# The statistical analysis of geochemical compositions

- 264 Downloads
- 57 Citations

## Abstract

The analysis and interpretation of compositional data, such as major oxide compositions of rocks, has been traditionally plagued by the so-called constant-sum or closure problem. Particular difficulties have been the lack of a satisfactory, interpretable covariance structure and of rich, tractable, parametric classes of distributions on the simplex sample space. Consideration of logistic and logratio transformations between the simplex and Euclidan space has allowed the introduction of new concepts of covariance structure and of classes of logistic-normal distributions which have now opened up a substantial and meaningful array of statistical methodology for compositional data. From the motivation of a wide variety of practical geological problems we examine the range of possibilities with this new approach to the constant-sum problem.

### Key words

Closure closed number system logistic logratio geochemistry## Preview

Unable to display preview. Download preview PDF.

### References

- Aitchison, J., 1981a, A new approach to null correlations of proportions: Jour. Math. Geol., v. 13, p. 175–189.Google Scholar
- Aitchison, J., 1981b, Distributions on the simplex for the analysis of neutrality,
*in*, C. Taillie, G. P. Patil, and B. Baldessari (Eds.), Statistical distributions in scientific work: D. Reidel, Dordrecht, Holland, p. 147–156.Google Scholar - Aitchison, J., 1982, The statistical analysis of compositional data (with discussion): Jour. Roy. Stat. Soc. Ser. B., v. 44, p. 139–177.Google Scholar
- Aitchison, J., 1983, Principal component analysis of compositional data: Biometrika, v. 70, p. 57–65.Google Scholar
- Aitchison, J., 1984a, Reducing the dimensionality of compositional data sets: Jour. Math. Geol., v. 16, to appear.Google Scholar
- Aitchison, J., 1984b, A general class of distributions on the simplex: Jour. Roy. Stat. Soc. Ser. B., v. 46, to appear.Google Scholar
- Aitchison, J. and Bacon-Shone, J. H., 1984, A logcontrast approach to experiments with mixtures: Biometrika, v. 71, to appear.Google Scholar
- Aitchison, J. and Dunsmore, I. R., 1975, Statistical prediction analysis: Cambridge University Press.Google Scholar
- Aitchison, J. and Lauder, I. J., 1984, Kernel density estimation for compositional data: submitted to Appl. Stat.Google Scholar
- Aitchison, J. and Li, C. K. T., 1985, A new approach to classification from compositional data: submitted to Jour. Math. Geol.Google Scholar
- Aitchison, J. and Shen, S. M., 1980, Logistic-normal distributions: Some properties and uses: Biometrika, v. 67, p. 261–272.Google Scholar
- Aitchison, J. and Shen, S. M., 1984, Measurement error in compositional data: Jour. Math. Geol., v. 16, p. 637–650.Google Scholar
- Anderson, T. W., 1958, An Introduction to Multivariate Statistical Analysis: John Wiley & Sons, New York.Google Scholar
- Barker, D. S., 1978, Magmatic trends on alkali-iron-magnesium diagrams: Amer. Min., v. 63, p. 531–534.Google Scholar
- Becker, N. G., 1968, Models for the response of a mixture: Jour. Roy. Stat. Soc. Ser. B., v. 30, p. 349–358.Google Scholar
- Becker, N. G., 1978, Models and designs for experiments with mixtures: Aust. Jour. Stat., v. 20, p. 195–208.Google Scholar
- Butler, J. C., 1979, Trends in ternary petrologic variation diagrams—fact or fantasy?: Amer. Min., v. 64, p. 1115–1121.Google Scholar
- Chayes, F., 1960, On correlation between variables of constant sum: Jour. Geophys. Res., v. 65, p. 4185–4193.Google Scholar
