Clusterwise functional linear regression modelling by Gibbs sampler and information criterion

This paper focuses on detecting clusterwise heterogeneous regression patterns in functional regression models with scalar response. We demonstrate that the optimal partition of the observed sample units into the best number of clusters, as well as the best regression coefficient estimates, can be obtained by minimizing an empirical Bayesian information criterion. However, due to the prospective combinatorial explosion in computation, it is impractical to explore all possible partitions exhaustively to find the best clustering and regression solution. To tackle this difficulty, we develop a Gibbs sampling-based functional clustering and regression method to generate a Markov chain of candidate partitions with its equilibrium probability distribution corresponding to that induced by the empirical Bayesian information criterion. This effectively overcomes the combinatorial explosion in clustering and allows for a consistent estimation of the best partition as well. Least absolute deviation method is used for clusterwise functional regression estimation to enhance robustness against nonnormal random errors. With both simulation and real data analysis, our method exhibits better performance compared to the existing methods. • Stochastic optimization by Gibbs sampling for functional regression and clustering. • Empirical Bayesian information criterion (eBIC) for cluster number selection. • Applications to two studies: 2023 China air pollution and sorghum flowering time.