  Spatially distributed functional data are prevalent in many statistical
applications such as meteorology, energy forecasting, census data, disease
mapping, and neurological studies. Given their complex and high-dimensional
nature, functional data often require dimension reduction methods to extract
meaningful information. Inverse regression is one such approach that has become
very popular in the past two decades. We study the inverse regression in the
framework of functional data observed at irregularly positioned spatial sites.
The functional predictor is the sum of a spatially dependent functional effect
and a spatially independent functional nugget effect, while the relation
between the scalar response and the functional predictor is modeled using the
inverse regression framework. For estimation, we consider local linear
smoothing with a general weighting scheme, which includes as special cases the
schemes under which equal weights are assigned to each observation or to each
subject. This framework enables us to present the asymptotic results for
different types of sampling plans over time such as non-dense, dense, and
ultra-dense. We discuss the domain-expanding infill (DEI) framework for spatial
asymptotics, which is a mix of the traditional expanding domain and infill
frameworks. The DEI framework overcomes the limitations of traditional spatial
asymptotics in the existing literature. Under this unified framework, we
develop asymptotic theory and identify conditions that are necessary for the
estimated eigen-directions to achieve optimal rates of convergence. Our
asymptotic results include pointwise and $L_2$ convergence rates. Simulation
studies using synthetic data and an application to a real-world dataset confirm
the effectiveness of our methods.
