发表机构
Complutense University of Madrid; Universidad Pública de Navarra（马德里康普顿斯大学; 纳瓦拉公立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对矩阵结构数据的聚合需求，形式化定义了矩阵聚合算子（MAO），分析其可分解性与对称性，并提出最大熵全局覆盖指标（MEGCIs）用于聚类质量评估。
AI 中文摘要
聚合理论传统上集中于定义在向量上的算子。然而，许多应用——包括多准则决策、群体决策、模糊规则基分类系统以及重叠/分组指标——需要聚合自然结构化为隶属度矩阵的信息（例如，一组对象与一族模糊集相互作用的情况）。尽管如此，尚未有针对此类算子的正式框架被提出，部分原因在于将矩阵展平为向量的常见做法（这会丢弃结构信息），部分原因在于依赖于可分解算子来顺序聚合行和列。本文通过形式化矩阵聚合算子（MAO）的概念来填补这一空白。我们分析了MAO的可分解性和对称性性质，表明某些算子无法以可分解形式表达，并考察了几种对称性概念。最后，我们引入了一族称为最大熵全局覆盖指标（MEGCIs）的MAO，基于组合分组函数和MEOWA算子提供了它们的构造方法，并通过一项广泛的计算研究展示了它们在聚类质量评估中的实用性。
英文摘要
Aggregation theory has traditionally focused on operators defined over vectors. However, many applications-including Multi-Criteria Decision Making, Group Decision Making, Fuzzy Rule-Based Classification Systems, and overlap/grouping indices-require aggregating information naturally structured as a matrix of membership degrees (e.g., where a set of objects interacts with a family of fuzzy sets). Despite this, no formal framework has been proposed for this class of operators, partly due to the common practice of flattening matrices into vectors (which discards structural information) and partly due to a reliance on decomposable operators that aggregate rows and columns sequentially. This paper addresses this gap by formalizing the notion of a matrix aggregation operator (MAO). We analyze the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry. Finally, we introduce a family of MAOs termed maximum entropy global coverage indices (MEGCIs), provide a construction method for them based on combining grouping functions and MEOWA operators, and illustrate their usefulness in cluster quality assessment through an extensive computational study.
