International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.8, p. 331
Section 3.8.4.2.2. The grand tour^{a}Department of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK |
The grand tour is a method of animating the parallel-coordinates plot to examine it from all possible viewpoints. Consider a 3D data plot using orthogonal axes: a grand tour takes 2D sections through these data and displays them in parallel-coordinates plots in a way that explores the entire space in a continuous way. The former is important, because the data can be seen from all points of view, and the latter allows the user the follow the data without abrupt discontinuities. This concept was devised by Asimov (1985) and further developed by Wegman (1990). In more than three dimensions it becomes a generalized rotation of all the coordinate axes. A d-dimensional tour is a continuous geometric transformation of a d-dimensional coordinate system such that all possible orientations of the coordinate axes are eventually achieved. The algorithm for generating a smooth and complete view of the data is described by Asimov (1985).
To do this, the restriction of p = 3 in the MMDS calculation is relaxed to 6, so that there is now a 6D data set with six orthogonal axes. The choice of six is somewhat arbitrary – more can be used, but six is sufficient to see whether the clustering is maintained without generating unduly complex plots and requiring extensive computing resources. The data are plotted as a parallel-coordinates plot. The grand-tour method is then applied by a continuous geometric transformation of the 6D coordinate system such that all possible orientations of the axes are achieved. Each orientation is reproduced as a parallel-coordinates plot using six axes.
Figs. 3.8.9(j) and (k) show an example from the clustering of the 13 aspirin samples using PXRD data. Fig. 3.8.9(j) shows the default parallel-coordinates plot. Fig. 3.8.9(k) shows alternative views of the data taken from the grand tour. In Fig. 3.8.9(j) there appears to be considerable overlap between clusters in the 4th, 5th and 6th dimensions (X4, X5 and X6), but the alternative view given in Fig. 3.8.9(k) show that the clustering is actually well defined in all six dimensions (Barr, Dong & Gilmore, 2009).
References
Asimov, D. (1985). The grand tour: a tool for viewing multidimensional data. SIAM J. Sci. Stat. Comput. 6, 128–143.Google ScholarBarr, G., Dong, W. & Gilmore, C. J. (2009). PolySNAP3: a computer program for analysing and visualizing high-throughput data from diffraction and spectroscopic sources. J. Appl. Cryst. 42, 965–974.Google Scholar
Wegman, E. J. (1990). Hyperdimensional data analysis using parallel coordinates. J. Am. Stat. Assoc. 85, 664–675.Google Scholar