AI 中文总结
针对神经科学和环境科学等领域中多变量功能数据的重叠聚类结构，提出基于潜在因子模型的方法，通过建立载荷矩阵可识别性等，开发估计程序并建立定理，经数值研究和应用展示了方法性能。
AI 中文摘要
在神经科学和环境科学等应用中，数据通常自然地建模为多变量功能数据，且常呈现重叠聚类结构。现有功能数据聚类方法通常要求成员互斥，无法捕捉这种结构。我们提出一种基于潜在因子模型的方法，含功能因子和编码潜在重叠聚类成员关系的实值载荷矩阵。在温和条件下，我们建立了载荷矩阵在置换下的可识别性，确保重叠聚类结构可恢复。我们开发了估计聚类数量和相关聚类成员关系的程序，涉及解决算子中的无穷维回归问题，其解用希尔伯特 - 施密特算子空间上的内积表征并用实值矩阵表示。该公式便于进行严格的渐近分析，我们建立了中心极限定理以促进对重叠聚类成员关系的统计推断。我们通过数值研究和对功能磁共振成像数据的应用展示了我们方法的性能。
英文摘要
In applications such as neuroscience and environmental science, data are naturally modeled as multivariate functional data and often exhibit overlapping cluster structure. Existing clustering methods for functional data typically impose mutually exclusive memberships and therefore fail to capture such structure. We propose a latent factor model based approach with functional factors and a real-valued loading matrix that encodes potentially overlapping cluster memberships. Under mild conditions, we establish identifiability of the loading matrix up to permutation, ensuring that the overlapping cluster structure is recoverable up to label switching. We develop a procedure for estimating both the number of clusters and the associated cluster memberships. This involves solving an infinite-dimensional regression problem in operators, whose solution is characterized using the inner product on the space of Hilbert-Schmidt operators and expressed in terms of real-valued matrices. This formulation enables rigorous asymptotic analysis, and we establish a central limit theorem to facilitate statistical inference on overlapping cluster memberships. We demonstrate the performance of our method using numerical studies and an application to functional magnetic resonance imaging data.
Comments54 pages, 10 figures
