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Basic statistics for distributional symbolic variables: a new metric-based approach

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  • Antonio Irpino
  • Rosanna Verde

Abstract

In data mining it is usual to describe a group of measurements using summary statistics or through empirical distribution functions. Symbolic data analysis (SDA) aims at the treatment of such kinds of data, allowing the description and the analysis of conceptual data or of macrodata summarizing classical data. In the conceptual framework of SDA, the paper aims at presenting new basic statistics for distribution-valued variables, i.e., variables whose realizations are distributions. The proposed measures extend some classical univariate (mean, variance, standard deviation) and bivariate (covariance and correlation) basic statistics to distribution-valued variables, taking into account the nature and the variability of such data. The novel statistics are based on a distance between distributions: the $$\ell _2$$ ℓ 2 Wasserstein distance. A comparison with other univariate and bivariate statistics presented in the literature points out some relevant properties of the proposed ones. An application on a clinic dataset shows the main differences in terms of interpretation of results. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Antonio Irpino & Rosanna Verde, 2015. "Basic statistics for distributional symbolic variables: a new metric-based approach," 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. 9(2), pages 143-175, June.
    • Handle: RePEc:spr:advdac:v:9:y:2015:i:2:p:143-175
    DOI: 10.1007/s11634-014-0176-4
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    References listed on IDEAS

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    1. Ginestet, Cedric E. & Simmons, Andrew & Kolaczyk, Eric D., 2012. "Weighted Frechet means as convex combinations in metric spaces: Properties and generalized median inequalities," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1859-1863.
    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. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
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    1. Luis Lorenzo & Javier Arroyo, 2022. "Analysis of the cryptocurrency market using different prototype-based clustering techniques," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-46, December.
    2. Francisco de A. T. Carvalho & Antonio Irpino & Rosanna Verde & Antonio Balzanella, 2022. "Batch Self-Organizing Maps for Distributional Data with an Automatic Weighting of Variables and Components," Journal of Classification, Springer;The Classification Society, vol. 39(2), pages 343-375, July.
    3. 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.
    4. Dias, Sónia & Brito, Paula & Amaral, Paula, 2021. "Discriminant analysis of distributional data via fractional programming," European Journal of Operational Research, Elsevier, vol. 294(1), pages 206-218.

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