相对Wasserstein空间深度用于分布数据中的簇数选择
Relative Wasserstein Spatial Depth for Cluster Number Selection in Distributional Data
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- Columbia University（哥伦比亚大学）
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中文总结 AI 辅助
本文提出相对Wasserstein空间深度（RWSD）作为分布数据聚类中簇数选择的标准，与Wasserstein K-means或K-medians结合，理论保证一致性，模拟和实际数据中优于轮廓分数和Davies-Bouldin指数。
中文摘要 AI 辅助
当代许多科学领域中的数据，如图像、媒体流、生物医学组学，自然地建模为Wasserstein空间中的概率分布，而非欧几里得空间中的点。因此，对分布进行聚类的方法需求很高，因为它们提供了对相似对象进行分组的探索性框架。在Wasserstein空间中对分布值数据进行聚类通常需要预先选择簇的数量。我们提出相对Wasserstein空间深度（RWSD）作为最优簇数的选择标准。它比较每个分布在其被分配的簇中的深度与其在竞争簇中的深度。我们将此标准与Wasserstein K-means或提出的K-medians算法配对。在适当的假设下，我们建立了簇中心和所选最优簇数的一致性。我们还给出了两阶段抽样下的错误率，并刻画了RWSD在任何固定划分下的稳健性。在大量模拟中，RWSD在不均匀分离、异质离散、重尾和异常值情况下选择了预期的簇数，而轮廓分数或Davies-Bouldin指数常常失败。对MNIST图像和流式细胞术数据的应用说明了RWSD的实际价值。在某些情况下，当簇不是测地线凸的且缺乏有意义的深度中心时，RWSD可能失败。
英文摘要
Contemporary data in many scientific domains, such as images, media streams, biomedical omics, are naturally modeled as probability distributions in Wasserstein space instead of points in Euclidean space. Consequently, clustering methods for distributions are in high demand, as they provide exploratory frameworks to group objects with similarity. Clustering distribution-valued data in Wasserstein space often requires choosing the number of clusters in advance. We propose Relative Wasserstein Spatial Depth (RWSD) as a selection criterion for optimal number of clusters. It compares the depth of each distribution in its assigned cluster with its depth in competing clusters. We pair this criterion with Wasserstein K-means or a proposed K-medians algorithm. Under suitable assumptions, we establish consistency of the cluster centers and of the selected optimal number of clusters. We also give an error rate under two-stage sampling and characterize the robustness of RWSD under any fixed partition. In extensive simulations, RWSD selects the intended number of clusters under unequal separations, heterogeneous dispersion, heavy tails, and outliers, where silhouette score or the Davies-Bouldin Index often fails. Applications to MNIST images and flow cytometry data illustrate the practical value of RWSD. In some cases where a cluster is not geodesically convex and lacks a meaningful depth center, the RWSD can fail.