- Chayes, F., 1962, Numerical correlation and petrographic variation: Jour. Geol., v. 70, p. 440–452.Google Scholar
- Chayes, F., 1971, Ratio correlation: University of Chicago Press, Illinois.Google Scholar
- Chayes, F., 1983, Detecting nonrandom associations between proportions by tests of remaining-space variables: Jour. Math. Geol., v. 15, p. 197–206.Google Scholar
- Chayes, F. and Kruskal, W., 1966, An approximate statistical test for correlations between proportions: Jour. Geol., v. 74, p. 692–702.Google Scholar
- Connor, R. J. and Mosimann, J. E., 1969, Concepts of independence for proportions with a generalization of the Dirichlet distribution: Jour. Amer. Stat. Assoc., v. 64, p. 194–206.Google Scholar
- Cornell, J. A., 1981, Experiments with mixtures: John Wiley & Sons, New York.Google Scholar
- Cox, D. R., 1971, A note on polynomial response functions for mixtures: Biometrika, v. 58, p. 155–159.Google Scholar
- Dawid, A. P., 1976, Properties of diagnostic data distributions: Biometrics, v. 32, p. 647–658.Google Scholar
- Iddings, J. P., 1903, Chemical composition of igneous rocks: U.S. Geol. Survey Prof. Pap. 18.Google Scholar
- James, I. R., 1981, Distributions associated with neutrality properties for random proportions,
*in*, C. Taillie, G. P. Patil, and B. Baldessari (Eds.), Statistical distributions in scientific work: D. Reidel, Dordrecht, Holland, p. 125–136.Google Scholar - Kork, J. O., 1977, Examination of the Chayes-Kruskal procedure for testing correlations between proportions: Jour. Math. Geol., v. 9, p. 543–562.Google Scholar
- Krumbein, W. C., 1962, Open and closed number systems in stratigraphic mapping: Bull. Amer. Assoc. Pet. Geol., v. 46, p. 2229–2245.Google Scholar
- Lauder, I. J., 1978, Computational problems in predictive diagnosis: Compstat 1978, p. 186–192.Google Scholar
- Le Maitre, R. W., 1968, Chemical variation within and between volcanic rock series—a statistical approach: Jour. Pet., v. 9, p. 220–252.Google Scholar
- Le Maitre, R. W., 1982, Numerical petrography: Elsevier, Amsterdam.Google Scholar
- McAlister, D., 1879, The law of the geometric mean: Proc. Roy. Soc., v. 29, p. 367.Google Scholar
- Miesch, A. T., 1969, The constant sum problem in geochemistry,
*in*D. F. Merriam (Ed.), Computer Applications in the earth sciences: Plenum Press, New York, p. 161–167.Google Scholar - Morrison, D. F., 1976, Multivariate statistical methods: New York, McGraw-Hill.Google Scholar
- Pearson, K., 1897, Mathematical contributions to the theory of evolution. On a form of spurious correlations which may arise when indices are used in the measurement of organs: Proc. Roy. Soc., v. 60, p. 489–498.Google Scholar
- Reyment, R., 1977, Presidential address to the International Association for Mathematical Geology: Jour. Math. Geol., v. 9, p. 451–454.Google Scholar
- Sarmanov, O. V. and Vistelius, A. B., 1959, On the correlation of percentage values: Dokl. Akad. Nauk. SSSR, v. 126, p. 22–25.Google Scholar
- Stephens, M. A., 1974, EDF statistics for goodness of fit and some comparisons: Jour. Amer. Stat. Assoc., v. 69, p. 730–737.Google Scholar
- Webb, W. M. and Briggs, L. I., 1966, The use of principal component analysis to screen mineralogical data: Jour. Geol., v. 74, p. 716–720.Google Scholar
- Wilks, S. S., 1938, The large-sample distribution of the likelihood ratio for testing composite hypotheses: Ann. Math. Stat., v. 9, p. 60–62.Google Scholar
- Zukav, G., 1979, The dancing Wu-Li masters: Bantam, New York.Google Scholar