5

17 B B B B B 4 B 8 B B 7 B 6 B 5 B 9 A 8 A A A 4 4 A 4 A A 5 A 5 A 6 7 A 5 7 A 5 8 A 6 A 9 A 6 4 A 9 A A 5 5 A 6 A 5 A 9 A 6 A 4 A 4 A 7 A 7 A A 4 5 A 9 A 7 A 6 8 A 4 7 A A 6 5 A 6 6 A 7 A 7 4 A A A 5 A 4 A 4 8 A 6 A A 4 9 A 6 A 4 6 A 5 A 4 A A 6 A 6 A 5 A 5 C 5 C A 6 9 C 4 C 6 C C 8 C C 7 A 4 A 6 A A 7 A 5 9 A A 4 D D 6 A 8 D 4 D 5 A 5 6 D 7 D D A 4 A A 7 A 8 A A 5 A 5 4 A A 7 A 7 A 8 Sammon Map with Each School Labeled by its Component Identifier 6

18 S C A L T N S C S C G A A L G A F L F L F L P A V A P A P A N C N J V A P A N C N J T N M D V A V A N C G A D E V A V A V A D C M A P A N Y M A N J C T M A C A I N Y M A M A M A P A M E M E M E P A P A P A N Y IN M I IN M I M O IL IL W I IN M A M A M A M A C T N H V T N Y N Y M N M N C A IA O H O C A C A C A V T W I M N IA IA A Z C O O C A C A C A W A C O A Z O O W A O W A C O Sammon Map with Each School Labeled by its Geographical Location 7

19 Sammon Map with Each School Labeled by its Designation ( Public () or Private () ) 8

20 Sammon Map with Each School Labeled by its Cost 9

21 Sammon Map with Each School Labeled by its Academic Quality

22 A45 A65 A66 A7 C5 C C8 C C7 C NC VA VA VA PA NY CT MA CA I A9 GA A A6 DC MA A68 A5 A4 A MO MA NY NY (a) Identifier (b) State (c) Public or private (d) Cost NC VPI VAW&M Penn Cornell Yale Harvard Stanford Brown Emory Georgetown Tufts Wash BC NY Barnard (e) Academics (f) School name Six Panels Showing Zoomed Views of Schools that are Neighbors of Tufts niversity

23 Benefits of Visualization Adjacency (overlap) data provides local information only E.g., which schools are Maryland s overlaps? With visualization, global information is more easily conveyed E.g., which schools are similar to Maryland?

24 Benefits of Visualization -- continued Within group (strongly connected component) and between group relationships are displayed at same time A variety of what-if questions can be asked and answered using maps Based on this concept, a web-based DSS for college selection is easy to envision

25 Conclusions The approach represents a nice application of shortest paths to data visualization The resulting maps convey more information than is immediately available in The Fiske Guide Visualization encourages what-if analysis of the data Can be applied in other settings (e.g., web-based recommender systems) 4