There is a rich literature on clustering functional data with applications to time-series modeling, trajectory data, and even spatio-temporal applications. However, existing methods routinely perform global clustering that enforces identical atom values within the same cluster. Such grouping may be inadequate for high-dimensional functions, where the clustering patterns may change between the more dominant high-level features and the finer resolution local features. While there is some limited literature on local clustering approaches to deal with the above problems, these methods are typically not scalable to high-dimensional functions, and their theoretical properties are not well-investigated. Focusing on basis expansions for high-dimensional functions, we propose a flexible non-parametric Bayesian approach for multi-resolution clustering. The proposed method imposes independent Dirichlet process (DP) priors on different subsets of basis coefficients that ultimately results in a product of DP mixture priors inducing local clustering. We generalize the approach to incorporate spatially correlated error terms when modeling random spatial functions to provide improved model fitting. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed for implementation. We show posterior consistency properties under the local clustering approach that asymptotically recovers the true density of random functions. Extensive simulations illustrate the improved clustering and function estimation under the proposed method compared to classical approaches. We apply the proposed approach to a spatial transcriptomics application where the goal is to infer clusters of genes with distinct spatial patterns of expressions. Our method makes an important contribution by expanding the limited literature on local clustering methods for high-dimensional functions with theoretical guarantees.

翻译：函数数据聚类在时间序列建模、轨迹数据乃至时空应用领域已有丰富文献。然而，现有方法通常执行全局聚类，强制同一簇内原子值完全相同。对于高维函数而言，此类分组可能并不充分，因为聚类模式可能在更显著的高层特征与更精细的局部特征之间发生变化。尽管已有少量文献提出局部聚类方法以应对上述问题，但这些方法通常难以扩展到高维函数，且其理论性质尚未得到充分研究。聚焦于高维函数的基展开方法，我们提出一种灵活的非参数贝叶斯多分辨率聚类方法。该方法在不同基系数子集上施加独立的狄利克雷过程先验，最终形成诱导局部聚类的狄利克雷过程混合先验乘积。我们将该方法推广至随机空间函数建模中，通过纳入空间相关误差项以提升模型拟合效果。我们开发了高效的马尔可夫链蒙特卡洛算法用于实现。我们证明了局部聚类方法下的后验一致性性质，该方法能渐近恢复随机函数的真实密度。大量仿真实验表明，与传统方法相比，所提方法在聚类和函数估计方面均有显著改进。我们将所提方法应用于空间转录组学领域，其目标是推断具有不同空间表达模式的基因簇。本方法通过拓展具有理论保证的高维函数局部聚类方法的有限文献，做出了重要贡献。