Multidimensional Techniques for Compositional Data Analysis

  • Surendra P. VermaEmail author


This chapter based on our recent research should represent a novel approach, because, so far, no book has been written on this subject in Earth Sciences, except the widely known classic work of Aitchison (The statistical analysis of compositional data. Chapman and Hall, London, UK, 1986) and a little-known recent book in Spanish by Verma (Análisis estadístico de datos composicionales. Universidad Nacional Autónoma de México, CDMX, 2016). The serious problems with the use of compositional data and ternary diagrams are pointed out. The main objective is to replace ternary diagrams from statistically coherent alternatives based on log-ratio transformations, which has been clearly achieved. The advantage of the fulfillment of the assumption of multi-normality is documented. The methodology presented can be applied to other science or engineering fields. The chapter concludes with a visual explanation of bivariate discordant outliers, which has direct implication with the multivariate discordancy.


  1. Agrawal, S. (1999). Geochemical discrimination diagrams: A simple way of replacing eye-fitted boundaries with probability based classifier surfaces. Journal of the Geological Society of India, 54, 335–346.Google Scholar
  2. Agrawal, S., & Verma, S. P. (2007). Comment on “Tectonic classification of basalts with classification trees” by Pieter Vermeesch (2006). Geochimica et Cosmochimica Acta, 71, 3388–3390.CrossRefGoogle Scholar
  3. Agrawal, S., Guevara, M., & Verma, S. P. (2004). Discriminant analysis applied to establish major-element field boundaries for tectonic varieties of basic rocks. International Geology Review, 46, 575–594.CrossRefGoogle Scholar
  4. Agrawal, S., Guevara, M., & Verma, S. P. (2008). Tectonic discrimination of basic and ultrabasic rocks through log-transformed ratios of immobile trace elements. International Geology Review, 50, 1057–1079.CrossRefGoogle Scholar
  5. Aitchison, J. (1981). A new approach to null correlations of proportions. Mathematical Geology, 13, 175–189.CrossRefGoogle Scholar
  6. Aitchison, J. (1982). The statistical analysis of compositional data (with discussion). Journal of the Royal Statistical Society, Series B (Statistical Methodology), 44, 137–177.Google Scholar
  7. Aitchison, J. (1984). Statistical analysis of geochemical compositions. Mathematical Geology, 16, 531–564.CrossRefGoogle Scholar
  8. Aitchison, J. (1986). The statistical analysis of compositional data. London, UK: Chapman and Hall.CrossRefGoogle Scholar
  9. Appelo, C. A. J., & Postma, D. (2005). Geochemistry, groundwater and pollution. Rotterdam: A.A. Balkema.Google Scholar
  10. Armstrong-Altrin, J. S., & Verma, S. P. (2005). Critical evaluation of six tectonic setting discrimination diagrams using geochemical data of Neogene sediments from known tectonic settings. Sedimentary Geology, 177, 115–129.CrossRefGoogle Scholar
  11. Arnórsson, S. (2000). Isotopic and chemical techniques in geothermal exploration, development and use. Sampling methods, data handling, interpretation (p. 351). Vienna: International Atomic Energy Agency.Google Scholar
  12. Barnett, V., & Lewis, T. (1994). Outliers in statistical data. Chichester: Wiley.Google Scholar
  13. Becke, F. (1897). Gesteine der Columbretes. Tschermaks Mineralogische und Petrographische Mitteilungen, 16, 308–336.Google Scholar
  14. Bevington, P. R., & Robinson, D. K. (2003). Data reduction and error analysis for the physical sciences. Boston: McGraw Hill.Google Scholar
  15. Bhatia, M. R. (1983). Plate tectonics and geochemical composition of sandstones. Journal of Geology, 91, 611–627.CrossRefGoogle Scholar
  16. Borgheresi, M., Buccianti, A., Di Benedetto, F., & Vaughan, D. J. (2013). Application of compositional techniques in the field of crystal chemistry: A case study of luzonite, a Sn-bearing mineral. Mathematical Geosciences, 45, 183–206.CrossRefGoogle Scholar
  17. Buccianti, A. (2013). Is compositional data analysis a way to see beyond the illusion? Computers & Geosciences, 50, 165–173.CrossRefGoogle Scholar
