Exploratory data analysis for interval compositional data

  • Karel HronEmail author
  • Paula Brito
  • Peter Filzmoser
Regular Article


Compositional data are considered as data where relative contributions of parts on a whole, conveyed by (log-)ratios between them, are essential for the analysis. In Symbolic Data Analysis (SDA), we are in the framework of interval data when elements are characterized by variables whose values are intervals on \(\mathbb {R}\) representing inherent variability. In this paper, we address the special problem of the analysis of interval compositions, i.e., when the interval data are obtained by the aggregation of compositions. It is assumed that the interval information is represented by the respective midpoints and ranges, and both sources of information are considered as compositions. In this context, we introduce the representation of interval data as three-way data. In the framework of the log-ratio approach from compositional data analysis, it is outlined how interval compositions can be treated in an exploratory context. The goal of the analysis is to represent the compositions by coordinates which are interpretable in terms of the original compositional parts. This is achieved by summarizing all relative information (logratios) about each part into one coordinate from the coordinate system. Based on an example from the European Union Statistics on Income and Living Conditions (EU-SILC), several possibilities for an exploratory data analysis approach for interval compositions are outlined and investigated.


Interval data Symbolic data analysis Aitchison geometry on the simplex Orthonormal coordinates Outlier detection Principal component analysis 

Mathematics Subject Classification

62H25 62H99 



Karel Hron gratefully acknowledges the support of the grant COST Action CRoNoS IC1408 and the grants IGA_PrF_2015_013, IGA_PrF_2016_025 Mathematical Models of the Internal Grant Agency of the Palacky University in Olomouc. Peter Filzmoser was supported by the K-project DEXHELPP through COMET - Competence Centers for Excellent Technologies, supported by BMVIT, BMWFI and the province Vienna. The COMET program is administrated by FFG. Finally, this work is financed also by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalization – COMPETE 2020 Programme within project «POCI-01-0145-FEDER-006961», and by National Funds through the FCT - Fundação para a Ci\(\hat{\mathrm{e}}\)ncia e a Tecnologia (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013.


