Measures of Incomparability and of Inequality and Their Applications

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.