Cluster analysis with regression of non-Gaussian functional data on covariates

Cluster analysis with functional data often imposes normality assumptions on outcomes and is typically carried out without covariates or supervision. However, nonnormal functional data are frequently encountered in practice, and unsupervised learning, without directly tying covariates to clusters, often makes the resulting clusters less interpretable. To address these issues, we propose a new semiparametric transformation functional regression model, which enables us to cluster nonnormal functional data in the presence of covariates. Our model incorporates several unique features. First, it omits the normality assumptions on the functional response, which adds more flexibility to the modelling. Second, our model allows clusters to have distinct relationships between functional responses and covariates, and thus makes the clusters formed more interpretable. Third, unlike various competing methods, we allow the number of clusters to be unspecified and data-driven. We develop a new method, which combines penalized likelihood and estimating equations, to estimate the number of clusters, regression parameters, and transformation functions simultaneously; we also establish the large-sample properties such as consistency and asymptotic normality. Simulations confirm the utility of our proposed approach. We use our proposed method to analyze Chinese housing market data and garner some interesting findings.