Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 175

populations that are spread over two or more dimensions, which do not have a
natural ordering. Intuitively the only possibility is to use a similar strategy to GRTS,
avoiding this problem by mapping the two-dimensional locations onto one dimen-
sion to preserve some spatial structure as described in Sect. 7.4 . Nothing else seems
possible in this direction. In other words, it is difficult to modify the second-order
inclusion probabilities while preserving fixed

ˀ k s.
Bondesson and Grafstr ¨ m( 2011 ) tried to address this issue by defining a
procedure that appears to be only in one dimension, but that actually explores
two dimensions of spatial units. They extended Sampford ' s method (1967) (which
is widely used in π

ps plans) to select unequal probability samples over small strata
when the inclusion probabilities can also have non-integer sums within strata. The
sample results are spread over populations that are spatially stratified into small
strata with inclusion probability sums greater than or equal to 1. On the basis of this
result, Grafstr¨m( 2012 ) proposed a method called spatially correlated Poisson
sampling (SCPS). This is a modification of the correlated Poisson sampling
(CPS) method introduced by Bondesson and Thorburn ( 2008 ) to select units with
unequal inclusion probabilities.
CPS is based on a list sequential criterion of the random decisions that deter-
mines whether each unit k should be included in the sample, according to proba-
bilities adapted at each step to create correlations between the indicator variable of
the unit visited ( I k ) and the indicator variables relative to all the other units of the
population ( I l , where l6 ¼ k) . It is clearly more suitable to have negative correlation
between the indicator variables of the units that are closer to those visited and
selected. In fact, in the presence of negative correlation, units that are close in
distance will rarely simultaneously appear in the sample. The procedure is so
general that any design without replacement with fixed first-order inclusion prob-
abilities can be implemented by CPS, and it only depends only on the rule for
updating the selection probabilities after each unit is visited.
The sequential nature of the list means that we first decide the sampling outcome
for the first unit of a (possibly previously randomly sorted) list, then for the second
unit, and so on until n units have been selected. If unit 1 is included with probability
π

ð 1 ¼ π 1 , we set I 1 ¼1, otherwise I 1 ¼0. After each step, the inclusion probabil-
ities for the remaining units in the list are updated according to a specific rule. We
start with

ðÞ
k

π ¼ π k , for k 1. At step t , the values of I 1 , I 2 , ... , I t -1 are known, and

ð t 1 Þ

we select unit t with probability π . We update the generic unit k t+ 1

t

w ðÞ

ðÞ
k

ð t 1 Þ ð t 1 Þ

kt , where w ðÞ

according to π ¼ π I t π

kt are weights that depend

t

k

on I 1 , I 2 , , I t -1 but not on I t (Bondesson and Thorburn 2008 ).

The inclusion probability vector is gradually updated in, at most, N steps, until it
becomes the vector of inclusion indicators. The set of weights w ðÞ

...

kt depends only on
the random selection outcome of the previous units, and do not have to be positive.
If they are positive, they typically induce a negative correlation between the
indicator variables. If they are negative, the correlation will be positive (Grafstr ¨ m
2012 ). Considering that the implementation is a list sequential algorithm and the