$K$-means clustering for sparsely observed longitudinal data

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.