Abstract
It is always hard to try a positioning. Nevertheless, it may help interested readers to find their way through the jungle of concepts, relations, and equations of this text. Certainly, partial order has to do with graph theory in discrete mathematics, as its visualization is a digraph and questions like connectivity or identification of articulation points and of separated subsets are typical of graph theory; see, e.g., Wagner and Bodendiek 1989) and Patil and Taillie (2004). There is also a connection to the network domain, as partial order constitutes a directed graph, which is one of the characteristics of networks. In our applications here, there is always a matrix, which quantifies the multi-indicator system, the data matrix. With or without the interim step of deriving the rank matrix, we arrive at a partial order. Once, however, the poset is derived, it may be analyzed as a mathematical object on its own right.
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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.
Bertin strategy. http://en.wikipedia.org/wiki/Jacques_Bertin.
Fattore, M. (2008). Hasse diagrams, poset theory and fuzzy poverty measures. Riv. Int. Sci. Soc., 1, 63–75.
Gansner, E.R., Koutsofios, E. and North, S. (2009). Drawing graphs with dot. http://www.graphviz.org/pdf/dotguide.pdf.
Habib, M., Moehring, R.H. and Steiner, G. (1988). Computing the bump number is easy. Order, 5, 107–129.
Ivanciuc, T., Ivanciuc, O. and Klein, D.J. (2005). Posetic quantitative superstructure/activity relationships (QSSARs) for chlorobenzenes. J. Chem. Inf. Model., 45, 870–879.
Ivanciuc, T., Ivanciuc, O. and Klein, D.J. (2006a). Modeling the bioconcentration factors and bioaccumulation factors of polychlorinated biphenyls with posetic quantitative super-structure/activity relationships (QSSAR). Mol. Divers., 10, 133–145.
Ivanciuc, T., Ivanciuc, O. and Klein, D.J. (2006b). Prediction of environmental properties for chlorophenols with posetic quantitative super-structure/property relationships (QSSPR). Int. J. Mol. Sci., 7, 358–374.
Klein, D.J. (1986). Chemical graph-theoretic cluster expansions. Int. J. Quant. Chem., 20, 153–171.
Klein, D.J. and Bytautas, L. (2000). Directed reaction graphs as posets. Match, 42, 261–290.
Luther, B., Bruggemann, R. and Pudenz, S. (2000). An approach to combine cluster analysis with order theoretical tools in problems of environmental pollution. Match, 42, 119–143.
Mucha, H. (2002). Clustering techniques accompanied by matrix reordering techniques. In K. Voigt and G. Welzl (Eds.), Order theoretical tools in environmental sciences – order theory (Hasse diagram technique) meets multivariate statistics (pp. 129–140). Aachen: Shaker-Verlag.
Myers, W.L. and Patil, G.P. (2010). Partial order and rank range runs for compositional complexes in landscape ecology and image analysis, with applications to restoration, remediation, and enhancement. Environ. Ecol. Stat., 17, 411–436.
Myers, W.L., Patil, G.P. and Cai, Y. (2006). Exploring patterns of habitat diversity across landscapes using partial ordering. In R. Bruggemann and L. Carlsen (Eds.), Partial order in environmental sciences and chemistry (pp. 309–325). Berlin: Springer.
Neggers, J. and Kim, H.S. (1998). Basic posets. Singapore: World Scientific Publishing.
Newlin, J.T. and Patil, G.P. (2010). Application of partial order to bridge engineering, stream channel assessment, and infra structure management. Environ. Ecol. Stat., 17, 437–454.
Patil, G.P. and Taillie, C. (2004). Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization. Environ. Ecol. Stat., 11, 199–228.
Rademaker, M., De Baets, B. and De Meyer, H. (2008). New operations for informative combinations of two partial orders with illustrations on pollution data. Comb. Chem. High Throughput Screen., 11(9), 745–755.
Saltelli, A. and Annoni, P. (2010). How to avoid a perfunctory sensitivity analysis. Environ. Model. Softw., 25, 1508–1517.
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., et al. (2008). Global sensitivity analysis – The primer. Chichester: Wiley.
Sørensen, P.B., Bruggemann, R., Thomsen, M., Gyldenkaerne, S. and Kjaer, C. (2009). How to guide and assess risk reduction using risk characterization indicators. Am. J. Appl. Sci., 6(6), 1255–1263.
Sørensen, P.B., Mogensen, B.B., Carlsen, L. and Thomsen, M. (2000). The influence on partial order ranking from input parameter uncertainty – Definition of a robustness parameter. Chemosphere, 41, 595–600.
Sørensen, P.B., Mogensen, B.B., Gyldenkaerne, S. and Rasmussen, A.G. (1998). Pesticide leaching assessment method for ranking both single substances and scenarios of multiple substance use. Chemosphere, 36(10), 2251–2276.
Stanley, R.P. (1986). Enumerative combinatorics (Vol. I). Monterey: Wadsworth and Brooks/Cole.
Syslo, M.M. (1985). A graph-theoretic approach to the jump-number problem. In I. Rival (Ed.), Graphs and order (pp. 185–215). Dordrecht: D. Reidel Publishing Company.
Wagner, K. and Bodendiek, R. (1989). Graphentheorie I: Anwendungen auf Toplogie, Gruppentheorie und Verbandstheorie. Mannheim: BI Wissenschaftsverlag.
Wasserman, S. and Faust, K. (2009). Social network analysis – Methods and applications. Cambridge: Cambridge University Press.
Welzl, G., Voigt, K. and Bruggemann, R. (2002). Order theory meets statistics – Environmental statistics. In K. Voigt and G. Welzl (Eds.), Order theoretical tools in environmental sciences – order theory (Hasse diagram technique) meets multivariate statistics (pp. 41–54). Aachen: Shaker-Verlag.
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Brüggemann, R., Patil, G.P. (2011). Partial Order and Related Disciplines. In: Ranking and Prioritization for Multi-indicator Systems. Environmental and Ecological Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8477-7_16
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