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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:
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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.
Annoni, P., Fattore, M. and Bruggemann, R. (2012). A multi-criteria fuzzy approach for analyzing poverty structure. Statistica & Applicazioni, accepted.
Bruggemann, R. and Bartel, H. (1999). A theoretical concept to rank environmentally significant chemicals. J. Chem. Inf. Comput. Sci., 39(2), 211–217. CrossRef
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. CrossRef
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.
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. CrossRef
- Hasse Diagrams Based on Transformed Data Matrices
Ganapati P. Patil
- Springer New York
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