  1. Aitchison J (1986) The statistical analysis of compositional data. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  2. Aitchison J, Greenacre M (2002) Biplots for compositional data. J R Stat Soc Ser C (Appl Stat) 51(4):375–392MathSciNetCrossRefzbMATHGoogle Scholar
  3. Aitchison J, Ng KW (2005) The role of perturbation in compositional data analysis. Stat Model 5:173–185MathSciNetCrossRefzbMATHGoogle Scholar
  4. Alfons A, Templ M (2013) Estimation of social exclusion indicators from complex surveys: the R package laeken. J Stat Softw 54(15):1–25CrossRefGoogle Scholar
  5. Billheimer D, Guttorp P, Fagan W (2001) Statistical interpretation of species composition. J Am Stat Assoc 96:1205–1214MathSciNetCrossRefzbMATHGoogle Scholar
  6. Billard L, Diday E (2003) From the statistics of data to the statistics of knowledge: symbolic data analysis. J Am Stat Assoc 98(462):470–487MathSciNetCrossRefGoogle Scholar
  7. Bock H-H, Diday E (eds) (2000) Analysis of symbolic data, exploratory methods for extracting statistical information from complex data. Springer, HeidelbergzbMATHGoogle Scholar
  8. Brito P, Duarte Silva AP (2012) Modelling interval data with Normal and Skew-Normal distributions. J Appl Stat 39(1):3–20MathSciNetCrossRefGoogle Scholar
  9. Bro R (1997) PARAFAC. Tutorial and applications. Chemometr Intell Lab Syst 38:149–171CrossRefGoogle Scholar
  10. Cazes P, Chouakria A, Diday E, Schektman Y (1997) Extensions de l’Analyse en Composantes Principales à des données de type intervalle. Rev Stat Appl 24:5–24Google Scholar
  11. Chouakria A, Cazes P, Diday E (2000) Symbolic principal component analysis. In: Bock HH, Diday E (eds) Analysis of symbolic data, exploratory methods for extracting statistical information from complex data. Springer, Heidelberg, pp 200–212Google Scholar
  12. Diday E, Noirhomme-Fraiture M (eds) (2008) Symbolic data analysis and the SODAS software. Wiley, ChichesterzbMATHGoogle Scholar
  13. Di Palma AM, Filzmoser P, Gallo M, Hron K (2015) A robust CP model for compositional data(Submitted) Google Scholar
  14. Eaton ML (1983) Multivariate statistics. A vector space approach. John Wiley & Sons, New YorkzbMATHGoogle Scholar
  15. Egozcue JJ, Pawlowsky-Glahn V, Mateu-Figueras G, Barceló-Vidal V (2003) Isometric logratio transformations for compositional data analysis. Math Geol 35:279–300MathSciNetCrossRefzbMATHGoogle Scholar
  16. Egozcue JJ, Pawlowsky-Glahn V (2005) Groups of parts and their balances in compositional data analysis. Math Geol 37:795–828MathSciNetCrossRefzbMATHGoogle Scholar
  17. Egozcue JJ, Pawlowsky-Glahn V (2006) Simplicial geometry for compositional data. In: Buccianti A, Mateu-Figueras G, Pawlowsky-Glahn V (eds) Compositional data analysis in the geosciences: from theory to practice. Geological Society, Special Publications, London, pp 145–160Google Scholar
  18. Filzmoser P, Hron K (2008) Outlier detection for compositional data using robust methods. Math Geosci 40(3):233–248CrossRefzbMATHGoogle Scholar
  19. Filzmoser P, Hron K, Reimann C (2009) Principal component analysis for compositional data with outliers. Environmetrics 20(6):621–632MathSciNetCrossRefGoogle Scholar
  20. Filzmoser P, Hron K (2009) Correlation analysis for compositional data. Math Geosci 41(8):905–919MathSciNetCrossRefzbMATHGoogle Scholar
  21. Filzmoser P, Hron K, Reimann C (2012) Interpretation of multivariate outliers for compositional data. Comput Geosci 39:77–85CrossRefGoogle Scholar
  22. Filzmoser P, Hron K (2011) Robust statistical analysis. In: Pawlowsky-Glahn V, Buccianti A (eds) Compositional data analysis: theory and applications. Wiley, Chichester, pp 59–72CrossRefGoogle Scholar
  23. Fišerová E, Hron K (2011) On interpretation of orthonormal coordinates for compositional data. Math Geosci 43:455–468CrossRefGoogle Scholar
  24. Engle MA, Gallo M, Schroeder KT, Geboy NJ, Zupancic JW (2014) Three-way compositional analysis of water quality monitoring data. Environ Ecol Stat 21(3):565–581MathSciNetCrossRefGoogle Scholar
  25. Giordani P, Kiers HAL (2006) A comparison of three methods for Principal Component Analysis of fuzzy interval data. Comput Stat Data Anal, special issue “The Fuzzy Approach to Statistical Analysis” 51(1):379–397Google Scholar
  26. Kojadinovic I, Holmes M (2009) Tests of independence among continuous random vectors based on Cramér-von Mises functionals of the empirical copula process. J Multivar Anal 100:1137–1154CrossRefzbMATHGoogle Scholar
  27. Kroonenberg EM (1983) Three-mode principal component analysis: theory and applications. DSWO, LeidenGoogle Scholar
  28. Kroonenberg EM, De Leeuw J (1980) Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45:69–97MathSciNetCrossRefzbMATHGoogle Scholar
  29. Lauro C, Palumbo F (2005) Principal component analysis for non-precise data. In: Vichi M et al (eds) New developments in classification and data analysis. Springer, Heidelberg, pp 173–184CrossRefGoogle Scholar
  30. Mateu-Figueras G, Pawlowsky-Glahn V (2008) A critical approach to probability laws in geochemistry. Math Geosci 40:489–502CrossRefzbMATHGoogle Scholar
  31. Moore RE (1966) Interval analysis. Prentice Hall, New JerseyzbMATHGoogle Scholar
  32. Morrison DF (1990) Multivariate statistical methods, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  33. Neto EAL, De Carvalho FAT (2008) Centre and range method for fitting a linear regression model to symbolic intervalar data. Comput Stat Data Anal 52(3):1500–1515CrossRefzbMATHGoogle Scholar
  34. Neto EAL, De Carvalho FAT (2010) Constrained linear regression models for symbolic interval-valued variables. Comput Stat Data Anal 54(2):333–347MathSciNetCrossRefzbMATHGoogle Scholar
  35. Noirhomme-Fraiture M, Brito P (2011) Far beyond the classical data models: symbolic data analysis. Stat Anal Data Min 4(2):157–170MathSciNetCrossRefGoogle Scholar
  36. Palarea-Albaladejo J, Martín-Fernández JA (2012) Dealing with distances and transformations for fuzzy c-means clustering of compositional data. J Classifi 29:144–169MathSciNetCrossRefzbMATHGoogle Scholar
  37. Pavlačka O (2013) Note on the lack of equality between fuzzy weighted average and fuzzy convex sum. Fuzzy Sets Syst 213:102–105MathSciNetCrossRefzbMATHGoogle Scholar
  38. Pawlowsky-Glahn V, Egozcue JJ (2001) Geometric approach to statistical analysis on the simplex. Stoch Environ Res Risk Assess 15:384–398CrossRefzbMATHGoogle Scholar
  39. Pawlowsky-Glahn V, Egozcue JJ, Tolosana-Delgado R (2015a) Modeling and analysis of compositional data. Wiley, ChichesterGoogle Scholar
  40. Pawlowsky-Glahn V, Egozcue JJ, Lovell D (2015b) Tools for compositional data with a total. Stat Model 15:175–190MathSciNetCrossRefGoogle Scholar
  41. Rousseeuw PJ, Ruts I, Tukey JW (1999) The bagplot: a bivariate boxplot. Am Stat 53(4):382–387Google Scholar
  42. Seber GAF (1984) Multivariate observations. Wiley, New YorkGoogle Scholar
  43. Teles P, Brito P (2015) Modeling interval time series with space-time processes. Commun Stat Theory Methods 44(17):3599–3627MathSciNetCrossRefzbMATHGoogle Scholar
  44. Wang H, Guan R, Wu J (2012) CIPCA: complete-information-based principal component analysis for interval-valued data. Neurocomputing 86:158–169CrossRefGoogle Scholar
  45. Zuccolotto P (2007) Principal components of sample estimates: an approach through symbolic data analysis. Stat Methods Appl 16(2):173–192MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical Analysis and Applications of MathematicsPalacký UniversityOlomoucCzech Republic
  2. 2.FEP and LIAAD INESC TECUniversidade do PortoPortoPortugal
  3. 3.Institute of Statistics and Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria

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