Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 179

LPM 1 randomly chooses the first unit k, and then the closer unit l (if two or more
units have the same distance from k , the method randomly chooses between them).
If k is the nearest neighbor of l , then the inclusion probabilities are updated as
follows. If

π k + π l < 1, then

8
<

π l
π k þ π l

ð 0, π k þ π l Þ with probability

¼

k l

π ; π ; ð 7 : 18 Þ

π k
π k þ π l

:

ð π k þ π l ,0 Þ with probability

or, if π k + π l 1, then

8
<

1 π l
2 π k π l

ð 1, π k þ π l 1 Þ with probability

¼

k l

π ; π : ð 7 : 19 Þ

1 π k
2 π k π l

:

ð π k þ π l 1, 1 Þ with probability

The expected number of computations for this algorithm is at worst proportional to
N 3 , and at best proportional to N 2 .
LPM 2 is very similar to LPM 1, but the inclusion probabilities are always
updated using Eqs. ( 7.18 ) and ( 7.19 ) without the restriction that the two units should
be mutually nearest neighbors. The expected number of computations needed to
select a sample is proportional to N 2 .
According to some simulated comparisons, LPM and SCPS produce samples
that are much more spatially balanced than the GRTS design. Moreover, LPM
1 appears to be slightly better than LPM 2 for several sample sizes, and for equal or
unequal inclusion probabilities (Grafstr ¨ m et al. 2012 ).
With regard to variance estimation, LPM produces some second-order inclusion
probabilities

π kl that are equal (or very close) to 0, for pairs of units that are close
according to the distance matrix. This is the case for all the algorithms that search
for a spatially balanced sample, for example, SCPS and GRTS. Therefore, it is not
possible to return to a situation in which a design-based variance estimator of the
HT estimator is feasible. However, it is always possible to use other estimators such
as the local neighborhood variance estimator [see Sect. 7.4 , Eq. ( 7.5 )]. This
estimator was originally suggested for GRTS (Stevens and Olsen 2003 ), but also
produces promising results for SCPS and LPM.
The LPM1 and LPM2 functions for selecting LPM samples have been
implemented in R in the BalancedSampling package. The selected samples
are mapped in Fig. 7.7 .

> n < - 100
> N < - 1000
> set.seed(200694)
> p¼rep(n/N,N)
> X < - cbind(framepop$xc,framepop$yc)