Metrics based on average distance between sets

Abstract

This paper presents a distance function between sets based on an average of distances between their elements. The distance function is a metric if the sets are non-empty finite subsets of a metric space. It includes the Jaccard distance as a special case, and can be generalized by using the power mean so as to also include the Hausdorff metric on finite sets. It can be extended to deal with non-null measurable sets, and applied for measuring distances between fuzzy sets and between probability distributions. These distance functions are useful for measuring similarity between data in computer science and information science. In instructional systems design and information retrieval, for example, they are likely to be useful for analyzing and processing text documents that are modeled as hierarchical collections of sets of terms. A distance measure of learners' knowledge is also discussed in connection with quantities of information. © 2012 The Author(s).