Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 307

The previous predictors have been defined in a very general context. However,
the focus of this topic is mainly related to sampling procedures for spatial units. The
predictive approach outlined in this section has not been defined for sampling
spatial units. For this reason, it is important that statisticians extend the predictive
approach to spatially distributed data. In the following discussion, we present some
ideas that could represent a basis for future research.
Consider the spatial error model in the reduced form defined in Eq. (1.52)

:

Þ 1

2 I

Y ¼ X β þ ð I ˁ W ʵ ʵ / N 0, ˃ ð 12 : 18 Þ

For Model ( 12.18 ), the superpopulation in Eq. ( 12.2 ) can be extended to

E ξ Y ðÞ¼ X β

ð : Þ

12 19

Þ 1 Þ 1 B t

Var ξ Y ðÞ¼ V ¼ ð I B ʣ ʵ ð I ;

assuming that

Þ 1

1 ; ˃ 2 ; ...; ˃ n 2 I,

where ʣ ʵ ¼ diag ˃ ð I B exists. If Var ðÞ¼˃

ð Þ h i . We can derive the BLU spatial predictor for
Model ( 12.19 ). This last approach can be applied to all the spatial models that were
described in Sect. 1.4.3.2 . Future research should be devoted to developing these
basic ideas.

Þ 1

2

B t Var ʵ Y ðÞ¼˃ ð I B I

12.3 Spatial Interpolation as a Predictive Approach
for Finite Populations

The model-based approach to sampling can be briefly summarized as estimating the
target variable y for the non-observed units, given the values of the same variable
for units from the selected sample.
Now, assume that we are dealing with spatial units and a variable that has
meaningful values at every spatial unit within a region. Then, given the values of
that variable at a set of sample points, we can use spatial interpolation methods to
predict its value for every un-sampled spatial unit. In fact, spatial interpolation is
the procedure for estimating the value of properties at un-sampled sites within an
area covered by existing observations (Waters 1989 ).
Therefore, the spatial interpolation problem can be briefly formulated as follows.
After selecting a set of spatial units (i.e., points or areas), the aim is to identify the
function which best represents the entire surface, and that provides the best possible
values of the variable of interest in other points or areas for which no observations
are available.
It is evident that this framework is very similar to the one described in Sect. 12.2 .
In this sense, spatial interpolation methods can be viewed as a tool for making
inferences using a model-based approach for sampling spatial units.