This work explores visualization methods for high-dimensional data using projections with the C2 Virtual Reality technology. The methods being explored can be likened to examining an object by looking at its shadows, for example you could distinguish between a pine tree and a red maple tree by looking only at the shadows cast on the ground by the two (especially during the winter months of the year!).
Projections have a history in multivariate analysis: union-intersection principle for constructing hypothesis tests, obtaining robust estimators for variance-covariance matrices by using 1-dimensional robust statistics on projections from high-dimensional spaces, non-parametric regression. From the earliest stages of statistical graphics, projections were used to obtain low-dimensional summaries of high-dimensional data. As a tool for general visualization the theory of using projections has just recently been crystallized.
One major breakthrough in using projections for visualizing higher dimensions was made by Asimov (1985) in his work on the grand tour. The grand tour in an abstract sense shows a viewer all possible projections in a continuous stream (which could be considered to be moving planes through p-space). Several possibilities for ``showing all possible projections'' were explored in the original work, but the most successful method to arise from it is based on interpolating between random planes.
Interpolation between projections is carefully documented in a article Buja, Cook, Asimov, Hurley (1997). This work is actually fundamental to the visualization process because it allows any sequence of projections to be viewed in a continuous manner. Continuity is vital for perceiving structure, analogous to the fact that moving objects are easier to discern from a background than still objects.
While the random walk approach provided by the grand tour is good for giving a general overview of the data, it is not useful for revealing finer detailed features of the data. Generally the finer structure of the data can only be seen in a small proportion of all possible projections. Several directions have been explored to discover and help focus on important finer details of data structure: (1) restricting motion to important subspaces, for example that provided by a few principal components (Hurley, 1990) and (2) directing the sequence of the projections by a guidance mechanism such as projection pursuit (Cook, Buja, Cabrera, Hurley, 1995). In the latter method a numerical measure of the importance of a projection is used to increase the frequency with which we select revealing projections over unrevealing projections.
The binding tool to the grand tour and the guided tour mechanisms for viewing projections is, as a final stage in the exploratory process, manual control by the user: to explore the neighborhood of interesting projections as provided by the grand or guided tour by manipulating the contribution of particular variables or combinations of variables. These have been developed recently. Examples of use can be seen in Buja, Cook, Swayne (1995) and technical details are to appear in Cook and Buja (1997).
Our work involves exploring the use of these controls in the C2 environment. Briefly the C2 Virtual Reality environment is a structure that has 3 to 4 walls. Projections of a 3-dimensional object (or collection of objects) are displayed on the wall, in stereo and users need 3d-viewing glasses to perceive the full structure. Perspective is changed as the user moves virtually through around and through the object. Several users can view the same object provided one user controls the movement.
Various directions of the use the C2 need to be explored for statistical graphics applications (some more immediately than others):
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