¹University of Milano-Bicocca, Department of Statistics and Quantitative Methods, Italy. Professor. Email: marco.fattore@unimib.it
²University of Milano-Bicocca, Department of Statistics and Quantitative Methods, Italy. Professor. Email: alberto.arcagni@unimib.it
³Company SmartStat, Italy. Researcher. Email: stefanobarberis@hotmail.it

In this paper, we develop a visualization process for partial orders derived from considering many numerical indicators on a statistical population. The issue is relevant, particularly in the field of socio-economic evaluation, where explicitly taking into account incomparabilities among individuals proves much more informative than adhering to classical aggregative and compensative approaches, which collapse complexity into unidimensional rankings. We propose a process of visual analysis based on a combination of tools and concepts from partial order theory, multivariate statistics and visual design. We develop the process through a real example, based on data pertaining to regional competitiveness in Europe.

Key words: Partial Order, Hasse Diagrams, Self-Organizing Map, Visualization.

En este artículo, se desarrolla un proceso de visualización para órdenes parciales derivados al considerar varios indicadores numéricos de una población estadística. Esto es relevante parcialmente en el área de la evaluación socioedonómica donde se toma en cuenta explicitamente incomparabilidades entre los individuos y resulta ser más informativo que la agregación clásica y compensativa que colapsa la complejidad en rankings unidimensionales. Se propone un proceso de análisis visual basado en la combinación herramientas y conceptos de la teoría de orden parcial, estadítica multivariada y diseño usual. Se desarrollo el proceso a través de un ejemplo real, basado en datos de competitividad regional en Europa

Palabras clave: diagramas Hasse, mapa autoorganizado, orden parcial, visualización.

Texto completo disponible en PDF

References

1. Al-Sharrah, G. (2014), Ranking Hazarduous chemicals with a heuristic approach to reduce isolated objectds in Hasse Diagrams, 'Multi-indicator Systems and Modelling in Partial Order', Springer-Verlag, New York, p. 217-235.         [ Links ]

2. Annoni, P. & Dijkstra, L. (2013), EU Regional Competitiveness Index - RCI 2013, Joint Research Centre of the European Commission, Luxembourg.         [ Links ]

3. Barthélemy, J.P., Flament, C. & Monjardet, B. (1982), 'Ordered sets and social sciences', Ordered Sets 83,, 721-758.         [ Links ]

4. Bruggemann, R. & Carlsen, L. (2014), 'Incomparable - what now? Absorption of incomparabilities by cluster and fuzzy-methods', Quality and Quantity. forthcoming.         [ Links ]

5. Davey, B. A. & Priestley, H. A. (2002), Introduction to Lattices and Order, Cambridge University Press.         [ Links ]

6. Dijkstra, L., Annoni, P. & Kozovska, K. (2011), A new European Regional Competitiveness Index: Theory, Methods and Findings, 02/2011, European Union Regional Policy Working Papers. (RCI 2010).         [ Links ]

7. Fattore, M. & Arcagni, A. (2014), PARSEC: an R package for poset-based evaluation of multidimensional poverty, 'Multi-indicator Systems and Modelling in Partial Order', Springer-Verlag, New York, p. 317-330.         [ Links ]

8. Fattore, M., Bruggemann, R. & Owsinski, J. (2011), Using poset theory to compare fuzzy multidimensional material deprivation across regions, 'New Perspectives in Statistical Modeling and Data Analysis', Springer-Verlag.         [ Links ]

9. Fattore, M., Maggino, F. & Colombo, E. (2012), From composite indicators to partial order: Evaluating socio-economic phenomena through ordinal data, 'Quality of Life in Italy: Research and Reflections, Social Indicators Research Series 48', Springer.         [ Links ]

10. Fattore, M., Maggino, F. & Greselin, F. (2011), 'Socio-economic evaluation with ordinal variables: Integrating counting and poset approaches', Statistica & Applicazioni, 31-42. Special Issue.         [ Links ]

11. Freese, R. (2004), Automated lattice drawing, 'Concept Lattices', Springer-Verlag, Berlin, p. 112-127.         [ Links ]

12. Kohonen, T. (2001), Self-Organizing Maps, Springer.         [ Links ]

13. R Core Team, (2013), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. *http://www.R-project.org/         [ Links ]

14. Rival, I. (1984), Graphs and Order. The Role of Graphs in the Theory of Ordered Sets and its Applications, Vol. 147 of Nato Science Series C, Springer, Dordrecht.         [ Links ]

15. Rival, I. (1985), The diagram, 'Graphs and Order', D.Reidel Publishing Company, p. 103-133.         [ Links ]

16. Rival, I. (1989), 'Graphical data structures for ordered sets', Algorithms and Order 255, 3-31.         [ Links ]

17. Sen, A. (1992), Inequality Reexamined, Oxford University Press.         [ Links ]

18. Tsakovski, S. & Simeonov, V. (2008), Hasse Diagrams as Explanatory Tool in Environmental Data Mining: A Case Study, 'Multicriteria Ordering and Ranking: Partial Orders, Ambiguities and Applied Issues', Systems Research Institute Polish Academy of Sciences, Warsaw, p. 50-68.         [ Links ]

19. Tsakovski, S. & Simeonov, V. (2011), 'Hasse diagram technique as exploratory tool in sediment pollution assessment', Journal of Chemometrics 25(5), 254-261. doi: 10.1002/cem.1381:1-8.         [ Links ]

20. Wehrens, R. & Buydens, L. M. C. (2007), 'Self- and super-organizing maps in R: The Kohonen Package', Journal of Statistical Software 21(5).         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: