Dimensional Stacking

 
Figure 4

Several recent techniques have emerged which involve projecting high dimensional data by embedding dimensions within other dimensions.  In the 1-D case [MIH:91], one starts by discretizing the ranges of each dimension and assigning an ordering to the dimensions (dimensions are said to have unique ``speeds'').  A background color is also associated with each speed. The next step is to divide the screen into C₀ vertical strips, where C₀ is the cardinality of the dimension with the slowest speed.  The strips are colored according to that speed.  Each of these strips is then divided into C₁ strips and colored accordingly.  This is repeated until all dimensions have been embedded and the data value associated with each cell can be plotted on the vertical axis.

In 2-D [LEB:90], an analogous technique called Dimensional Stacking involves recursively embedding images defined by a pair of dimensions within pixels of a higher-level image. Unlike the previous system, however, data is not restricted to functions, thus making this technique amenable to a wider range of data types.  In Worlds within Worlds [BESHERS:93], each location in a 3-D space may in turn contain a 3-D space which the user may investigate in a hierarchical fashion.  The most detailed level may contain surfaces, solids, or point data.

XmdvTool requires three types of information to project data using dimensional stacking.  The first is the cardinality (number of buckets) for each dimension. The range of values for each dimension is then decomposed into that many equal sized subranges.  The second type of information needed is the ordering for the dimensions, from outer-most (slowest) to inner-most (fastest). Dimensions are assumed to alternate in orientation. The last piece of information used is the minimum size for the plotted data item (the system will increase this value if the entire image can fit within the view area).  Each data point then maps into a unique bucket, which in turn maps to a unique location in the resulting image.  If the image generated exceeds the size of the view area, scroll bars are automatically generated to allow panning.  A key is provided in a separate window to help users understand the order of embedding, and grid lines of varying intensity provide assistance in interpreting transitions between buckets at different levels in the hierarchy.

The sparseness of the data set of Figure 1 makes uncovering relationships difficult using Dimensional Stacking. Figure 4 shows a denser set consisting of 3-D drill hole data with a fourth dimension representing the ore grade found at the location (more than 8000 data points).  Longitude and latitude are mapped to the outer dimensions, each with cardinality 10. Depth and ore grade map to the inner dimensions (ore grade is the vertical orientation), with cardinality 10 and 5, respectively. There is a clear region in which the ore grade improves with depth, and other places where digging had stopped prior to the ore grade falling significantly.  By adjusting the cardinalities and ranges for the various dimensions, a more detailed view of the data may be obtained [WAR:CGI].

The hierarchical techniques are best suited for fairly dense data sets and do rather poorly with sparse data.  This is due to the fact that each possible data point is allocated a specific screen location (with overlaps avoidable by careful discretization of dimensions), and as the dimension of the data increases, the screen space needed expands rapidly.  In contrast, the techniques described earlier generally do well with sparse data over high numbers of dimensions, though scatterplots are constrained somewhat it the maximum manageable dimension.  The major problem with hierarchical methods is the difficulty in determining spatial relationships between points in non-adjacent dimensions.  Two points which in fact are quite close in N-space may project to screen locations which are quite far apart.  This is somewhat alleviated by providing users with the ability to rapidly change the nesting characteristics and discretization of the dimensions.

We have extended flat dimensional stacking  to hierarchical dimensional stacking . In the hierarchical dimensional stacking display, a block within the extent of a cluster is assigned the color of this cluster. Its density is determined by the distance between it and the mean of the cluster, that is, the longer the lighter. Movie 4 is a multiresolutional cluster display of hierarchical dimensional stacking.

References

[BESHERS:93]:  Beshers, C., Feiner, S..  AutoVisual: rule-based design of interactive multivariate visualizations.   IEEE Computer Graphics and Applications, Vol. 13, No. 4, pp. 41-49, 1993.

[LEB:90]:  LeBlanc, J., Ward, M.O., Wittels, N..  Exploring N-dimensional databases.  Proceedings of Visualization '90, pp. 230 - 237, 1990.

[MIH:91]:  Mihalisin, T., Gawlinski, E., Timlin, J., and Schwegler, J..  Visualizing multivariate functions, data, and distributions. IEEE Computer Graphics and Applications, Vol. 11, pp. 28 - 37, 1991.

[WAR:CGI]:  Ward, M.O., LeBlanc, J.T., Tipnis, R..  N-Land: a graphical tool for exploring N-dimensional data. Proceedings of CG International '94, 1994.

[WARD:94]:  M. Ward.  Xmdvtool: Integrating multiple methods for visualizing multivariate data.  Proc. of Visualization '94, p. 326-33, 1994.