  18. Buccianti, A., Mateau-Figueras, G., & Pawlowsky-Glahn, V. (2006). Compositional data analysis in the geosciences: From theory to practice (p. 212). London: Geological Society of London Special Publication 262.Google Scholar
  19. Butler, J. C. (1979). Trends in ternary petrologic variation diagrams—Fact or fantasy? American Mineralogist, 64, 1115–1121.Google Scholar
  20. Chayes, F. (1960). On correlation between variables of constant sum. Journal of Geophysical Research, 65, 4185–4193.CrossRefGoogle Scholar
  21. Chayes, F. (1971). Ratio correlation. A manual for students of petrology and geochemistry. Chicago and London: The University of Chicago Press.Google Scholar
  22. Dickinson, W. R., Beard, L. S., Brakenridge, G. R., Erjavec, J. I., Ferguson, R. C., Inman, K. F., et al. (1983). Provenance of North American Phanerozoic sandstones in relation to tectonic setting. Geological Society of America Bulletin, 94, 222–235.CrossRefGoogle Scholar
  23. Egozcue, J. J., & Pawlowsky-Glahn, V. (2005). Groups of parts and their balances in compositional data analysis. Mathematical Geology, 37, 795–828.CrossRefGoogle Scholar
  24. Egozcue, J. J., & Pawlowsky-Glahn, V. (2006). Simplicial geometry for compositional data. In A. Buccianti, G. Mateu-Figueras, & V. Pawlowsky-Glahn (Eds.), Compositional data analysis in the geosciences: From theory to practice (pp. 145–159). London: The Geological Society of London Special Publication.Google Scholar
  25. Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., & Barceló-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35, 279–300.CrossRefGoogle Scholar
  26. Filzmoser, P., Hron, K., & Reimann, C. (2009). Univariate statistical analysis of environmental (compositional) data: Problems and possibilities. Science of the Total Environment, 407, 6100–6108.CrossRefGoogle Scholar
  27. Fung, W.-K. (1988). Critical values for testing in multivariate statistical outliers. Journal of Statistical Computation and Simulation, 30, 195–212.CrossRefGoogle Scholar
  28. Gasparik, T. (2003). Phase diagrams for geoscientists: An atlas of the earth’s interior. Berlin: Springer.CrossRefGoogle Scholar
  29. Hall, A. (1996). Igneous petrology. Essex, England: Longman.Google Scholar
  30. Howard, J. L. (1994). A note on the use of statistics in reporting detrital clastic compositions. Sedimentology, 41, 747–753.CrossRefGoogle Scholar
  31. Hron, K., Filzmoser, P., & Thompson, K. (2012). Linear regression with compositional explanatory variables. Journal of Applied Statistics, 39, 1115–1128.CrossRefGoogle Scholar
  32. Jennings, L. W., & Young, D. M. (1988). Extended critical values of the multivariate extreme deviate test for detecting a single spurious observation. Communications in Statistics—Simulation and Computation, 17, 1359–1573.CrossRefGoogle Scholar
  33. Le Maitre, R. W., Streckeisen, A., Zanettin, B., Le Bas, M. J., Bonin, B., Bateman, P., et al. (2002). Igneous rocks. A classification and glossary of terms: Recommendations of the International Union of Geological Sciences Subcommission of the Systematics of Igneous Rocks. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  34. Martín-Fernández, J. A., Barceló-Vidal, C., Pawlowsky-Glahn, V., Kovács, L. O., & Kovács, G. P. (2005). Subcompositional patterns in Cenozoic volcanic rocks of Hungary. Mathematical Geology, 37, 729–752.CrossRefGoogle Scholar
  35. Morrison, D. F. (1990). Multivariate statistical methods. New York: McGraw-Hill Publishing Co.Google Scholar
  36. Nicholls, J., & Russell, J. K. (1990). Modern methods of igneous petrology: Understanding magmatic processes. In P. H. Ribbe (Ed.), Reviews in mineralogy (p. 314). Mineralogical Society of America.Google Scholar
  37. Ottonello, G. (1997). Principles of geochemistry. New York: Columbia University Press.Google Scholar
  38. Parent, S.-E., Parent, L. E., Egozcue, J. J., Rozane, D.-E., Hernandes, A., Lapointe, L., et al. (2013). The plant ionome revisited by the nutrient balance concept. Frontiers in Plant Science, 4.
