International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.8, pp. 326-327

Section 3.8.2.5. Generation of the correlation and distance matrices

C. J. Gilmore,a G. Barra and W. Donga*

aDepartment of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK
Correspondence e-mail:  chris@chem.gla.ac.uk

3.8.2.5. Generation of the correlation and distance matrices

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Using equation (3.8.3)[link], a correlation matrix is generated in which a set of n patterns is matched with every other to give a symmetric (n × n) correlation matrix ρ with unit diagonal. The matrix ρ can be converted to a Euclidean distance matrix, d, of the same dimensions via[{\bf{d}} = 0.5\left({1.0 - \boldrho } \right) \eqno(3.8.7)]or a distance-squared matrix,[{\bf{D}} = 0.25{\left({1 - \boldrho } \right)^2} \eqno(3.8.8)]for each entry i, j in d, [0.0 \le {d_{ij}} \le 1.0]. A correlation coefficient of 1.0 translates to a distance of 0.0, a coefficient of −1.0 to 1.0, and zero to 0.5. There are other methods of generating a distance matrix from ρ (see, for example, Gordon, 1981[link], 1999[link]), but we have found this to be both simple and as effective as any other.

For other purposes a dissimilarity matrix s is also needed, whose elements are defined via[s_{ij} = 1 - d_{ij}/d^{\max } ,\eqno(3.8.9)]where dmax is the maximum distance in matrix d. A dissimilarity matrix, δ, is also generated with elements[\delta _{ij} = d_{ij}/d_{ij}^{\max}. \eqno(3.8.10)]In some cases it can be advantageous to use I1/2 in the distance-matrix generation; this can enhance the sensitivity of the clustering to weak peaks (Butler et al., 2019[link]).

References

Butler, B. M., Sila, A., Nyambura, K. D., Gilmore, C. J., Kourkoumelis, N. & Hillier, S. (2019). Pre-treatment of soil X-ray powder diffraction data for cluster analysis. Geoderma, 337, 413–424.Google Scholar
Gordon, A. D. (1981). Classification, 1st ed., pp. 46–49. London: Chapman and Hall.Google Scholar
Gordon, A. D. (1999). Classification, 2nd ed. Boca Raton: Chapman and Hall/CRC.Google Scholar








































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