Summary
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.
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Notes
1 Legend has it that young people are most likely to do something when they are forbidden from doing so … rather than when they encouraged.
2 Since then Parallel Coordinates has occasionally enjoyed rediscovery by other “inventors” … confirming the old adage that “success has many parents while failure is an orphan”.
3 For some applications, we will see later that, different inter-axes distances are more advantageous.
4 It has been reported that in a rare Ecumenical manifestation, the Chief Rabbinate of Israel and the Vatican are preparing a non-denominational blessing for Dimensionality.
5 So far there at 17 1/2 known ways of proving of this, but rush to contribute yours since the grand prize for the most elegant proof has not yet been awarded.
6 To iterate is human, but to recurse is divine.
References
E.W. Bassett. Ibm’s ibm fix. Industrial Computing, 14(41):23–25, 1995.
V. G. Boltyanskii. Envelopes, R.B.Brown translator (original in Russian). Pergamon Press, New York, 1964.
A. Buja, D. Cook, D. Asimov, and C. Hurley. Theory and computational methods for dynamic projections in high-dimensional data visualization. http:/www.research.att.com/andreas/dynamic-projections.ps.gz, unpublished manuscript.
A. Chatterjee. Visualizing Multidimensional Polytopes and Topologies for Tolerances. Ph.D. Thesis USC, 1995.
T. Chomut. Exploratory Data Analysis in Parallel Coordinates. M.Sc. Thesis, UCLA Comp. Sc. Dept., 1987.
H. S. M. Coxeter. The Real Projective Plane. Third Edition, Springer-Verlag, New York, 1992.
P. Ehrenfest. In What Way Does it Become Manifest in the Fundamental Laws of Physics that Space has Three Dimensions?, — Appeared in 1917 — in Collected Scientific Papers of P. Ehrenfest, Klein, M.J. ed., 400–409. North-Holland Publishing Company, Interscience Publishers, New York, 1959.
J. Eickemeyer. Visualizing p-flats in N-space using Parallel Coordinates. Ph.D. Thesis UCLA, 1992.
R. Finsterwalder. A Parallel Coordinate Editor as a Visual Decision Aid in Multi-Objective Concurrent Control Engineering Environment 119–122. IFAC CAD Contr. Sys., Swansea, UK, 1991.
C. Gennings, K. S. Dawson, W. H. Carter, and R. H. Myers. Interpreting plots of a multidimensional dose-response surface in a parallel coordinate systems. Biometrics, 46:719–35, 1990.
C.K. Hung. A New Representation of Surfaces Using Parallel Coordinates. Submitted for Publication, 1998.
A. Inselberg. N-Dimensional Graphics, Part I — Lines and Hyperplanes, in IBM LASC Tech. Rep. G320-2711, 140 pages. IBM LA Scientific Center, 1981.
A. Inselberg. The plane with parallel coordinates. Visual Computer, 1:69–97, 1985.
A. Inselberg. Parallel Coordinates: A Guide for the Perplexed, in Hot Topics Proc. of IEEE Conf. on Visualization, 35–38. IEEE Comp. Soc., Los Alamitos, CA, 1996.
A. Inselberg. Visual data mining with parallel coordinates. J. of Comp. Stat., 13-1:47–64, 1998.
A. Inselberg and B. Dimsdale. Parallel Coordinates: A Tool For Visualizing Multidimensional Geometry, in Proc. of IEEE Conf. on Vis.’ 90, 361–378. IEEE Comp. Soc., Los Alamitos, CA, 1990.
A. Inselberg and B. Dimsdale. Multidimensional lines i: Representation. SIAM J. of Applied Math., 54-2:559–577, 1994.
A. Inselberg and B. Dimsdale. Multidimensional lines ii: Proximity and applications. SIAM J. of Applied Math., 54-2:578–596, 1994.
A. Inselberg, M. Reif, and T. Chomut. Convexity algorithms in parallel coordinates. J. ACM, 34:765–801, 1987.
D. A. Keim and H. P. Kriegel. Visualization techniques for mining large databases: A comparison. Trans. Knowl. and Data Engr., 8-6:923–938, 1996.
A. R. Martin and M. O. Ward. High dimensional brushing for interactive exploration of multivariate data, Proc. IEEE Conf. on Visualization, Atlanta, GA, 271–278. IEEE Comp. Soc., Los Alamitos, CA, 1995.
T. Matskewich and Y. Brenner. Tight error bounds for fitting points by perturbed flats. To be submitted for publication, 1998.
B. H. McCormick, T. A. Defanti, and M. D. Brown. Visualization in Scientific Computing. Computer Graphics 21-6, ACM SIGGRAPH, New York, 1987.
M. Schall. Diamond and ice: Visual exploratory data analysis tools. Perspective, J. of OAC at UCLA, 18(2):15–24, 1994.
C. Schmid and H. Hinterberger. Comparative Multivariate Visualization Across Conceptually Different Graphic Displays, in Proc. of 7th SSDBM. IEEE Comp. Soc., Los Alamitos, CA, 1994.
D. F. Swayne, D. Cook, and A. Buja. XGobi: Interactive Dynamic Graphics in the X Window System. JCGS, 7-1, 113–130, 1998.
J. L. Synge and Schild A. Tensor Calculus 2nd Edit. University of Toronto Press, Toronto, 1956.
L. S. Tierney. LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics, Exer. 10.15. Wiley, New York, 1990.
E. R. Tufte. The Visual Display of Quantitative Information. Graphic Press, Connecticut, 1983.
E. R. Tufte. Envisioning Information. Graphic Press, Connecticut, 1990.
E. R. Tufte. Visual Explanation. Graphic Press, Connecticut, 1996.
L Tweedie and R. Spence. The prosection matrix: A tool to support the interactive exploration of statistical analysis. J. of Comp. Stat., 13-1:65–76, 1998.
M. O. Ward. XmdvTool: integrating multiple methods for visualizing multivariate data, Proc. IEEE Conf. on Visualization, San Jose, CA, 326–333. IEEE Comp. Soc., Los Alamitos, CA, 1994.
E. Wegman and Qiang L. High dimensional clustering using parallel coordinates and the grand tour. J. Comp. Science & Stat., 28:352–360, 1997.
P.C. Wong and Bergeron R.D. 30 Years of Multidimensional Multivariate Visualization in Scientific Visualization: Overviews, Methodologies & Techniques, G.M. Nelson, H. Mueller and H. Hagen (Eds.) 3–33. IEEE Comp. Soc., Los Alamitos, CA, 1997.
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Inselberg, A. Don’t panic … just do it in parallel!. Computational Statistics 14, 53–77 (1999). https://doi.org/10.1007/PL00022705
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DOI: https://doi.org/10.1007/PL00022705