  39. Pawlowsky-Glahn, V., & Egozcue, J. J. (2006). Compositional data and their analysis: An introduction. In A. Buccianti, G. Mateu-Figueras, & V. Pawlowsky-Glahn (Eds.), Compositional data analysis in the geosciences: From theory to practice (pp. 1–10). London: The Geological Society of London Special Publication.Google Scholar
  40. Pearce, J. A., & Cann, J. R. (1973). Tectonic setting of basic volcanic rocks determined using trace element analyses. Earth and Planetary Science Letters, 19, 290–300.CrossRefGoogle Scholar
  41. Pearson, K. (1897). Mathematical contribution to the theory of evolution—On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London, 60, 489–502.CrossRefGoogle Scholar
  42. Philip, G. M., Skilbeck, C. G., & Watson, D. F. (1987). Algebraic dispersion fields on ternary diagrams. Mathematical Geology, 19, 171–181.CrossRefGoogle Scholar
  43. Ragland, P. C. (1989). Basic analytical petrology. New York: Oxford University Press.Google Scholar
  44. Rivera-Gómez, M. A., & Verma, S. P. (2016). Testing of multidimensional tectonomagmatic discrimination diagrams on fresh and hydrothermally altered rocks. Geologica Carpathica, 67, 195–208 + Supplement i–cxiii (113 pages).Google Scholar
  45. Rollinson, H. R. (1993). Using geochemical data: Evaluation, presentation, interpretation. Essex: Longman Scientific Technical.Google Scholar
  46. Rosales-Rivera, M., Díaz-González, L., & Verma, S. P. (2018). A new online computer program (BiDASys) for ordinary and uncertainty weighted least-squares linear regressions: Case studies from food chemistry. Revista Mexicana de Ingeniería Química, 17, 507–522.CrossRefGoogle Scholar
  47. Roser, B. P., & Korsch, R. J. (1986). Determination of tectonic setting of sandstone-mudstone suites using SiO2 content and K2O/Na2O ratio. Journal of Geology, 94, 635–650.CrossRefGoogle Scholar
  48. Shoemaker, D. P., Garland, C. W., & Nibler, J. W. (1996). Experiments in physical chemistry. New York: McGraw Hill.Google Scholar
  49. Spear, F. S. (1995). Metamorphic phase equilibria and pressure-temperature-time paths. Washington, DC: Mineralogical Society of America.Google Scholar
  50. Taylor, S. R., & McLennan, S. M. (1985). The continental crust: Its composition and evolution. Oxford: Blackwell Scientific.Google Scholar
  51. Tolosana-Delgado, R., Palomera-Roman, R., Gimeno-Torrente, D., Pawlowsky-Glahn, V., & Thió-Henestrosa, S. (2002). A first approach to the classification of basalts using trace elements. In U. Bayer, H. Burger, & W. Skala (Eds.), VIII Annual Conference of the International Association for Mathematical Geology (pp. 435–440). Berlin: Selbstverlag der Alfred-Wegener-Stiftung.Google Scholar
  52. Verma, S. P. (1997). Sixteen statistical tests for outlier detection and rejection in evaluation of international geochemical reference materials: Example of microgabbro PM-S. Geostandards Newsletter: The Journal of Geostandards and Geoanalysis, 21, 59–75.CrossRefGoogle Scholar
  53. Verma, S. P. (2010). Statistical evaluation of bivariate, ternary and discriminant function tectonomagmatic discrimination diagrams. Turkish Journal of Earth Sciences, 19, 185–238.Google Scholar
  54. Verma, S. P. (2012a). Geochemometrics. Revista Mexicana de Ciencias Geológicas, 29, 276–298.Google Scholar
  55. Verma, S. P. (2012b). Application of multi-dimensional discrimination diagrams and probability calculations to acid rocks from Portugal and Spain. Comunicações Geológicas, 99, 79–93.Google Scholar
  56. Verma, S. P. (2013). Application of 50 multi-dimensional discrimination diagrams and significance tests to decipher compositional similarities and differences between Hawaiian and Icelandic volcanism. International Geology Review, 55, 1553–1572.CrossRefGoogle Scholar
