On asymmetric matching between sets

A comparison of two objects may be viewed as an attempt to determine the degree to which they are similar or different in a given sense. Defining a good measure of proximity, or else similarity or dissimilarity between objects is very important in practical tasks as well as theoretical achievements. Each object is usually represented as a point in Cartesian coordinates, and therefore the distance between points reflects similarities between respective objects. In general, the space is assumed to be Euclidean, and a distance between points assigns a nonnegative number. From another point of view the concept of symmetry underlies essentially all theoretical treatments of similarity. Tversky (1977) provided empirical evidence of asymmetric similarities and argued that similarity should not be treated as a symmetric relation. According to Tversky’s consideration, an object is described by sets of features instead of geometric points in a metric space. In this paper we propose a new measure of remoteness between sets of nominal values. Instead of considering distance between two sets, we introduce the measures of perturbation type 1 of one set by another. The consideration is based on set-theoretic operations and the proposed measure describes changes of the second set after adding the first set to it, or vice versa. The measure of sets’ perturbation returns a value from [0, 1], and it must be emphasized that this measure is not symmetric in general. The difference between 1 and the sum of these two measures of perturbation of a pair of sets can be understood as Jaccard’s extended similarity measure. In this paper several mathematical properties of the measure of sets’ perturbation are studied, and interpretation of proximity is explained by the comparison of selected measures.