Agriculture Reference
In-Depth Information
+n¼ length(samp)
+ index
¼
1:N
+ incl
¼
rep(0,n)
+ for(i in 1:N) {
+ near
<
- which(diss[i,samp]
¼¼
min(diss[i,samp]))
+ incl[near]
¼
incl[near] + pi[i]
+}
+ var(incl)
+}
>
ds
<
- as.matrix(dist(framepop[,2:3]))
>
pik
<
- rep(n/N,N)
>
spbalance(ds,pik,framesrs$id)
[1] 0.3290909
>
framepop
<
- cbind(framepop,strataid2
¼
floor(framepop$xc*10)*10+
+ floor(framepop$yc*10))
>
kk
<
- (1/table(framepop$strataid2))[as.numeric(framepop$
+ $strataid2)+1]
>
spbalance(ds,kk,str$id)
[1] 0.1111589
>
spbalance(ds,pik,(1:1000)[srs_grts¼¼1])
[1] 0.1430303
Note that the spatial balance index can be also calculated with the command sb of
the package BalancedSampling. The reference manual can be downloaded from
The results are, of course, consistent with the variance of the area of the poly-
gons, because our population is not concentrated in some zones, but is randomly
distributed across the entire study region.
The GRTS design produces a sample with specified
π
k
s, so the HT estimator
[or its continuous population analog (Cordy
1993
; Stevens
1997
)] can be applied to
estimate the population characteristics.
Stevens (
1997
) provided exact expressions for the
π
kl
s in a particular case of
GRTS. However, these expressions unfortunately prevent the proper use of vari-
ance estimators based on the HT or Yates-Grundy-Sen estimators (see Eqs.
1.27
and
1.30
) because they tend to be unstable if some
π
kl
s are very close to zero
(Stevens and Olsen
2004
).
Stevens and Olsen (
2003
) proposed a contrast-based variance estimator for the
GRTS design that has some similarities to the smoothed estimator (Overton and
Stehman
1993
). For an RTS design, Overton and Stehman (
1993
) considered a
smoothed
contrast-based estimator of the form
V
SMO
¼
X
k
wd
k
y
k
y
k
2
, where
wd
k
s are a set of weights and
y
k
(i.e., the smoothed value for
y
k
) is assumed to be a
weighted mean of the values observed at a point and its nearest neighbors in the
tessellation. In Stevens and Olsen (
2003
), the single
contrast y
k
y
k
2
was
replaced with an average of several
contrasts
over a local neighborhood, which is
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