1. Multi-Dimensional Data Visualization
cs5764: Information Visualization
Chris North

2. Review What is the Visualization Pipeline?
What are the steps of Visual Mapping?
What is the Info Vis Mantra?

3. Information Types Multi-dimensional: databases,… 1D: timelines,…
2D: maps,…
3D: volumes,…
Hierarchies/Trees: directories,…
Networks/Graphs: web, communications,…
Document collections: digital libraries,…

4. The Simple Stuff
Univariate
Bivariate
Trivariate

5. Univariate Dot plot Bar chart (item vs. attribute) Tukey box plot
Histogram

6. Bivariate
Scatterplot

7. Trivariate
3D scatterplot, spin plot
2D plot + size (or color…)

8. Multi-Dimensional Data
Each attribute defines a dimension
Small # of dimensions easy
Data mapping, Cleveland’s rules
What about many dimensional data? n-D
What does 10-D space look like?

9. Projection
map n-D space onto 2-D screen

10. Glyphs: Chernoff Faces
10 Parameters:
Head Eccentricity
Eye Eccentricity
Pupil Size
Eyebrow Slope
Nose Size
Mouth Vertical Offset
Eye Spacing
Eye Size
Mouth Width
Mouth Openness

11. Glyphs: Stars d1 d2 d7 d3 d6 d4 d5

12. Multiple Views with Brushing-and-linking

13. Scatterplot Matrix All pairs of attributes Brushing and linking

14. … on steroids

15. Different Arrangements of Axes
Axes are good
Lays out all points in a single space
“position” is 1st in Cleveland’s rules
Uniform treatment of dimensions
Space > 3D ?
Must trash orthogonality

16. Parallel Coordinates
Inselberg, “Multidimensional detective” (parallel coordinates)

17. Parallel Coordinates
Bag cartesian
(0,1,-1,2)= x y z w

18. Star Plot 1 8 2 7 3 4 6 5
Parallel Coordinates with axes arranged radially

19. Star Coordinates
Kandogan, “Star Coordinates”

20. Star Coordinates Cartesian Star Coordinates
P=(v1,v2,v3,v4,v5,v6,v7,v8)
P=(v1, v2) d1 d1 d8 d2 v3 v4 p v2 v1 v5 v2 d7 d3 d2 v1 p v8 v6 d6 v7
Mapping:
Items → dots
Σ attribute vectors → (x,y) d4 d5

21. Analysis

22. Table Lens
Rao, “Table Lens”

23. FOCUS / InfoZoom
Spenke, “FOCUS”

24. VisDB
Keim, “VisDB”

25. Pixel Bar Charts
Keim

26. Comparison of Techniques

27. Comparison of Techniques
ParCood: <1000 items, <20 attrs
Relate between adjacent attr pairs
StarCoord: <1,000,000 items, <20 attrs
Interaction intensive
TableLens: similar to par-coords
more items with aggregation
Relate 1:m attrs (sorting), short learn time
Visdb: 100,000 items with 10 attrs
Items*attrs = screenspace, long learn time, must query
Spotfire: <1,000,000 items, <10 attrs (DQ many)
Filtering, short learn time

28. Multi-Dimensional Functions
cs5764: Information Visualization
Chris North

29. Multi-Dimensional Functions
y = f(x1, x2, x3, …, xn)
Continuous:
E.g. y = x13 + 2x22 - 9x3
Discrete:
xi are uniformly sampled in a bounded region
E.g. xi = [0,1,2,…,100]
E.g. measured density in a 3D material under range of pressures and room temperatures.

30. Relations vs. Functions
R(A, B, C, D, E, F)
All dependent variables (1 ind.var.?)
Sparse points in multi-d dep.var. space
Functions:
R(A, B, C, D, E, F, Y) : Y=f(A, B, C, D, E, F)
Many independent variables
Defined at every point in multi-d ind.var. space (“onto”)
Huge scale: 6D with 10 samples/D = 1,000,000 data points

31. Multi-D Relation Visualizations…
Don’t work well for multi-D functions
Example:
Parallel coords
5D func sampled on 1-9 for all ind.vars.

