  In this paper, we propose a novel model to analyze serially correlated
two-dimensional functional data observed sparsely and irregularly on a domain
which may not be a rectangle. Our approach employs a mixed effects model that
specifies the principal component functions as bivariate splines on
triangulations and the principal component scores as random effects which
follow an auto-regressive model. We apply the thin-plate penalty for
regularizing the bivariate function estimation and develop an effective EM
algorithm along with Kalman filter and smoother for calculating the penalized
likelihood estimates of the parameters. Our approach was applied on simulated
datasets and on Texas monthly average temperature data from January year 1915
to December year 2014.
