  In this article, we propose a penalized clustering method for large scale
data with multiple covariates through a functional data approach. In the
proposed method, responses and covariates are linked together through
nonparametric multivariate functions (fixed effects), which have great
flexibility in modeling a variety of function features, such as jump points,
branching, and periodicity. Functional ANOVA is employed to further decompose
multivariate functions in a reproducing kernel Hilbert space and provide
associated notions of main effect and interaction. Parsimonious random effects
are used to capture various correlation structures. The mixed-effect models are
nested under a general mixture model, in which the heterogeneity of functional
data is characterized. We propose a penalized Henderson's likelihood approach
for model-fitting and design a rejection-controlled EM algorithm for the
estimation. Our method selects smoothing parameters through generalized
cross-validation. Furthermore, the Bayesian confidence intervals are used to
measure the clustering uncertainty. Simulation studies and real-data examples
are presented to investigate the empirical performance of the proposed method.
Open-source code is available in the R package MFDA.
