In longitudinal data analysis, observation points of repeated measurements over time often vary among subjects except in well-designed experimental studies. Additionally, measurements for each subject are typically obtained at only a few time points. From such sparsely observed data, identifying underlying cluster structures can be challenging. This paper proposes a fast and simple clustering method that generalizes the classical $k$-means method to identify cluster centers in sparsely observed data. The proposed method employs the basis function expansion to model the cluster centers, providing an effective way to estimate cluster centers from fragmented data. We establish the statistical consistency of the proposed method, as with the classical $k$-means method. Through numerical experiments, we demonstrate that the proposed method performs competitively with, or even outperforms, existing clustering methods. Moreover, the proposed method offers significant gains in computational efficiency due to its simplicity. Applying the proposed method to real-world data illustrates its effectiveness in identifying cluster structures in sparsely observed data.

翻译：在纵向数据分析中，除精心设计的实验研究外，重复测量随时间变化的观测点通常在不同个体间存在差异。此外，每个个体的测量通常仅在少数时间点获得。从这类稀疏观测数据中识别潜在的聚类结构可能具有挑战性。本文提出了一种快速而简单的聚类方法，将经典的$k$-均值方法推广至稀疏观测数据中的聚类中心识别。所提方法采用基函数展开对聚类中心进行建模，为从碎片化数据中估计聚类中心提供了有效途径。与经典$k$-均值方法类似，我们建立了所提方法的统计一致性。通过数值实验，我们证明所提方法在性能上与现有聚类方法相当甚至更优。此外，得益于其简洁性，所提方法在计算效率上具有显著优势。将所提方法应用于实际数据，验证了其在稀疏观测数据中识别聚类结构的有效性。