Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 189

Fig. 7.9 Spatial distribution of EMAP lakes population ( left ) partitioned into four strata, and the
semivariogram ( right ) of the acid neutralizing capacity (ANC) variable
non-homogeneity in the spatial structures of the phenomenon. In these circum-
stances, it is typical to assume that the non-homogeneity can be approximated by a
set of local homogeneous zones, and solved by partitioning the study region.
To this purpose, we used a very simple K -means clustering algorithm (Everitt
et al. 2011 ) on the two coordinates x 1 and x 2 . This splits the population into four
strata (see Fig. 7.10 ), assuming that each of them will have a semivariogram that is
much more easily interpreted. The resulting semivariograms were effectively
monotonically increasing in each stratum (see Fig. 7.10 ). This can be especially
seen in Stratum 3, where the weight is significant for both the population size N and
the variance of the target variable y (i.e., ANC). Note that in the following exercise,
the allocation of sampling units was fixed to approximately the population size of
each stratum.
The main conclusion we can reach from the results in Table 7.5 is that GRTS
finds the partition and uses it to better select the units, leading to a sensible gain in
efficiency when compared with an SRS design. When n ¼100, it is not convenient
to use GRTS on a predefined stratification when compared with conventional
stratified SRS, with the sample size allocated to each stratum proportional to its
population size. This is because there are no more groupings of the units to find and
exploit.
Conversely, the CUBE method does not provide any interesting results when the
data are not stratified. However, its performance increases if a partition is provided
such that in each group y has some relationship with the coordinates, as confirmed
by the semivariograms in each stratum. In particular, this occurs when the selection
is constrained to respect the second-order moments of each coordinate, and the
dispersion over space, which seems to be a peculiar feature of these kinds of data.
The two different DUST methods did not obtain good results when applied to the
non-stratified population, but they did when spatial stratification was introduced.