×

zbMATH — the first resource for mathematics

On criteria for measures of compositional difference. (English) Zbl 0970.86531
Summary: Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It is shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible log ratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.

MSC:
86A32 Geostatistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aitchison, J., 1986a, The Statistical Analysis of Compositional Data: Chapman and Hall, London. · Zbl 0688.62004
[2] Aitchison, J., 1986b, CODA: A Microcomputer Package for the Statistical Analysis of Compositional Data: Chapman and Hall, London. · Zbl 0688.62004
[3] Aitchison, J., 1990, Comment on ?Measures of Variability for Geological Data? by D. F. Watson and G. M. Philip: Math. Geol., v. 22, p. 223-226. · doi:10.1007/BF00891826
[4] Aitchison, J., 1991, Delusions of Uniqueness and Ineluctability: Math. Geol., v. 23, p. 275-277. · doi:10.1007/BF02066299
[5] Aitchison, J., 1992, Principles of Compositional Data Analysis, in Anderson, T. W., Olkin, I. and Fang, K. T. (Eds.), Proceedings of the International Symposium of Multivariate Analysis and its Applications, Institute of Mathematical Statistics. To appear. · Zbl 0970.86531
[6] Lehmann, E. L., 1959, Testing Statistical Hypotheses: New York, Wiley.
[7] Watson, D. F., 1990, Reply to Comment on ?Measures of Variability for Geological Data? by D. F. Watson and G. M. Philip: Math. Geol., v. 22, p. 227-231. · doi:10.1007/BF00891827
[8] Watson, D. F., 1991, Reply to ?Delusions of Uniqueness and Ineluctability? by J. Aitchison: Math. Geol., v. 23, p. 279. · doi:10.1007/BF02066300
[9] Watson, D. F., and Philip, G. M., 1989, Measures of Variability for Geological Data: Math. Geol., v. 21, p. 233-254. · doi:10.1007/BF00893217
[10] Woronow, A., and Love, K. M., 1990, Quantifying and Testing Differences Among Means of Compositional Data Suites: Math. Geol., v. 22, p. 837-852. · doi:10.1007/BF00890666
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.