×

zbMATH — the first resource for mathematics

A critical approach to probability laws in geochemistry. (English) Zbl 1153.86338
Summary: Probability laws in geochemistry have been a major issue of concern over the last decades. The lognormal on the positive real line or the additive logistic normal on the simplex are two classical laws of probability to model geochemical data sets due to their association with a relative measure of difference. This fact is not fully exploited in the classical approach when viewing both the positive real line and the simplex as subsets of real space with the induced geometry. But it can be taken into account considering them as real linear vector spaces with their own structure. This approach implies using a particular geometry and a measure different from the usual ones. Therefore, we can work with the coordinates with respect to an orthonormal basis. It could be shown that the two mentioned laws are associated with a normal distribution on the coordinates. In this contribution both approaches are compared, and a real data set is used to illustrate similarities and differences.

MSC:
86A99 Geophysics
92E99 Chemistry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahrens LH (1953) A fundamental law of geochemistry. Nature 172:1148 · doi:10.1038/1721148a0
[2] Aitchison J (1982) The statistical analysis of compositional data (with discussion). J R Stat Soc Ser B (Stat Method) 44(2):139–177 · Zbl 0491.62017
[3] Aitchison J (1986) The statistical analysis of compositional data. Chapman & Hall, London. Reprinted in 2003 with additional material by The Blackburn Press, 416 p · Zbl 0688.62004
[4] Aitchison J (1997) The one-hour course in compositional data analysis or compositional data analysis is simple. In: Pawlowsky-Glahn V (ed) Proceedings of IAMG’97. International Center for Numerical Methods in Engineering (CIMNE), Barcelona, pp 3–35
[5] Aitchison J, Brown JAC (1957) The lognormal distribution. Cambridge University Press, Cambridge, 176 p · Zbl 0081.14303
[6] Agterberg F (2005) High-and low-value tails of frequency distributions in geochemistry and mineral resource evaluation. Geophys Res Abstr 7:03405. SRef-ID: 1607-7962/gra/EGU05-A-03405
[7] Barceló-Vidal C. (1996) Mixturas de datos composicionales. PhD, Universitat Politècnica de Catalunya, Barcelona, 261 p
[8] Billheimer D, Guttorp P, Fagan W (2001) Statistical interpretation of species composition. J Am Stat Assoc 96(456):1205–1214 · Zbl 1073.62573 · doi:10.1198/016214501753381850
[9] Clark I, Harper WV (2000) Practical geostatistics 2000. Ecosse North America, Columbus, 342 p · Zbl 1038.86008
[10] Crow EL, Shimizu K (1988) Lognormal distributions. Theory and applications. Marcel Dekker, New York, 387 p · Zbl 0644.62014
[11] Eaton ML (1983) Multivariate statistics. A vector space approach. Wiley, New York. 512 p · Zbl 0587.62097
[12] Egozcue JJ, Pawlowsky-Glahn V (2005) Groups of parts and their balances in compositional data analysis. Math Geol 37(7):795–828 · Zbl 1177.86018 · doi:10.1007/s11004-005-7381-9
[13] Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal C (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35(3):279–300 · Zbl 1302.86024 · doi:10.1023/A:1023818214614
[14] Galton F (1879) The geometric mean, in vital and social statistics. Proc R Soc Lond 29:365–366 · doi:10.1098/rspl.1879.0060
[15] Herdan G (1960) Small particle statistics. Butterwoths, London
[16] Martín-Fernández JA, Barceló-Vidal C, Pawlowsky-Glahn V (1998) A critical approach to non-parametric classification of compositional data. In: Rizzi A, Vichi M, Bock HH (eds) Advances in data science and classification. Springer, Berlin, pp 49–56 · Zbl 1052.62531
[17] Martín-Fernández JA, Bren M, Barceló-Vidal C, Pawlowsky-Glahn V (1999) A measure of difference for compositional data based on measures of divergence. In: Lippard SJ, Næss A, Sinding-Larsen R (eds) Proceedings of IAMG’99. Tapir, Trondheim, pp 211–216
[18] Martín-Fernández JA, Barceló-Vidal C, Pawlowsky-Glahn V (2003) Dealing with zeros and missing values in compositional data sets. Math Geol 35(3):253–278 · Zbl 1302.86027 · doi:10.1023/A:1023866030544
[19] Mateu-Figueras G (2003) Models de distribució sobre el símplex: PhD, Universitat Politècnica de Catalunya, Barcelona, 202 p
[20] Mateu-Figueras G, Pawlowsky-Glahn V, Martín-Fernández JA (2002) Normal in R+ vs lognormal in R. In: Burger H, Wolfdietrich S (eds) Terra nostra, Proceedings of IAMG’02, vol. 3, pp 305–310
[21] Mateu-Figueras G, Pawlowsky-Glahn V, Barceló-Vidal C (2005) The additive logistic skew-normal distribution on the simplex. Stoch Environ Res Risk Assess (SERRA) 19(3):205–214 · doi:10.1007/s00477-004-0225-1
[22] McAlister D (1879) The law of geometric mean. Proc R Soc Lond 29:367–376 · JFM 11.0163.04 · doi:10.1098/rspl.1879.0061
[23] Palarea-Albaladejo J, Martín-Fernández JA, Gómez-García JA (2007) Parametric approach for dealing with compositional rounded zeros. Math Geol 39(7):625–645 · Zbl 1130.86001 · doi:10.1007/s11004-007-9100-1
[24] Pawlowsky-Glahn V (2003) Statistical modelling on coordinates. In: Thió-Henestrosa S, Martín-Fernández JA (eds) Proceedings of CoDaWork’03. Universitat de Girona, CD-ROM
[25] Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis on the simplex. Stoch Environ Res Risk Assess (SERRA) 15(5):384–398 · Zbl 0987.62001 · doi:10.1007/s004770100077
[26] Pawlowsky-Glahn V, Egozcue JJ (2002) BLU estimators and compositional data. Math Geol 34(3):259–274 · Zbl 1031.86007 · doi:10.1023/A:1014890722372
[27] Pearson K (1897) Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc R Soc Lond LX:489–502 · JFM 28.0209.02
[28] Thompson RN, Esson J, Dunham AC (1972) Major element chemical variation in the Eocene lavas of the Isle of Skye, Scotland. J Petrol 13:219–253. Cited in Aitchison (1986)
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.