Groupmost

Groupmost is a term used in data analysis to designate the most representative member of a group. Given a finite group G and a distance or dissimilarity function d defined on its members, the groupmost is the element m in G that minimizes the total within-group dissimilarity S(m) = sum_{x in G} d(m, x). When distances are symmetric and nonnegative, this selects the member most central to the group, in the sense of minimizing travel or communication cost to others.

Formally, groupmost(G, d) = argmin_{m in G} sum_{x in G} d(m, x). If multiple members achieve the same minimum, a tie-break rule is applied, such as selecting the member with the smallest index or smallest sum of pairwise distances to others.

Relation to medoid: The groupmost is equivalent to the medoid of the cluster, a representative member used in exemplar-based clustering. Unlike the centroid, which may not belong to the group, the groupmost is always a member of the group.

Computation: Computing the groupmost requires pairwise dissimilarities; a straightforward approach evaluates S(m) for each m in G, yielding O(n^2) distance evaluations for a group of size n. For large data sets, approximate or incremental methods can reduce cost.

Example: Consider G = {A, B, C} with d(A,B)=2, d(A,C)=3, d(B,C)=4. S(A)=5, S(B)=6, S(C)=7, so the groupmost is A.

Applications: groupmost is used for data summarization, prototype selection, and explainable representations in clustering and social network analysis.

See also: Medoid; Centroid; Exemplar; Prototype.