We consider the problem of clustering functional data according to their covariance structure. We contribute a soft clustering methodology based on the Wasserstein–Procrustes distance, where the in-between cluster variability is penalized by a term proportional to the entropy of the partition matrix. In this way, each covariance operator can be partially classified into more than one group. Such soft classification allows for clusters to overlap, and arises naturally in situations where the separation between all or some of the clusters is not well-defined. We also discuss how to estimate the number of groups and to test for the presence of any cluster structure. The algorithm is illustrated using simulated and real data. An R implementation is available in the Appendix S1.

中文翻译：

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基于 Wasserstein-Procrustes 度量的函数数据的协方差软聚类

我们考虑根据协方差结构对函数数据进行聚类的问题。我们贡献了一种基于 Wasserstein-Procrustes 距离的软聚类方法，其中聚类之间的变异性受到与分区矩阵的熵成比例的项的惩罚。这样，每个协方差算子可以部分地分为多个组。这种软分类允许聚类重叠，并且在所有或一些聚类之间的分离没有明确定义的情况下自然出现。我们还讨论了如何估计组的数量并测试是否存在任何集群结构。使用模拟和真实数据来说明该算法。附录 S1 提供了R实现。