Don’t panic … just do it in parallel!

Abstract

Parallel coordinates is a methodology for visualizing N-dimensional geometry and multivariate problems. In this self-contained up-to-date overview the aim is to clarify salient points causing difficulties, and point out more sophisticated applications and uses in statistics which are marked by **. Starting from the definition of the parallel-axes multidimensional coordinate system, where a point in Euclidean N-space RN is represented by a polygonal line, it is found that a point ↔ line duality is induced in the Euclidean plane R2. This leads to the development in the projective, P2, rather than the Euclidean plane. Pointers on how to minimize the technical complications and avoid errors are provided. The representation (i.e. visualization) of 1-dimensional objects is obtained from the envelope of the polygonal lines representing the points on their points. On the plane R2 there is a inflection-point ↔ cusp, conies ↔ conies and other potentially useful dualities. A line l ⊂ RN is represented by N − 1 points with a pair of indices in [1, 2, …, N]. This representation also enables the visualization and computation of proximity properties like the minimum distance between pairs of lines [18]. The representation of objects of dimension ≥ 2 is obtained recursively. Specifically, the representation of a p-flat, a plane of dimension 2 ≤ p ≤ N − 1 in RN is obtained from the (p−1)-flats it contains, and which are obtained from the (p−2)-flats and so on all the way down from the points (0-dimensional); hence the recursion. A p-flat is represented by p-points each with (p+1) indices. This is the key message: ** high-dimensional objects may be visualized recursively, in terms of their higher dimensional components, rather than directly from their points. Further, this process is robust so that “near” p-flats are also detected in the same way and very useful tight error bounds are available. The representation of a smooth hypersurface in RN is obtained as the envelope of the tangent hyperplanes. The set of points obtained in this way visually reveal properties like convexity, whether the surface is developable, or ruled. A simpler but ambiguous representation for hypersurfaces is also given together with modeling applications of an algorithm for computing and displaying interior, exterior or surface points.