32. Typically want to encode ind.vars. as spatial attrs

33. 1-D: Easy
b = f(a)
a  x
b  y b a

34. 2-D: Easy
c = f(a, b)
Height field:
a  x
b  y
c  z c b a

35. 2-D: Easy
c = f(a, b)
Heat map:
a  x
b  y
c  color b a c

36. 3-D: Hard d = f(a, b, c) Color volume: a  x b  y c  z d  color
What’s inside? c b a

37. 4D: Really Hard y = f(x1, x2, x3, x4, …, xn)
What does a 5D space look like?
Approaches:
Hierarchical axes (Mihalisin)
Nested coordinate frames (Worlds within Worlds)
Slicing (HyperSlice)
Radial Focus+Context (PolarEyez, Sanjini)

38. Hierarchical Axes 1D view of 3D function: (Mihalisin et al.)
f(x1, x2, x3) x3 x2 x1

39. as in TableLens 5D
9 samp/D

41. Hierarchical Axes 2D view of 4D function (using heat maps)
y = f(x1, x2, x3, x4)
Discrete: xi = [0,1,2,3,4] x3 x1 x2
y = f(x1,x2,0,0)
as color x4

42. Hierarchical Axes Scale? For more dimensions:
6d = 3 levels in the 2d approach
10 samples/d = 1,000,000 data points = 1 screen
For more dimensions:
zoom in on “blocks”
reorder dimensions

43. 5D 9 sample/D

44. Nested Coordinate Frames
Feiner, “Worlds within Worlds”

45. Slicing
Van Wijk, “HyperSlice”

46. Radial Focus+Context Jayaraman, “PolarEyez” infovis.cs.vt.edu x3 x2 x4

47. Comparison Hierarchical axes (Mihalisin):
Nested coordinate frames (Worlds in Worlds)
Slicing (HyperSlice):
Radial Focus+Context (PolarEyez)

48. Comparison Hierarchical axes (Mihalisin):
< 6d by 10 samples, ALL slices, view 2d at a time
Nested coordinate frames (Worlds in Worlds)
< 5-8d, continuous, no overview, 3d hardware
Slicing (HyperSlice):
< 10d by 100 samples, 2d slices
Radial Focus+Context (PolarEyez)
< 10d by 1000 samples, overview, all D uniform, rays

49. cs5764: Information Visualization Chris North
Dynamic Queries
cs5764: Information Visualization
Chris North

50. HomeFinder

51. Spotfire

52. Limitations Scale: “AND” queries only
Scatterplot screen space: 10,000 – 1,000,000
Data structures & algorithms: < 50,000
Poor screen drawing on Filter-out
A Solution: Query Previews!
“AND” queries only
Arbitrary boolean queries?
A solution: Filter Flow

53. erases items underneath too
DQ Algorithm
Idea: incremental algorithm
only deal with data items that changed state
When slider moves:
Calculate slider delta
Search in data structure for data items in the delta region
If slider moved inward (filter out):
Erase data items from visualization
Else slider moved outward (filter in):
Draw data items on visualization
Problem!
Overlapped items,
erases items underneath too

54. DQ Data Structures (1) Sorted array of the data for each slider
Need counter for each data item = # sliders that filter it
Attribute Explorer visualizes these counters too!
O(delta)
Year:
Delta

55. DQ Data Structures (2) Multi-dimensional data structure
E.g.: K-d tree, quad-tree, …
Recursively split space, store in tree structure
Enables fast range search, O()

56. DQ Data Structures (2) Multi-dimensional data structure
E.g.: K-d tree, quad-tree, …
Recursively split space, store in tree structure
Enables fast range search, O(logn)
Delta

57. Erasure Problem Each pixel has counter = number of items Z-buffer?
Can visualize this for density!
Z-buffer?
Redraw local area only

58. Filter-Flow
Betty
Catherine
Edna
Freda
Grace
Hilda
Judy
Marcus
Tom

59. Influence/Attribute Explorer
Tweedie, Spence, “Externalizing Abstract Mathematical Models” (Influence/Attribute Explorer)

60. Query Previews
Doan, “Query Previews”