IDEAS home Printed from https://ideas.repec.org/a/spr/testjl/v28y2019i3d10.1007_s11749-019-00670-6.html
   My bibliography  Save this article

Compositional data: the sample space and its structure

Author

Listed:
  • Juan José Egozcue

    (Universitat Politècnica de Catalunya)

  • Vera Pawlowsky-Glahn

    (Universitat de Girona)

Abstract

The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison’s seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain.

Suggested Citation

  • Juan José Egozcue & Vera Pawlowsky-Glahn, 2019. "Compositional data: the sample space and its structure," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 599-638, September.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:3:d:10.1007_s11749-019-00670-6
    DOI: 10.1007/s11749-019-00670-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11749-019-00670-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11749-019-00670-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Martín-Fernández, J.A. & Hron, K. & Templ, M. & Filzmoser, P. & Palarea-Albaladejo, J., 2012. "Model-based replacement of rounded zeros in compositional data: Classical and robust approaches," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2688-2704.
    2. Hugh Chipman & Hong Gu, 2005. "Interpretable dimension reduction," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(9), pages 969-987.
    3. K. Hron & P. Filzmoser & K. Thompson, 2012. "Linear regression with compositional explanatory variables," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(5), pages 1115-1128, November.
    4. J. L. Scealy & A. H. Welsh, 2011. "Regression for compositional data by using distributions defined on the hypersphere," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(3), pages 351-375, June.
    5. Joanna Morais & Christine Thomas-Agnan & Michel Simioni, 2018. "Using compositional and Dirichlet models for market share regression," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(9), pages 1670-1689, July.
    6. John Aitchison & Michael Greenacre, 2002. "Biplots of compositional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 375-392, October.
    7. Juan José Egozcue & Vera Pawlowsky-Glahn & Matthias Templ & Karel Hron, 2015. "Independence in Contingency Tables Using Simplicial Geometry," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(18), pages 3978-3996, September.
    8. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
    9. Peter Filzmoser & Karel Hron & Matthias Templ, 2012. "Discriminant analysis for compositional data and robust parameter estimation," Computational Statistics, Springer, vol. 27(4), pages 585-604, December.
    10. Michael Greenacre, 2008. "Measuring subcompositional incoherence," Economics Working Papers 1106, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2011.
    11. Shorrocks, A F, 1980. "The Class of Additively Decomposable Inequality Measures," Econometrica, Econometric Society, vol. 48(3), pages 613-625, April.
    12. Jane Fry & Tim Fry & Keith McLaren, 2000. "Compositional data analysis and zeros in micro data," Applied Economics, Taylor & Francis Journals, vol. 32(8), pages 953-959.
    13. Zachary D Kurtz & Christian L Müller & Emily R Miraldi & Dan R Littman & Martin J Blaser & Richard A Bonneau, 2015. "Sparse and Compositionally Robust Inference of Microbial Ecological Networks," PLOS Computational Biology, Public Library of Science, vol. 11(5), pages 1-25, May.
    14. Wei Lin & Pixu Shi & Rui Feng & Hongzhe Li, 2014. "Variable selection in regression with compositional covariates," Biometrika, Biometrika Trust, vol. 101(4), pages 785-797.
    15. Billheimer D. & Guttorp P. & Fagan W.F., 2001. "Statistical Interpretation of Species Composition," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1205-1214, December.
    16. David Lovell & Vera Pawlowsky-Glahn & Juan José Egozcue & Samuel Marguerat & Jürg Bähler, 2015. "Proportionality: A Valid Alternative to Correlation for Relative Data," PLOS Computational Biology, Public Library of Science, vol. 11(3), pages 1-12, March.
    17. Jiajia Chen & Xiaoqin Zhang & Shengjia Li, 2017. "Multiple linear regression with compositional response and covariates," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(12), pages 2270-2285, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Salvador Linares-Mustar'os & Maria `Angels Farreras-Noguer & N'uria Arimany-Serrat & Germ`a Coenders, 2022. "New financial ratios based on the compositional data methodology," Papers 2210.11138, arXiv.org.
    2. María Dolores Esteban & María José Lombardía & Esther López-Vizcaíno & Domingo Morales & Agustín Pérez, 2020. "Small area estimation of proportions under area-level compositional mixed models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 793-818, September.
    3. Pol Jofre-Campuzano & Germà Coenders, 2022. "Compositional Classification of Financial Statement Profiles: The Weighted Case," JRFM, MDPI, vol. 15(12), pages 1-17, November.
    4. Anna Maria Fiori & Francesco Porro, 2023. "A compositional analysis of systemic risk in European financial institutions," Annals of Finance, Springer, vol. 19(3), pages 325-354, September.
    5. Alvis Cabrera & Lyvia Biagi & Aleix Beneyto & Ernesto Estremera & Iván Contreras & Marga Giménez & Ignacio Conget & Jorge Bondia & Josep Antoni Martín-Fernández & Josep Vehí, 2023. "Validation of a Probabilistic Prediction Model for Patients with Type 1 Diabetes Using Compositional Data Analysis," Mathematics, MDPI, vol. 11(5), pages 1-17, March.
