Agriculture Reference
In-Depth Information
updated inclusion probabilities are known, the probability function for CPS can be
written as (Grafstr¨m
2010b
)
k
x
k
1
x
k
,
x 2
0
Þ
Y
ð
k
1
Þ
ð
k
1
Þ
N
Pr
I
s
¼
x
ð
k2U
π
1
π
½
;
1
:
ð
7
:
16
Þ
k
The correlation is thus introduced using the weights
w
ðÞ
kt
. Note that different
strategies for choosing the weights were investigated by Grafstr
¨
m(
2010a
).
Those strategies exclusively depended on the order of the units in the list, assigning
as much weight as possible to units in order. In some different sampling situations,
these strategies provided very promising results in terms of a low variance for the
HT ratio estimator. The gain was greater if there was some trend in the
y
k
/
x
k
over the
ordered population. When it is difficult to find a natural ordering of the population
in terms of spatial units, the weights can be considered as functions of the distance
(
d
kt
) between the visited unit
t
and the unit
k
whose first-order inclusion probability
must be updated.
Following this logic, SCPS (Grafstr¨m
2012
) is based on two different strategies
for choosing the weights: maximal weights and Gaussian preliminary weights. The
maximal weights strategy produces samples of fixed size, if the inclusion probabil-
ities sum to an integer. After a decision on the unit
t
, we give as much weight as
possible to the closest of the
k
¼
t
+1
,
,N
units
,
then, we give as much weight as
possible to the second closest unit, and so on. If the distance between the units is
equal, the weight is equally distributed on those units. A unit is not weighted if it is
possible to weight a closer unit. To preserve the fixed first-order inclusion proba-
bilities, the weight that we can give to a unit is limited by
...
!
!
:
ð
t
1
Þ
ð
t
1
Þ
ð
t
1
Þ
ð
t
1
Þ
1
π
;
π
π
1
π
w
ðÞ
k
k
k
1
π
k
min
kt
min
;
ð
7
:
17
Þ
ð
t
1
Þ
ð
t
1
Þ
ð
t
1
Þ
ð
t
1
Þ
1
π
π
π
t
t
t
t
One very interesting property is that the maximal weights strategy locally balances
the sample size, like a form of spatial maximal stratification without fixed and
accurate boundaries. This local property can be better appreciated by showing that,
if the study region is partitioned into two strata,
A
and
B
, so that units within the
same stratum are always closer than units belonging to different strata,
X
l2A
π
l
and
X
l2B
π
l
are asymptotically equal to
n
A
and
n
B
, respectively. Then the maximal
weights method will approximately select
n
A
units from
A
and
n
B
from
B
(Grafstr¨m
2012
, Theorem 1). Therefore, the procedure locally satisfies the
theoretical basis of the
spatial balance
index suggested in Sect.
7.4
.
In the second strategy, weights are controlled by a Gaussian distribution cen-
tered on the position of unit
k.
Note that it performs worse than the maximal weights
method.
It is worth noting that the second-order inclusion probabilities may change when
the order of the units in the list changes, both for maximal weights and Gaussian
Search WWH ::
Custom Search