Abstract
Usually, there are only two stages of comparability between two objects: they are comparable or incomparable (see, for instance, the theory of partially ordered sets). The same holds with respect to equality/inequality. In this publication, measures of incomparability u ij and of inequality v ij between two objects g i and g j with m attributes with respect to the relation ≤ are introduced. Based on these definitions the (non-metric) distance measure \( {a}_{ij}=\frac{1}{2}\left({u}_{ij}+{v}_{ij}\right) \) with maximal possible values \( m+1+\left[\frac{m}{2}\right]\cdot \left(m-\left[\frac{m}{2}\right]\right) \) is proposed. The distance matrix A = (a ij ) will be used for clustering starting from the corresponding complete graph 〈g〉 (g – number of objects), whose edges g i –g j are valued by a ij . The result of the classification consists of a set of complete subgraphs, where, for instance, the objective function of compactness of a cluster is based on all pairwise distances of its members. The same edge-valued graph is used to construct a transitive-directed tournament. Thus, a unique seriation of the objects can be obtained which can also be used for further interpretation of the data. For illustrative purposes, an application to environmental chemistry with only a small data set is considered.
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Bartel, HG., Mucha, HJ. (2014). Measures of Incomparability and of Inequality and Their Applications. In: Brüggemann, R., Carlsen, L., Wittmann, J. (eds) Multi-indicator Systems and Modelling in Partial Order. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8223-9_3
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DOI: https://doi.org/10.1007/978-1-4614-8223-9_3
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