  57. Verma, S. P. (2015). Present state of knowledge and new geochemical constraints on the central part of the Mexican Volcanic Belt and comparison with the Central American Volcanic Arc in terms of near and far trench magmas. Turkish Journal of Earth Sciences, 24, 399–460.CrossRefGoogle Scholar
  58. Verma, S. P. (2016). Análisis estadístico de datos composicionales. CDMX: Universidad Nacional Autónoma de México.Google Scholar
  59. Verma, S. P., & Agrawal, S. (2011). New tectonic discrimination diagrams for basic and ultrabasic volcanic rocks through log-transformed ratios of high field strength elements and implications for petrogenetic processes. Revista Mexicana de Ciencias Geológicas, 28, 24–44.Google Scholar
  60. Verma, S. P., & Armstrong-Altrin, J. S. (2013). New multi-dimensional diagrams for tectonic discrimination of siliciclastic sediments and their application to Precambrian basins. Chemical Geology, 355, 117–133.CrossRefGoogle Scholar
  61. Verma, S. P., & Armstrong-Altrin, J. S. (2016). Geochemical discrimination of siliciclastic sediments from active and passive margin settings. Sedimentary Geology, 332, 1–12.CrossRefGoogle Scholar
  62. Verma, S. P., & Díaz-González, L. (2012). Application of the discordant outlier detection and separation system in the geosciences. International Geology Review, 54, 593–614.CrossRefGoogle Scholar
  63. Verma, S. P., & Quiroz-Ruiz, A. (2006a). Critical values for six Dixon tests for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geológicas, 23, 133–161.Google Scholar
  64. Verma, S. P., & Quiroz-Ruiz, A. (2006b). Critical values for 22 discordancy test variants for outliers in normal samples up to sizes 100, and applications in science and engineering. Revista Mexicana de Ciencias Geológicas, 23, 302–319.Google Scholar
  65. Verma, S. P., & Rivera-Gómez, M. A. (2013). Computer programs for the classification and nomenclature of igneous rocks. Episodes, 36, 115–124.Google Scholar
  66. Verma, S. P., & Rivera-Gómez, M. A. (2017). Transformed major element based multidimensional classification of altered volcanic rocks. Episodes, 40, 295–303.CrossRefGoogle Scholar
  67. Verma, S. P., & Verma, S. K. (2013). First 15 probability-based multi-dimensional discrimination diagrams for intermediate magmas and their robustness against post-emplacement compositional changes and petrogenetic processes. Turkish Journal of Earth Sciences, 22, 931–995.CrossRefGoogle Scholar
  68. Verma, S. P., Torres-Alvarado, I. S., & Sotelo-Rodríguez, Z. T. (2002). SINCLAS: Standard igneous norm and volcanic rock classification system. Computers & Geosciences, 28, 711–715.CrossRefGoogle Scholar
  69. Verma, S. P., Torres-Alvarado, I. S., & Velasco-Tapia, F. (2003). A revised CIPW norm. Schweizerische Mineralogische und Petrographische Mitteilungen, 83, 197–216.Google Scholar
  70. Verma, S. P., Guevara, M., & Agrawal, S. (2006a). Discriminating four tectonic settings: Five new geochemical diagrams for basic and ultrabasic volcanic rocks based on log-ratio transformation of major-element data. Journal of Earth System Science, 115, 485–528.CrossRefGoogle Scholar
  71. Verma, S. P., Díaz-González, L., Sánchez-Upton, P., & Santoyo, E. (2006b). OYNYL: A new computer program for ordinary, York, and New York least-squares linear regressions. WSEAS Transactions on Environment and Development, 2, 997–1002.Google Scholar
  72. Verma, S. P., Díaz-González, L., & González-Ramírez, R. (2009). Relative efficiency of single-outlier discordancy tests for processing geochemical data on reference materials and application to instrumental calibration by a weighted least-squares linear regression model. Geostandards and Geoanalytical Research, 33, 29–49.CrossRefGoogle Scholar
