Multivariate and High Dimensional Visualizations

1. Multivariate and High Dimensional Visualizations Robert Herring

2. Articles Covered • Visualizing the Behavior of Higher Dimensional Dynamical Systems • Rainer Wegenkittl, Helwig Loffelmann, and Eduard Groller • Multivariate Visualization Using Metric Scaling • Pak Chung Wong and R. Daniel Bergeron

3. Problem Addressed • Information gathered often contains multiple variables to be studied • Most visualization techniques focus on discrete statistical characteristics • These techniques are ill suited for visualizing continuous flow in high-dimensional space from dynamical systems • Statistical visualizations typically are not designed to show integral curves within a high-dimensional phase space Visualizing the Behavior of Higher Dimensional Dynamical Systems

4. Visualizing Multidimensional Data • Multivariate data sets becoming common • Data is either discrete of continuous • Data can be spatially coherent or spatially incoherent • Data sets may consist of a collection of sampled data • Each sample is an n-dimensional data item • Can be sampled from m-dimensional space • Lmn data set Visualizing the Behavior of Higher Dimensional Dynamical Systems

5. Visualizing Multidimensional Data • Two important goals • Identification of individual parameters • Detection of regions and correlation of variables • Methods for visualizing high-dimensional data • Attribute Mapping • Geometric Coding • Sonification • Reduction of Dimension • Parallel Coordinates Visualizing the Behavior of Higher Dimensional Dynamical Systems

6. Attribute Mapping • Use geometric primitives, planes, etc. • Most commonly used attribute used in attribute mapping is color • Most common color models RGB and HLS • Advantages • Easy calculation/interpretation, many people familiar with color mapping • Disadvantages • No unique order, requires legend • Can only encode 3 variables • 8% of population has some form of color blindness Visualizing the Behavior of Higher Dimensional Dynamical Systems

7. Geometric Coding • Use distinct geometric objects and map high-dimensional data to geometric features or attributes of these objects • Glyphs • Icons • Chernoff Faces • Data Jacks • m-Arm Glyph Visualizing the Behavior of Higher Dimensional Dynamical Systems

8. Geometric Coding • Glyphs • Utilized for interactive exploration of data sets • Generic term for graphical entity whose shape/appearance is modified by mapping data values to graphical attributes (length, shape, angle, color, transparency, et.) • Icons • Use icons as basic primitives • Attributes mapped to icon shape, color, and texture to map multiple variables Visualizing the Behavior of Higher Dimensional Dynamical Systems

9. Geometric Coding • Chernoff Faces • Uses stylized faces where variables influence appearance features like overall shape, mouth, eyes, nose, eyebrows, etc • Data Jacks • Three-dimensional shapes with four different limbs (length, color, etc modified) • m-Arm Glyph • Two-dimensional structure with m arms attached (thickness, angle from main axis, etc modified) Visualizing the Behavior of Higher Dimensional Dynamical Systems

10. Sonification • The use of sound to add a layer of dimensional that does not overload visual system • Sounds can vary over • Pitch • Volume • Pulse Visualizing the Behavior of Higher Dimensional Dynamical Systems

11. Reduction of Dimension • Focusing • Selecting subsets, reduction of dimension through projection • Examples are panning, zooming, and slicing • High dimensional data can be mapped to lower dimensions with other dimensions being represented via attributes • Linking • Showing multiple varying visualizations of data Visualizing the Behavior of Higher Dimensional Dynamical Systems

12. Parallel Coordinates • Represent dimensions as parallel axes orthogonal to a horizontal line uniformly spaced on display • Each data set corresponds to a polyline that traverses/intersects these parallel axes Visualizing the Behavior of Higher Dimensional Dynamical Systems

13. High Dimensional Dynamical Systems and Visualization • Many natural phenomenona can be described by differential equations • Each differential equation describes the change of one state variable • n differential equations define behavior of n state variables describing a n-dimensional dynamical system • n-dimensional vector from each sampled set of state variables • The discretized flow described by n differential equations forms a vector field of dimension n, where each vector itself is of dimension n • Lnn data set Visualizing the Behavior of Higher Dimensional Dynamical Systems

14. High Dimensional Dynamical Systems and Visualization • Dynamical system typically describes a complex but smooth flow • Behavior of flow determined by topology • To interpret behavior of system each point within n-space cannot be investigated by itself but seen in respect to its neighborhood • Derived from continuous flow field • Two basic approaches Visualizing the Behavior of Higher Dimensional Dynamical Systems

15. High Dimensional Dynamical Systems and Visualization • Neighboring information can be calculated from vector field (interpreting Jacobian matrix) and the derived data displayed in n-space • Directional information at each point in n-space may be projected to an m-dimensional data object describing some local feature Visualizing the Behavior of Higher Dimensional Dynamical Systems

16. High Dimensional Dynamical Systems and Visualization • Direct global flow visualization can be done by starting short integral curves (trajectories), which follow the flow, at the nodes of an n-dimensional regular grid • Features like separatrices can be detected visually by interpreting the flow directions of the trajectories. Visualizing the Behavior of Higher Dimensional Dynamical Systems

17. Extruded Parallel Coordinates • Instead of using same coordinate system for each sample, move parallel coordinate system along third spatial axis • Polylines viewed as cross sections of a moving plane with a complex surface that defines the trajectory Visualizing the Behavior of Higher Dimensional Dynamical Systems

18. Extruded Parallel Coordinates Visualizing the Behavior of Higher Dimensional Dynamical Systems

19. Extruded Parallel Coordinates • Geometry of surface can be generated and modified quickly • Clustering and correlation visually detectable • Convergence and divergence pbserved by varying the starting coordinates of the trajectory slightly Visualizing the Behavior of Higher Dimensional Dynamical Systems

20. Linking with Wings • Two dimensions of high-dimensional system selected and displayed as a two-dimensional trajectory within a base plane • Third dimension of display can now be use to display third variable over base trajectory • If resulting three-dimensional trajectory is connected with base trajectory, thought of as a wing on the base trajectory • Wing can be tilted at each point within a plane normal to base trajectory Visualizing the Behavior of Higher Dimensional Dynamical Systems

21. Linking with Wings Visualizing the Behavior of Higher Dimensional Dynamical Systems

22. Linking with Wings • Any number of wings can be added to display high-dimensional trajectories (occlusion problem) • Wings can be textured with a grid texture allowing exact measurement of wing dimensions • Self intersection can be a problem • Wing size chosen to be small with respect to size of base trajectories • Angles of wings kept must not be too big Visualizing the Behavior of Higher Dimensional Dynamical Systems

23. Three-dimensional Parallel Coordinates • Based on parallel coordinate method • One-dimensional spaces put together within two-dimensional space (planes) and linked with polylines • Positioning of planes is more flexible • Can be moved, rotated within three-dimensional space Visualizing the Behavior of Higher Dimensional Dynamical Systems

24. Three-dimensional Parallel Coordinates Visualizing the Behavior of Higher Dimensional Dynamical Systems

25. Three-dimensional Parallel Coordinates Visualizing the Behavior of Higher Dimensional Dynamical Systems

26. Results Visualizing the Behavior of Higher Dimensional Dynamical Systems

27. Results Visualizing the Behavior of Higher Dimensional Dynamical Systems

28. Results Visualizing the Behavior of Higher Dimensional Dynamical Systems

29. Results Visualizing the Behavior of Higher Dimensional Dynamical Systems

30. Results Visualizing the Behavior of Higher Dimensional Dynamical Systems

31. Problem Addressed • Large multivariate can be difficult to navigate • Need a low dimensional representation to easy navigation • Metric scaling used as basis for creating low-dimensional overview Multivariate Visualization Using Metric Scaling

