Unbalanced Longitudinal Data Clustering with a Copula Kernel Mixture Model

• Published in: Statistics and computing Vol. 35; no. 5
• Main Authors: Zhang, Xi, Murphy, Orla A., McNicholas, Paul D.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.10.2025
• Subjects: Algorithms; Clustering; Data analysis; Eigenvalues; Eigenvectors; Multivariate analysis; Parameters; Warping
• ISSN: 0960-3174; 1573-1375
• DOI: 10.1007/s11222-025-10650-6

Unbalanced longitudinal data appears commonly in practice, for example in cases where measurements are collected at different time points for different subjects and can therefore be sparse and/or irregularly sampled. Treating such data as functional enables smooth curve estimation and better handling of missing or irregularly spaced observations. Therefore, a Gaussian copula kernel mixture model (CKMM), based on functional data analysis, is proposed for clustering unbalanced multivariate longitudinal data. In this model, subject-specific warping matrices are included to account for irregularly spaced observations. A regularized functional eigen-decomposition is employed to estimate the copula correlation parameters, ensuring the smoothing procedure is integrated into clustering. Additionally, a functional gradient descent algorithm is implemented as an alternative to kernel density estimation to reduce computational complexity. An expectation-maximization-like algorithm is proposed to estimate marginal distributions, copula parameters, eigenfunctions, and eigenvalues in the CKMM. The performance of the CKMM is demonstrated through a simulation study and a data application. The proposed model exhibits superior performance compared to k-means with dynamic time warping, the growth mixture model, and functional high-dimensional data clustering.