  As high-dimensional and high-frequency data are being collected on a large
scale, the development of new statistical models is being pushed forward.
Functional data analysis provides the required statistical methods to deal with
large-scale and complex data by assuming that data are continuous functions,
e.g., a realization of a continuous process (curves) or continuous random
fields (surfaces), and that each curve or surface is considered as a single
observation. Here, we provide an overview of functional data analysis when data
are complex and spatially correlated. We provide definitions and estimators of
the first and second moments of the corresponding functional random variable.
We present two main approaches: The first assumes that data are realizations of
a functional random field, i.e., each observation is a curve with a spatial
component. We call them 'spatial functional data'. The second approach assumes
that data are continuous deterministic fields observed over time. In this case,
one observation is a surface or manifold, and we call them 'surface time
series'. For the two approaches, we describe software available for the
statistical analysis. We also present a data illustration, using a
high-resolution wind speed simulated dataset, as an example of the two
approaches. The functional data approach offers a new paradigm of data
analysis, where the continuous processes or random fields are considered as a
single entity. We consider this approach to be very valuable in the context of
big data.