  73. Verma, S. P., Rodríguez-Ríos, R., & González-Ramírez, R. (2010). Statistical evaluation of classification diagrams for altered igneous rocks. Turkish Journal of Earth Sciences, 19, 239–265.Google Scholar
  74. Verma, S. P., Pandarinath, K., Verma, S. K., & Agrawal, S. (2013). Fifteen new discriminant-function-based multi-dimensional robust diagrams for acid rocks and their application to Precambrian rocks. Lithos, 168–169, 113–123.CrossRefGoogle Scholar
  75. Verma, S. P., Cruz-Huicochea, R., Díaz-González, L., & Verma, S. K. (2015). A new computer program TecDIA for multidimensional tectonic discrimination of intermediate and acid magmas and its application to the Bohemian Massif, Czech Republic. Journal of Geosciences, 60, 203–218.CrossRefGoogle Scholar
  76. Verma, S. P., Rivera-Gómez, M. A., Díaz-González, L., & Quiroz-Ruiz, A. (2016). Log-ratio transformed major-element based multidimensional classification for altered high-Mg igneous rocks. Geochemistry, Geophysics, Geosystems, 17, 4955–4972.CrossRefGoogle Scholar
  77. Verma, S. K., Quiroz-Ruiz, A., & Armstrong-Altrin, J. A. (2017). Multidimensional classification of magma types for altered igneous rocks and application to their tectonomagmatic discrimination and igneous provenance of siliciclastic sediments. Lithos, 278, 321–330.CrossRefGoogle Scholar
  78. Verma, S. P., Verma, S. K., Rivera-Gómez, M. A., Torres-Sánchez, D., Díaz-González, L., Amezcua-Valdez, A., et al. (2018). Statistically coherent calibration of X-ray fluorescence spectrometry for major elements in rocks and minerals. Journal of Spectroscopy, 2018, 13, Article ID 5837214. Scholar
  79. Verma, S. P., Rosales-Rivera, M., Rivera-Gómez, M. A., & Verma, S. K. (2019a). Comparison of matrix-effect corrections for ordinary and uncertainty weighted linear regressions and determination of major element mean concentrations and total uncertainties of 62 international geochemical reference materials from wavelength-dispersive X-ray fluorescence spectrometry. In Colloquium Spectroscopicum Internationale XLI (CSI XLI) and I Latin-American Meeting on Laser Induced Breakdown Spectroscopy (I LAMLIBS). Mexico City.Google Scholar
  80. Verma, S. P., Díaz-González, L., & Rivera-Gómez, M. A. (2019b). New multidimensional classification scheme of altered igneous rocks from performance comparison of isometric and modified log-ratio transformations of major elements. Geochemical Transactions, submitted.Google Scholar
  81. Weltje, G. J. (2006). Ternary sandstone composition and provenance: An evaluation of the ‘Dickinson model’. In A. Buccianti, G. Mateu-Figueras, & V. Pawlowsky-Glahn (Eds.), Compositional data analysis in the geosciences: From theory to practice (pp. 79–99). London: The Geological Society of London.Google Scholar
  82. Wilks, S. S. (1963). Multivariate statistical outliers. Sankhya, 25, 407–426.Google Scholar
  83. Yang, Z.-F., Li, J., Jiang, Q.-B., Xu, F., Guo, S.-Y., Li, Y., & Zhang, J. (2019). Using major element logratios to recognize compositional patterns of basalt: Implications for source lithological and compositional heterogeneities. Journal of Geophysical Research Solid Earth, 124. Scholar
  84. Young, D. A. (1998). N. L. Bowen and crystallization-differentiation: The evolution of a theory. Washington, DC: Mineralogical Society of America.Google Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Instituto de Energías RenovablesUniversidad Nacional Autónoma de MéxicoTemixcoMexico

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