In this paper, we advocate a novel measure for the purpose of checking the quality of a cluster partition for a sample into several distinct classes, and thus, determine the unknown value for the true number of clusters prevailing the provided set of data. Our objective leads us to the development of an approach through applying the multinomial distribution to the distances of data members, clustered in a group, from their respective cluster representatives. This procedure is carried out independently for each of the clusters, and the concerned statistics are combined together to design our targeted measure. Individual clusters separately possess the category-wise probabilities which correspond to different positions of its members in the cluster with respect to a typical member, in the form of cluster-centroid, medoid or mode, referred to as the corresponding cluster representative. Our method is robust in the sense that it is distribution-free, since this is devised irrespective of the parent distribution of the underlying sample. It fulfills one of the rare coveted qualities, present in the existing cluster accuracy measures, of having the capability to investigate whether the assigned sample owns any inherent clusters other than a single group of all members or not. Our measure's simple concept, easy algorithm, fast runtime, good performance, and wide usefulness, demonstrated through extensive simulation and diverse case-studies, make it appealing.

翻译：本文提出一种新度量，用于检验样本划分至多个不同类别的聚类质量，进而确定给定数据集中真实聚类数这一未知值。我们通过将多项分布应用于聚类组内数据成员与其对应聚类代表之间的距离来实现该目标。该过程独立应用于每个聚类，并将相关统计量合并以设计目标度量。每个聚类单独具有类别概率，这些概率对应于其成员相对于聚类中心（质心、中心点或众数，统称为对应聚类代表）的不同位置。本方法具有鲁棒性，因其无需依赖底层样本的原始分布——该方法独立于分布设计。它具备现有聚类精度度量中罕见的珍贵特性：能够检验分配样本是否包含除单一全体集群外的内在聚类结构。通过大量模拟与多样化案例研究验证，本度量具有概念简洁、算法易实现、运行快速、性能优良及广泛适用性等优势。