  A new model-based procedure is developed for sparse clustering of functional
data that aims to classify a sample of curves into homogeneous groups while
jointly detecting the most informative portions of domain. The proposed method
is referred to as sparse and smooth functional clustering (SaS-Funclust) and
relies on a general functional Gaussian mixture model whose parameters are
estimated by maximizing a log-likelihood function penalized with a functional
adaptive pairwise penalty and a roughness penalty. The former allows
identifying the noninformative portion of domain by shrinking the means of
separated clusters to some common values, whereas the latter improves the
interpretability by imposing some degree of smoothing to the estimated cluster
means. The model is estimated via an expectation-conditional maximization
algorithm paired with a cross-validation procedure. Through a Monte Carlo
simulation study, the SaS-Funclust method is shown to outperform other methods
already appeared in the literature, both in terms of clustering performance and
interpretability. Finally, three real-data examples are presented to
demonstrate the favourable performance of the proposed method. The SaS-Funclust
method is implemented in the $\textsf{R}$ package $\textsf{sasfunclust}$,
available online at https://github.com/unina-sfere/sasfunclust.
