Computational Methods for High-Dimensional Rotations in Data Visualization

There exist many methods for visualizing complex relations among variables of a multivariate dataset. For pairs of quantitative variables, the method of choice is the scatterplot. For triples of quantitative variables, the method of choice is 3D data rotations. Such rotations let us perceive structure among three variables as shape of point scatters in virtual 3D space. Although not obvious, three-dimensional data rotations can be extended to higher dimensions. The mathematical construction of high-dimensional data rotations, however, is not an intuitive generalization. Whereas three-dimensional data rotations are thought of as rotations of an object in space, a proper framework for their high-dimensional extension is better based on rotations of a low-dimensional projection in high-dimensional space. The term "data rotations" is therefore a misnomer, and something along the lines of "high-to-low dimensional data projections" would be technically more accurate. To be useful, virtual rotations need to be under interactive user control, and they need to be animated. We therefore require projections not as static pictures but as movies under user control. Movies, however, are mathematically speaking one-parameter families of pictures. This article is therefore about one-parameter families of low-dimensional projections in high-dimensional data spaces. We describe several algorithms for dynamic projections, all based on the idea of smoothly interpolating a discrete sequence of projections. The algorithms lend themselves to the implementation of interactive visual exploration tools of high-dimensional data, such as so-called grand tours, guided tours and manual tours.