Agriculture Reference
In-Depth Information
t
¼
X
s
y
k
þ
X
s
y
k
¼
t
y
s
þ t
y
s
:
ð
1
:
33
Þ
In other words, the population total is the sum of the sample total
t
y
s
and the
corresponding non-sample total
t
y
s
. Obviously, after the sample has been drawn, the
sample total
t
y
s
is known, and the estimation problem is reduced to predicting
t
y
s
given
t
y
s
. Given the superpopulation model
, the aim is to choose the
best
predictor
t
y
s
of
t
y
s
, and a sample
s
, so that we minimize the sample error,
t t
¼
t
y
s
t
y
s
.
Finally, it is possible to define a model-based prediction interval for
t
. Suppose
that an estimator
MSE
ʾ
ðÞ
can be derived from a sample
s
. A prediction interval for
t
at the level (1-
ʾ
ʱ
), given
s
,is
1
=
2
Þ
MSE
ʾ
ðÞ
t z
1
ʱ=
2
;
ð
1
:
34
Þ
ð
where
t
is the model unbiased predictor of
t,
and
z
is a standard Normal random
variable.
In this section, we have only introduced model-based survey sampling. For more
details see Royall and Herson (
1973a
,
b
), Valliant et al. (
2000
), and Chambers and
Clark (
2012
). Additionally, also Chap.
12
of this topic discusses this approach.
1.4 Statistics for Spatial Data
The main aim of this topic is to describe the most important spatial sampling
approaches for agricultural resources. To properly describe these methods, we
include a brief overview of the principal statistical model used for spatial data.
The narrative is mainly based on Cressie (
1993
), and Schabenberger and Gotway
(
2005
). The reader can refer to these texts for further details.
Generally speaking, the term
spatial
means that each item of data has a geo-
graphical reference; we know where each case occurs on a map. If the locations of
the observations as labels, the resulting data are called spatial data. In spatial data
analysis, the set of spatial locations are taken into account (see Chap.
3
for some
examples in the GIS framework). Spatial statistics is a field of spatial data analysis
in which the observations are modeled using random variables. In this section, we
first outline the main types of spatial data and describe the concept of spatial
dependence. We then summarize the foremost statistical model for spatial data.
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