Changing the Reference Measure in the Simplex and its Weighting Effects

Juan Jose Egozcue, Vera Pawlowsky-Glahn

Abstract


Standard analysis of compositional data under the assumption that the Aitchison geometry holds assumes a uniform distribution as reference measure of the space. Weighting of parts can be done changing the reference measure. The changes that appear in the algebraic-geometric structure of the simplex are analysed, as a step towards understanding the implications for elementary statistics of random compositions. Some of the standard tools in exploratory analysis of compositional data analysis, such as center, variation matrix and biplots are studied in some detail, although further research is still needed. The main conclusion is that down-weighting some parts is approaching the geometry of the corresponding subcomposition, thus preserving a kind of coherence between standard and down-weighted analyses.

Full Text:

PDF

References


Aitchison J (1983). “Principal component analysis of compositional data.” Biometrika, 70(1),

–65.

Aitchison J (1986). The Statistical Analysis of Compositional Data. Monographs on Statistics

and Applied Probability. Chapman & Hall Ltd., London (UK). (Reprinted in 2003 with

additional material by The Blackburn Press), London (UK). ISBN 0-412-28060-4. 416 p.

Aitchison J (1992). “On Criteria for Measures of Compositional Difference.” Mathematical

Geology, 24(4), 365–379.

Aitchison J, Barcel ́

o-Vidal C, Mart ́ın-Fern ́andez JA, Pawlowsky-Glahn V (2000). “Logratio

analysis and compositional distance.” Mathematical Geology, 32(3), 271–275. ISSN 0882-

Aitchison J, Egozcue JJ (2005). “Compositional data analysis: where are we and where should

we be heading?” Mathematical Geology, 37(7), 829–850.

Aitchison J, Greenacre M (2002). “Biplots for compositional data.” Journal of the Royal

Statistical Society, Series C (Applied Statistics), 51(4), 375–392.Austrian Journal of Statistics

Boogaart KGvd, Egozcue JJ, Pawlowsky-Glahn V (2010). “Bayes linear spaces.” SORT -

Statistics and Operations Research Transactions, 34(2), 201–222. ISSN 1696-2281.

Boogaart KGvd, Egozcue JJ, Pawlowsky-Glahn V (2014). “Bayes Hilbert Spaces.” Australian

and New Zealand Journal of Statistics, 56(2), 171–194. doi:10.1111/anzs.12074.

Boogaart KGvd, Tolosana-Delgado R (2013). Analysing compositional data with R. Springer,

Heidelberg. 280 pp.

Egozcue JJ (2009). “Reply to “On the Harker variation diagrams;...” by J. A. Cort ́es.” Math-

ematical Geosciences, 41(7), 829–834.

Egozcue JJ, Barcel ́

o-Vidal C, Mart ́ın-Fern ́andez JA, Jarauta-Bragulat E, D ́ıaz-Barrero JL,

Mateu-Figueras G (2011). “Elements of simplicial linear algebra and geometry.” In

Pawlowsky-Glahn and Buccianti (2011), pp. 141–157. 378 p.

Egozcue JJ, D ́ıaz-Barrero JL, Pawlowsky-Glahn V (2006). “Hilbert space of probabil-

ity density functions based on Aitchison geometry.” volume 22, pp. 1175–1182. DOI:

1007/s10114-005-0678-2.

Egozcue JJ, Lovell D, Pawlowsky-Glahn V (2013a). Testing compositional association. In:

Proceedings of the 5th Workshop on compositional data analysis, CoDaWork 2013, ISBN:

-3-200-03103-6. Pp 28–36.

Egozcue JJ, Pawlowsky-Glahn V (2005). “Groups of parts and their balances in compositional

data analysis.” Mathematical Geology, 37(7), 795–828.

Egozcue JJ, Pawlowsky-Glahn V (2006). “Simplicial geometry for compositional data.” In

Compositional Data Analysis in the Geosciences: From Theory to Practice, pp. 145–159.

Egozcue JJ, Pawlowsky-Glahn V (2011). “Basic concepts and procedures.” In Pawlowsky-

Glahn and Buccianti (2011), pp. 12–28. 378 p.

Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barcel ́o-Vidal C (2003). “Isometric

logratio transformations for compositional data analysis.” Mathematical Geology, 35(3),

–300. ISSN 0882-8121.

Egozcue JJ, Pawlowsky-Glahn V, Tolosana-Delgado R, Ortego MI, Boogaart KGvd (2013b).

“Bayes spaces: use of improper distributions and exponential families.” Revista de la Real

Academia de Ciencias Exactas, F ́ısicas y Naturales, Serie A, Matem ́

aticas (RACSAM),

, 475–486. DOI 10.1007/s13398-012-0082-6.

Filzmoser P, Hron K (2015). “Robust coordinates for compositional data using weighted

balances.” In Nordhausen, K. and Taskinen, S., (eds.), Modern Nonparametric, Robust

and Multivariate Methods. Springer, Heidelberg.

Fr ́echet M (1948). “Les ́el ́ements Al ́eatoires de Nature Quelconque dans une Espace Distanci ́e.”

Annales de l’Institut Henri Poincar ́e , 10(4), 215–308.

Lovell D, Pawlowsky-Glahn V, Egozcue JJ, Marguerat S, B ̈ahler J (2015). “Proportionality: A

Valid Alternative to Correlation for Relative Data.” PLoS Comput Biol, 11(3), e1004075.

doi:10.1371/journal.pcbi.1004075. URL http://dx.doi.org/10.1371%2Fjournal.

pcbi.1004075.

Mateu-Figueras G, Pawlowsky-Glahn V, Egozcue JJ (2013). “The normal distribution in

some constrained sample spaces.” SORT - Statistics and Operations Research Transactions,

(1), 29–56. ISSN 1696-2281.

Pawlowsky-Glahn V, Buccianti A (eds.) (2011). Compositional Data Analysis: Theory and

Applications. John Wiley & Sons. ISBN 978–0–470–71135–4. 378 p.

Pawlowsky-Glahn V, Egozcue JJ (2001). “Geometric approach to statistical analysis on the

simplex.” Stochastic Environmental Research and Risk Assessment (SERRA), 15(5), 384–

Pawlowsky-Glahn V, Egozcue JJ (2011). “Exploring Compositional Data with the Coda-

Dendrogram.” Austrian Journal of Statistics, 40(1 & 2), 103–113. ISSN 1026-597X.

Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015). Modeling and analysis of

compositional data. Statistics in practice. John Wiley & Sons, Chichester UK. ISBN

272 pp.




DOI: http://dx.doi.org/10.17713/ajs.v45i4.126

Refbacks

  • There are currently no refbacks.


@Matthias Templ (using Open Journal Systems) -- see previous editions at http://www.stat.tugraz.at/AJS/Editions.html