Measures of Incomparability and of Inequality and Their Applications

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 i j = 1 2 ( u i j + v i j ) $$ {a}_{ij}=\frac{1}{2}\left({u}_{ij}+{v}_{ij}\right) $$ with maximal possible values m + 1 + [ m 2 ] ⋅ ( m − [ m 2 ] ) $$ m+1+\left[\frac{m}{2}\right]\cdot \left(m-\left[\frac{m}{2}\right]\right) $$ 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.