Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 159

0.05994 1.63600 3.80900 4.67600 6.52100 22.50000
> summary(vorstr$area[vorstr$area > 0])*1000
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.4777 3.0220 4.3910 4.9380 6.3360 13.4900
> summary(vorgrts$area[vorgrts$area > 0])*1000
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.5521 2.7260 4.4780 4.9500 6.5000 12.9000
> var(vorsrs$area[vorsrs$area > 0])*1000
[1] 0.0154809
> var(vorstr$area[vorstr$area > 0])*1000
[1] 0.006867594
> var(vorgrts$area[vorgrts$area > 0])*1000
[1] 0.007740887

The maximal stratification method clearly performs better, and in this context it
can be considered a randomly aligned systematic sample, because the coordinates
of the population are well-spread generated in the quadrat [0, 1] 2 . A completely
different result might be obtained when dealing with finite spatially clustered
populations (as those in Sect. 7.8.1 ). In such populations a high spatial stratification
necessarily implies that N h ¼0 for many strata.
It is also clear that there is an appreciable gain from the spatial distribution of
GRTS when compared with SRS. In this example, the population is randomly
distributed over the plane. The index shows interesting results, but it strictly
depends on the spatial distribution of the population to be sampled.
As previously mentioned, the Voronoi polygon for unit k of a generic sample
s includes all the population units closer to k than to any other unit in the sample.

Let v k ¼ X i2VP ðÞ π i be the sum of the inclusion probabilities of the units in the k -th
Voronoi polygon VP( k ). Then, for any sample unit, we will have an expected value
E( v k ) ¼ 1. Additionally, all the v k s should be close to 1 for a spatially balanced
sample (Stevens and Olsen 2004 ). Thus, the index V( v k ) (the variance of the v k s)
can be used as a measure of the spatial balance of a sample. Obviously, a lower
value of V( v k ) implies a good spatially balanced sample.
Note that the distance between a pair is a basic concept in all these features, but
the spatial balance index appears to be directly related to the possibility that there is
a spatial stratification in the observed phenomenon. Besides, it is worth noting that
there are practical difficulties when directly design samples using the spatial
balance index for each sample s [ SB s ¼V s ( v k )], because it involves the

π k s. The
Voronoi polygons are based on the distance matrix, so a good solution could be to
define selection rules using the distance between sampled units (see Sect. 7.6 ).
It is not difficult to evaluate the spatial balance index in R with the following
function.

> spbalance < - function(diss,pi,samp) {
+N¼ length(pi)