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, pp. 326-327
Section 3.8.2.5. Generation of the correlation and distance matrices^{a}Department of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK |
Using equation (3.8.3), 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 viaor a distance-squared matrix,for each entry i, j in d, . 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, 1999), 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 viawhere d^{max} is the maximum distance in matrix d. A dissimilarity matrix, δ, is also generated with elementsIn some cases it can be advantageous to use I^{1/2} in the distance-matrix generation; this can enhance the sensitivity of the clustering to weak peaks (Butler et al., 2019).
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 ScholarGordon, 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