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Exploratory data analysis for interval compositional data

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

Abstract

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.

Keywords

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

Mathematics Subject Classification

62H25 62H99 

Notes

Acknowledgments

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.

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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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