Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 172

10 -8.361090e-13 5.833945e+00 -6.977493e+14

11 1.311451e-12 5.607455e+00 4.275764e+14

12 -3.384126e-12 -1.356765e+00 4.009203e+13

13 6.564749e-13 1.673957e+00 2.549918e+14

14 -1.530413e-24 -4.172359e-24 1.726296e+02

15 -6.507250e-25 -1.850151e-25 -7.156786e+01

7.6 Selection Methods Based on the Distance Between
Statistical Units

The motivations described in the last part of Sect. 7.2 can be considered reasonable
if we assume that increasing the distance between two units k and l increases the
difference between the values of the survey variable, y k yj j . In this situation, it is
clear [see Eq. ( 1.29 )] that the variance of the HT estimator will necessarily decrease
if we set high joint inclusion probabilities to pairs that have very different y values,
because they are far each other.
Arbia ( 1993 ) was inspired by purely model-based assumptions on the depen-
dence of the stochastic process that generates the data. He suggested, according to
the algorithm types identified by Till´ ( 2006 ), a draw-by-draw scheme called the
dependent areal units sequential technique (DUST). The properties of DUST can be
also analyzed within a design-based framework, because it respects the randomi-
zation principle.
The main argument for this method was that
“ it is intuitively clear that, when we have a clue of the spatial correlation structure

underlying the spatial phenomenon to be sampled, it is desirable to exploit this information
in the sampling design. In this way we could avoid duplicate information partly contained
in areas already sampled and we can economize sampling costs without loosing reliability
of the estimates” (Arbia 1993 ).

...

It is evident that this kind of situation can be easily solved by carefully using a
systematic sampling design. However, it is also clear that this selection criterion
needs an ordering and, when used in two dimensions, there is rarely a unique
ordering of spatial units. So, systematic sampling depends on some subjective or
approximate choice. For example, a typical way of solving this problem is to
superimpose a grid of points on the irregular areas (unordered), and sample those
areas (which contain a sample point of the grid) using the usual techniques.
However, the nature of the systematic sample is totally lost when used in this
way, and the technique does not safeguard against homogeneity of the y or clus-
tering. Besides, it creates several drawbacks such as making it difficult to control
the selection probabilities.
The DUST algorithm starts by randomly selecting a unit k . Then, at every step

t n , the algorithm updates the selection probabilities of any other unit ( l ) of the

population according to the rule

<

, where

ðÞ
l

ð t 1 Þ

d kl

is a tuning param-
eter used to control the distribution of the sample over the study region, and d kl is a

π ¼ π 1 ʻ ʻ

l