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Association measures for interval variables

Author

Listed:
  • M. Rosário Oliveira

    (Universidade de Lisboa)

  • Margarida Azeitona

    (Universidade de Lisboa)

  • António Pacheco

    (Universidade de Lisboa)

  • Rui Valadas

    (Universidade de Lisboa)

Abstract

Symbolic Data Analysis (SDA) is a relatively new field of statistics that extends conventional data analysis by taking into account intrinsic data variability and structure. Unlike conventional data analysis, in SDA the features characterizing the data can be multi-valued, such as intervals or histograms. SDA has been mainly approached from a sampling perspective. In this work, we propose a model that links the micro-data and macro-data of interval-valued symbolic variables, which takes a populational perspective. Using this model, we derive the micro-data assumptions underlying the various definitions of symbolic covariance matrices proposed in the literature, and show that these assumptions can be too restrictive, raising applicability concerns. We analyze the various definitions using worked examples and four datasets. Our results show that the existence/absence of correlations in the macro-data may not be correctly captured by the definitions of symbolic covariance matrices and that, in real data, there can be a strong divergence between these definitions. Thus, in order to select the most appropriate definition, one must have some knowledge about the micro-data structure.

Suggested Citation

  • M. Rosário Oliveira & Margarida Azeitona & António Pacheco & Rui Valadas, 2022. "Association measures for interval variables," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(3), pages 491-520, September.
    • Handle: RePEc:spr:advdac:v:16:y:2022:i:3:d:10.1007_s11634-021-00445-8
    DOI: 10.1007/s11634-021-00445-8
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    References listed on IDEAS

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    1. A. Pedro Duarte Silva & Peter Filzmoser & Paula Brito, 2018. "Outlier detection in interval data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(3), pages 785-822, September.
    2. Billard L. & Diday E., 2003. "From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 470-487, January.
    3. A. Silva & Paula Brito, 2015. "Discriminant Analysis of Interval Data: An Assessment of Parametric and Distance-Based Approaches," Journal of Classification, Springer;The Classification Society, vol. 32(3), pages 516-541, October.
    4. X. Zhang & B. Beranger & S. A. Sisson, 2020. "Constructing likelihood functions for interval‐valued random variables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(1), pages 1-35, March.
    5. Paulo Teles & Paula Brito, 2015. "Modeling Interval Time Series with Space–Time Processes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(17), pages 3599-3627, September.
    6. Paula Brito & A. Pedro Duarte Silva, 2012. "Modelling interval data with Normal and Skew-Normal distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(1), pages 3-20, March.
    7. Francisco Carvalho & Paula Brito & Hans-Hermann Bock, 2006. "Dynamic clustering for interval data based on L 2 distance," Computational Statistics, Springer, vol. 21(2), pages 231-250, June.
    8. Dias, Sónia & Brito, Paula, 2017. "Off the beaten track: A new linear model for interval data," European Journal of Operational Research, Elsevier, vol. 258(3), pages 1118-1130.
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