  Areal level spatial data are often large, sparse and may appear with
geographical shapes that are regular or irregular (e.g., postcode). Moreover,
sometimes it is important to obtain predictive inference in regular or
irregular areal shapes that is misaligned with the observed spatial areal
geographical boundary. For example, in a survey the respondents were asked
about their postcode, however for policy making purposes, researchers are often
interested to obtain information at the SA2. The statistical challenge is to
obtain spatial prediction at the SA2s, where the SA2s may have overlapped
geographical boundaries with postcodes. The study is motivated by a practical
survey data obtained from the Australian National University (ANU) Poll. Here
the main research question is to understand respondents' satisfaction level
with the way Australia is heading. The data are observed at 1,944 postcodes
among the 2,516 available postcodes across Australia, and prediction is
obtained at the 2,196 SA2s. The proposed method also explored through a
grid-based simulation study, where data have been observed in a regular grid
and spatial prediction has been done in a regular grid that has a misaligned
geographical boundary with the first regular grid-set. The real-life example
with ANU Poll data addresses the situation of irregular geographical boundaries
that are misaligned, i.e., model fitted with postcode data and hence obtained
prediction at the SA2. A comparison study is also performed to validate the
proposed method. In this paper, a Gaussian model is constructed under Bayesian
hierarchy. The novelty lies in the development of the basis function that can
address spatial sparsity and localised spatial structure. It can also address
the large-dimensional spatial data modelling problem by constructing knot based
reduced-dimensional basis functions.
