  Functional time series analysis, whether based on time of frequency domain
methodology, has traditionally been carried out under the assumption of
complete observation of the constituent series of curves, assumed stationary.
Nevertheless, as is often the case with independent functional data, it may
well happen that the data available to the analyst are not the actual sequence
of curves, but relatively few and noisy measurements per curve, potentially at
different locations in each curve's domain. Under this sparse sampling regime,
neither the established estimators of the time series' dynamics, nor their
corresponding theoretical analysis will apply. The subject of this paper is to
tackle the problem of estimating the dynamics and of recovering the latent
process of smooth curves in the sparse regime. Assuming smoothness of the
latent curves, we construct a consistent nonparametric estimator of the series'
spectral density operator and use it develop a frequency-domain recovery
approach, that predicts the latent curve at a given time by borrowing strength
from the (estimated) dynamic correlations in the series across time. Further to
predicting the latent curves from their noisy point samples, the method fills
in gaps in the sequence (curves nowhere sampled), denoises the data, and serves
as a basis for forecasting. Means of providing corresponding confidence bands
are also investigated. A simulation study interestingly suggests that sparse
observation for a longer time period, may be provide better performance than
dense observation for a shorter period, in the presence of smoothness. The
methodology is further illustrated by application to an environmental data set
on fair-weather atmospheric electricity, which naturally leads to a sparse
functional time-series.
