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Hasse Diagrams Based on Transformed Data Matrices

  • Rainer Brüggemann
  • Ganapati P. Patil
Chapter
Part of the Environmental and Ecological Statistics book series (ENES)

Abstract

We have seen in Chapter 5 as to why and how the structure of partial order (X, IB) can be related to properties of the data matrix. However, partial order with many objects can lead to messy Hasse diagrams with too many lines hiding the structure. What may be the reason for complexity in such diagrams? The number of objects |X| is not necessarily causing messy Hasse diagrams because chains of height |X| or antichains of width |X| certainly allow clear visualizations. There is another reason for complexity: In partial orders, we obtain either x < y or x || y even if the numerical difference ɛ between attribute values is small:

References

  1. Annoni, P., Fattore, M. and Bruggemann, R. (2008). Analyzing the structure of poverty by fuzzy partial order. In J. Owsinski and R. Bruggemann (Eds.), Multicriteria ordering and ranking: Partial orders, ambiguities and applied issues (pp. 107–124). Warsaw: Systems Research Institute Polish Academy of Sciences.Google Scholar
  2. Annoni, P., Fattore, M. and Bruggemann, R. (2012). A multi-criteria fuzzy approach for analyzing poverty structure. Statistica & Applicazioni, accepted.Google Scholar
  3. Bruggemann, R. and Bartel, H. (1999). A theoretical concept to rank environmentally significant chemicals. J. Chem. Inf. Comput. Sci., 39(2), 211–217.CrossRefGoogle Scholar
  4. Bruggemann, R., Bucherl, C., Pudenz, S. and Steinberg, C. (1999). Application of the concept of partial order on comparative evaluation of environmental chemicals. Acta Hydrochim. Hydrobiol., 27(3), 170–178.CrossRefGoogle Scholar
  5. Bruggemann, R. and Welzl, G. (2002). Order theory meets statistics – Hasse diagram technique. In K. Voigt and G. Welzl (Eds.), Order theoretical tools in environmental sciences – order theory (Hasse diagram technique) meets multivariate statistics (pp. 9–39). Aachen: Shaker-Verlag.Google Scholar
  6. De Baets, B. and De Meyer, H. (2003). On the existence and construction of T-transitive closures. Inf. Sci. 152, 167–179.CrossRefMATHGoogle Scholar
  7. Van de Walle, B., De Baets, B. and Kersebaum, K.C. (1995). Fuzzy multi-criteria analysis of cutting techniques in a nuclear dismantling project. Fuzzy Sets Syst., 74, 115–126.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of EcohydrologyLeibniz Institute of Freshwater Ecology and Inland FisheriesSchöneicheGermany
  2. 2.Center for Statistical Ecology and Environmental StatisticsPennsylvania State UniversityUniversity ParkUSA

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