1. Extension of Star Coordinates into Three Dimensions Nathan Cooprider University of Utah School of Computing coop@cs.utah.edu Robert Burton Brigham Young University Department of Computer Science rpburton@cs.byu.edu Abstract Conclusion Selected References Traditional Star Coordinates displays a multi-variate data set by mapping it to two Cartesian dimensions. This technique facilitates cluster discovery and multi-variate analysis, but binding to two dimensions hides features of the data. Three- dimensional Star Coordinates spreads out data elements to reveal features. This allows the user more intuitive freedom to explore and process the data sets. Three-dimensional Star Coordinates is implemented by extending the data structures and transformation facilities of traditional Star Coordinates. We have given high priority to maintaining the simple, traditional interface. We simultaneously extend existing features, such as scaling of axes, and add new features, such as system rotation in three dimensions. These extensions and additions enhance data visualization and cluster discovery. We use three examples to demonstrate the advantage of three-dimensional Star Coordinates over the traditional system. First, in an analysis of customer churn data, system rotation in three dimensions gives the user new insight into the data. Second, in cluster discovery of car data, the additional dimension allows the true shape of the data to be seen more easily. Third, in a multi-variate analysis of cities, the perception of depth increases the degree to which multi-variate analysis can occur. We have presented three-dimensional Star Coordinates as a valuable extension of two-dimensional Star Coordinates. It retains all the utility of traditional 2D Star Coordinates and has several advantages over the two-dimensional technique. First, system rotation allows for configurations of data to be maintained while considering different views. Second, the infinitely enlarged space of volumes relative to surfaces allows the structure of the data to be discovered more easily. Third, depth cues provide attribute references which may be used to perform more complex multi-variate analysis. These features significantly enhance cluster discovery and data analysis. These features have been provided by extending Kandogan’s implementation of Star Coordinates in significant ways. In traditional 2D Star Coordinates, the underlying representation of the data is inherently two- dimensional, the display is inherently two-dimensional, and the input device (mouse) is inherently two-dimensional. We overcome these limitations through a combination of methods and techniques. Three-dimensional Star Coordinates maintains the intuitiveness of two-dimensional Star Coordinates while providing and capitalizing on the new, three-dimensional aspects of the system. 1.Develop a three-dimensional visualization technique 2.System must be interactive 3.Build on a successful two-dimensional system, preserving the desirable features of the interface 1.R. Johnson, “Visualization of multi-dimensional data with vector fusion,” in IEEE Proceedings Visualization 2000, pp. 297–302 and 570, 2000. 2.E. Kandogan, “Star Coordinates: A multi-dimensional visualization technique with uniform treatment of dimensions,” in Proceedings of the IEEE Information Visualization Symposium, Late Breaking Hot Topics, pp. 4–8, 2000. 3.M. Tavanti and M. Lind, “2D vs 3D, implications on spatial memory,” in Proceedings of the Conference on Information Visualization ’01, 2001. 4.P. Hoffman, G. Grinstein, K. Marx, I. Grosse, and E. Stanley, “DNA visual and analytic data mining,” in VIS ’97: Proceedings of the 8th conference on Visualization ’97, pp. 437–ff., IEEE Computer Society Press, (Los Alamitos, CA, USA), 1997. 5.E. Kandogan, “Visualizing multi-dimensional clusters, trends, and outliers using Star Coordinates,” in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 107–116, ACM Press, 2001. Acknowledgments We thank Dr. Eser Kandogan of IBM’s Almaden Research Center for his cooperation, support of this project, and original work with Star Coordinates. We also thank John Regehr, Eric Eide, and the anonymous reviewers for their comments and suggestions. This work was supported by the Department of Computer Science at Brigham Young University. Goals Three Applications: Churn, Cars, and Cities Star Coordinates Kandogan describes Star Coordinates as “simply an extension of typical 2d and 3d scatter-plots to higher dimensions with normalization”. 2 It is a form of interactive multidimensional scaling. The system creates an axis for each component attribute and then maps the values of that attribute for the data to vectors along each axis. As explained in Kandogan's original work, 2 Star Coordinates plots a star by summing the vectors of each component attribute for the piece of data the star represents. Figure 2 illustrates the calculation of a single data star in Star Coordinates. Axes exist for each of the eight attributes, labeled C 1 through C 8. The d-labeled vectors are computed for each attribute based on the orientation of the axes and the mapping of values. The smallest value in the set for an attribute is mapped to the origin and the largest is mapped to the end of the attribute's axis. In this way, no data values are “out of range.” By computing the sum of the d vectors, Star Coordinates determines the location of star P. Our work extends this two-dimensional model into three dimensions. Figure 1. Computation for a single star location in Star Coordinates 2 (Used with author’s permission) Figure 2. Kandogan’s implementation of 2D Star Coordinates Figure 3. 3D Star Coordinates after rotating the system Figure 4. Two-dimensional Star Coordinates performing cluster analysis on churn data. Churned and non-churned customers are clustered together confusingly. Figure 5. Three-dimensional Star Coordinates performing cluster analysis on churn data. Churned customers are separated from non-churned customers. Figure 9. Three-dimensional Star Coordinates performing multi-variate analysis of cities, using all variables to compare cities Figure 8. Two-dimensional Star Coordinates performing multi- variate analysis of cities, limited to fewer variables by space constraints Figure 7. Three-dimensional Star Coordinates performing cluster analysis on car specification, revealing five clusters Figure 6. Two-dimensional Star Coordinates performing cluster analysis on car specification, revealing four clusters New York San Francisco Provo Salt Lake City San Jose Three-dimensional Star Coordinates extends traditional two- dimensional Star Coordinates in several ways: ● Stars distribute in a volume instead of a plane, giving users more space to exploit. ● Depth cues allow users to include more meaningful variables simultaneously in an analysis. ● Transformations are extended to three dimensions. ● System rotation is introduced as a powerful new transformation. Churn in telecommunication companies (Figures 4 and 5). Usage and plan information for 5000 customers is considered. “Churning” occurs when a customer cancels services. The data set is analyzed to discover why customers churn. Car specifications (Figures 6 and 7). Data from 406 cars manufactured world-wide. Analyzed to find clusters of similar cars. City attributes (Figures 8 and 9). Data from 329 cities evaluated based on user criteria. Goal is to discover city that maximizes desirable attributes while minimizing undesirable ones. Extended Extended Extended