The Principle of Working on Coordinates

Compositional Data Analysis: Theory and Applications

Volumn , Issue , 2011, Pages 29-42

  Mateu Figueras, Glòria ^a   Pawlowsky Glahn, Vera ^a   Díaz Barrero, José Luis ^b  

^a UNIVERSITY OF GIRONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Establishing role of coordinates in the simplex; Euclidean space structure of simplex orthonormal basis, applying Gram Schmidt orthonormalisation process to any basis; Mathematical statistics, and real analysis using coordinates, on orthonormal basis; Principle of working on coordinates standard statistical methods, to coordinates of any composition; Role of coordinates in statistics standard or classical statistics, in real space; Similar concepts for addressing random compositions; Simplex SD, a (D 1) dimensional subset of D dimensional real space; Standard statistical methodology traditionally developed on real space, sample space with different algebraic geometric structure; The simplex, notation used standard in compositional data analysis; Theory statistical modeling, principle of working on coordinates

Indexed keywords

EID: 84859424365     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9781119976462.ch3     Document Type: Chapter

References (28)

1. Aitchison J 1982 The statistical analysis of compositional data (with discussion). Journal of the Royal Statistical Society, Series B (Statistical Methodology) 44(2), 139-177.
2. Aitchison J 1983 Principal component analysis of compositional data. Biometrika 70(1), 57-65.
3. Aitchison J 1986 The Statistical Analysis of Compositional Data.Monographs on Statistics and Applied Probability. Chapman and Hall Ltd (reprinted 2003 with additional material by The Blackburn Press), London (UK). 416 p.
4. Aitchison J 2001 Simplicial inference. In Algebraic Methods in Statistics and Probability (ed. Viana MA and Richards DS), vol. 287. American Mathematical Society, Providence, RI (USA). pp. 1-22.
5. Aitchison J and Brown JAC 1957 The Lognormal Distribution. Cambridge University Press, Cambridge (UK). 176 p.
6. Aitchison J and Egozcue JJ 2005 Compositional data analysis: where are we and where should we be heading?. Mathematical Geology 37(7), 829-850.
7. Aitchison J and Greenacre M 2002 Biplots for compositional data. Applied Statistics 51(4), 375-392.
8. Aitchison J and Shen SM 1980 Logistic-normal distributions. Some properties and uses. Biometrika 67(2), 261-272.
9. Barceló-Vidal C 1996 Mixturas de Datos Composicionales. PhD thesis, Universitat Politecnica de Catalunya, Barcelona (Spain).
10. Barceló-Vidal C, Martín-Fernández JA and Pawlowsky-Glahn V 2001 Mathematical foundations of compositional data analysis. In Proceedings of IAMG'01-The VII Annual Conference of the International Association for Mathematical Geology (ed. Ross G). Kansas Geological Survey, Cancun (Mexico). 20 p.
11. Billheimer D, Guttorp P and FaganW2001 Statistical interpretation of species composition. Journal of the American Statistical Association 96(456), 1205-1214.
12. Eaton ML 1983 Multivariate Statistics. A Vector Space Approach. John Wiley & Sons, Ltd, New York, NY (USA).
13. Egozcue JJ and Pawlowsky-Glahn V 2005 Groups of parts and their balances in compositional data analysis. Mathematical Geology 37(7), 795-828.
14. Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G and Barceló-Vidal C 2003 Isometric logratio transformations for compositional data analysis. Mathematical Geology 35(3), 279-300.
15. Gueorguieva R, Rosenheck R and Zelterman D 2008 Dirichlet component regression and its applications to psychiatric data. Computational Statistics and Data Analysis 52, 5344-5355.
16. Hijazi RH and Jernigan RW 2009 Modelling compositional data using Dirichlet regression models. Journal of Applied Probability and Statistics 4(1), 77-91.
17. Kolmogorov 1933 Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (Germany).
18. Kolmogorov AN and Fomin SV 1957 Elements of the Theory of Functions and Functional Analysis, vols. 1 and 2. Dover Publications, Inc., Mineola, NY (USA).
19. Martín-Fernández J, Barceló-Vidal C and Pawlowsky-Glahn V 1998 A critical approach to nonparametric classification of compositional data. In Advances in Data Science and Classification. In Proceedings of the 6th Conference of the International Federation of Classification Societies (IFCS-98) (ed. Rizzi A, Vichi M and Bock H). Springer-Verlag, Berlin (Germany). pp. 49-56.
20. Mateu-Figueras G 2003 Models de distribució sobre el símplex. PhD thesis, Universitat Politecnica de Catalunya, Barcelona (Spain).
21. Mateu-Figueras G and Pawlowsky-Glahn V 2005 The Dirichlet distribution with respect to the Aitchison measure on the simplex-a first approach. In Proceedings of CoDaWork'05, The 2nd Compositional Data Analysis Workshop (ed. Mateu-Figueras G and Barceló-Vidal C). http://ima.udg.es/Activitats/CoDaWork05/. University of Girona, Girona (Spain).
22. Mateu-Figueras G and Pawlowsky-Glahn V 2007 The skew-normal distribution on the simplex. Communications in Statistics-Theory and Methods 36(9), 1787-1802.
23. Mateu-FiguerasGand Pawlowsky-GlahnV2008Acritical approach to probability laws in geochemistry. Mathematical Geosciences 40(5), 489-502.
24. Mateu-Figueras G, Pawlowsky-Glahn V and Barceló-Vidal C 2005 The additive logistic skew-normal distribution on the simplex. Stochastic Environmental Research and Risk Assessment (SERRA) 19(3), 205-214.
25. Pawlowsky-Glahn V 2003 Statistical modelling on coordinates. In Proceedings of CoDaWork'03, The 1st Compositional Data Analysis Workshop (ed. Thió-Henestrosa S and Martín-Fernández JA). http://ima.udg.es/Activitats/CoDaWork03/. University of Girona, Girona (Spain). CD-ROM.
26. Pawlowsky-Glahn V and Egozcue JJ 2001 Geometric approach to statistical analysis on the simplex. Stochastic Environmental Research and Risk Assessment (SERRA) 15(5), 384-398.
27. Pawlowsky-Glahn V and Egozcue JJ 2002 BLU estimators and compositional data. Mathematical Geology 34(3), 259-274.
28. Pearson K 1897 Mathematical contributions to the theory of evolution. On a form of spurious correlation which may arisewhen indices are used in the measurement of organs. Proceedings of the Royal Society of London LX, 489-502.