32. Metric Scaling • Start with set of n records with v variables and dissimilarities *rs measured between all pairs of records in n dimensional space • Configure graph of n vertices in d dimensional space • Each vertex represents one record • Distances drs measured all pairs of vertices in display space match *rs in variate space as closely as possible • Goal is to determine the dissimilarities between all pairs in v space and map them to coordinates in d dimensional display space Multivariate Visualization Using Metric Scaling

33. Metric Scaling Multivariate Visualization Using Metric Scaling

34. Data Dissimilarity Measurement • Compute dissimilarity between all pairs of input records • Euclidean distance in v space most common metric • Dissimilarity *rs between records r and s Multivariate Visualization Using Metric Scaling

35. Data Dissimilarity Measurement • Dataset with n records generates n x n real symmetric dissimilarity matrix Multivariate Visualization Using Metric Scaling

36. Recovery of Coordinates • Represent data as points in new p dimensional space where p <= n • Create inner product matrix from *rs in variate space, find its non-negative eigenvalues and the corresponding eigenvectors • Yield the Euclidean coordinates of n vertices in p dimensional space Multivariate Visualization Using Metric Scaling

37. Principal Coordinates • Let Euclidean coordinates of n vertices in n dimensional Euclidean space be a matrix • X = [x1, x2, …, xn] such that • xr = [xr1, …, xrn]T where r = 1, 2, …, n • Euclidean distance between vertices r and s (1) (2) Multivariate Visualization Using Metric Scaling

38. Principal Coordinates • Standardize data to have zero mean and unit variance, center of mass of the vertices is the origin Multivariate Visualization Using Metric Scaling

39. Principal Coordinates since Equation (2) becomes (3) Multivariate Visualization Using Metric Scaling

40. Principal Coordinates Similarly (4) Multivariate Visualization Using Metric Scaling

41. Principal Coordinates Furthermore from (4) (5) Multivariate Visualization Using Metric Scaling

42. Principal Coordinates Defining inner product matrix B such that Multivariate Visualization Using Metric Scaling

43. Principal Coordinates Substituting (3), (4), and (5) into (1) gives the inner product Matrix B in terms of drs Multivariate Visualization Using Metric Scaling

44. Principal Coordinates • Use principle components to recover Euclidean coordinates of the n dimensional space denoted by the matrix X from B = XXT • Since B is symmetric and positive semi-definite it has p positive eigenvalues • Let 7 be the eigenvalue matrix where the diagonals are the sorted eigenvalues (6) Multivariate Visualization Using Metric Scaling

45. Principal Coordinates • Let the corresponding normalized eigenvector of 7 be V • By definition of eigenvectors matrix B can be described as B = V7VT • Since there are only p positive eigenvalues B can be expressed as B = V171V1T = V1711/2711/2V1T • Where 71is the eigenvalue matrix with the diagonal with 1 – p eigenvalues and V1 is the corresponding eigenvalue of 71 (7) Multivariate Visualization Using Metric Scaling

46. Principal Coordinates • From (6) and (7) X = V1711/2 Multivariate Visualization Using Metric Scaling

47. Recovery of Coordinates • If eigenvalues are sorted in desending order the first principal component associated with the first eigenvalue is more important than the second • The distance between vertices r and s is • Where xr and xs are the distance vectors associated with points r and s respectively Multivariate Visualization Using Metric Scaling

48. Recovery of Coordinates • A smaller eigenvalue contributes much less weight to the distance drs • Smaller eigenvalues can be truncated with less error • Suppose d is selected as most significant eigenvalue to display data overview, the degree of accuracy of the approximation can be measured by Multivariate Visualization Using Metric Scaling

49. Strengths and Weaknesses • Data clustering Multivariate Visualization Using Metric Scaling

50. Strengths and Weaknesses • Display Density Multivariate Visualization Using Metric Scaling