    6. Joscha Krause & Jan Pablo Burgard & Domingo Morales, 2022. "Robust prediction of domain compositions from uncertain data using isometric logratio transformations in a penalized multivariate Fay–Herriot model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(1), pages 65-96, February.
    7. Janice L. Scealy, 2021. "Comments on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 68-70, March.
    8. Germ`a Coenders & N'uria Arimany Serrat, 2023. "Accounting statement analysis at industry level. A gentle introduction to the compositional approach," Papers 2305.16842, arXiv.org, revised Feb 2024.
    9. María Dolores Esteban & María José Lombardía & Esther López-Vizcaíno & Domingo Morales & Agustín Pérez, 2023. "Small area estimation of average compositions under multivariate nested error regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 651-676, June.
    10. Vasilii Erokhin & Tianming Gao & Anna Ivolga, 2020. "Structural Variations in the Composition of Land Funds at Regional Scales across Russia," Land, MDPI, vol. 9(6), pages 1-39, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jacob Fiksel & Scott Zeger & Abhirup Datta, 2022. "A transformation‐free linear regression for compositional outcomes and predictors," Biometrics, The International Biometric Society, vol. 78(3), pages 974-987, September.
    2. Jiajia Chen & Xiaoqin Zhang & Shengjia Li, 2017. "Multiple linear regression with compositional response and covariates," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(12), pages 2270-2285, September.
    3. Haixiang Zhang & Jun Chen & Zhigang Li & Lei Liu, 2021. "Testing for Mediation Effect with Application to Human Microbiome Data," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 13(2), pages 313-328, July.
    4. Tsagris, Michail & Preston, Simon & T.A. Wood, Andrew, 2016. "Improved classi cation for compositional data using the $\alpha$-transformation," MPRA Paper 67657, University Library of Munich, Germany.
    5. Michail Tsagris & Simon Preston & Andrew T. A. Wood, 2016. "Improved Classification for Compositional Data Using the α-transformation," Journal of Classification, Springer;The Classification Society, vol. 33(2), pages 243-261, July.
    6. Tsagris, Michail, 2014. "The k-NN algorithm for compositional data: a revised approach with and without zero values present," MPRA Paper 65866, University Library of Munich, Germany.
    7. Rieser, Christopher & Filzmoser, Peter, 2023. "Extending compositional data analysis from a graph signal processing perspective," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    8. M. Templ & K. Hron & P. Filzmoser, 2017. "Exploratory tools for outlier detection in compositional data with structural zeros," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(4), pages 734-752, March.
    9. Tsagris, Michail, 2015. "Regression analysis with compositional data containing zero values," MPRA Paper 67868, University Library of Munich, Germany.
    10. Andonie, Costel & Kuzmics, Christoph & Rogers, Brian W., 2019. "Efficiency-based measures of inequality," Journal of Mathematical Economics, Elsevier, vol. 85(C), pages 60-69.
    11. Casilda Lasso de la Vega & Ana Urrutia & Oscar Volij, 2011. "An Axiomatic Characterization Of The Theil Inequality Order," Working Papers 1103, Ben-Gurion University of the Negev, Department of Economics.
    12. Tugce, Cuhadaroglu, 2013. "My Group Beats Your Group: Evaluating Non-Income Inequalities," SIRE Discussion Papers 2013-49, Scottish Institute for Research in Economics (SIRE).
    13. Maria Cubel & Peter Lambert, 2002. "Progression-neutral income tax reforms and horizontal inequity," Journal of Economics, Springer, vol. 77(1), pages 1-8, December.
    14. Duo Jiang & Thomas Sharpton & Yuan Jiang, 2021. "Microbial Interaction Network Estimation via Bias-Corrected Graphical Lasso," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 13(2), pages 329-350, July.
    15. Teixidó Figueras, Jordi & Duro Moreno, Juan Antonio, 2012. "Ecological Footprint Inequality: A methodological review and some results," Working Papers 2072/203168, Universitat Rovira i Virgili, Department of Economics.
    16. Hoy, Michael & Huang, Rachel J., 2017. "Measuring discrimination using principles of stochastic dominance," Journal of Economic Theory, Elsevier, vol. 167(C), pages 39-52.
    17. Anna Maria Fiori & Francesco Porro, 2023. "A compositional analysis of systemic risk in European financial institutions," Annals of Finance, Springer, vol. 19(3), pages 325-354, September.
    18. Frankel, David M. & Volij, Oscar, 2011. "Measuring school segregation," Journal of Economic Theory, Elsevier, vol. 146(1), pages 1-38, January.
    19. Sorger, Gerhard & Stark, Oded, 2013. "Income redistribution going awry: The reversal power of the concern for relative deprivation," Journal of Economic Behavior & Organization, Elsevier, vol. 86(C), pages 1-9.
    20. Lorena Remuzgo & Jose Maria Sarabia, 2015. "A general factorial decomposition of the second Theil index of inequality with applications in environmental economics," Economics Bulletin, AccessEcon, vol. 35(2), pages 1369-1378.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:testjl:v:28:y:2019:i:3:d:10.1007_s11749-019-00670-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.