Agriculture Reference
In-Depth Information
3. Another possible choice that is widely used in agricultural surveys is that
U
is a
list of points, typically the corners of a regular grid overlaid on the survey
s
geographical domain. This represents a non-exhaustive population of the study
area, and is only the first-stage of sampling. In this case, X can be only
represented by a geographical nomenclature and by a design matrix of land
use classification codes obtained using previous land use maps, or by a classi-
fication of remotely sensed data where the C are simply the coordinates of each
point.
'
To efficiently use positional data in sample design, we need methods that cannot
be adapted from classical auxiliary variables techniques. This is, in part, a conse-
quence of the multivariate nature of the data, and the traditional design solutions
can only handle one size measure at a time (Bee et al.
2010
), e.g.
ps
(see Sect.
6.4
).
Additionally, to use some covariates, we always assume that there is at least an
approximate certain degree of dependence between a survey variable y and the set
X, even if the details are not specified. With regards to the set C, the distance matrix
contains the spatial information that emphasizes the importance of the sample
spread over the study region (which can be related to this dependence), and some
form of homogeneity between adjacent units.
In design-based sampling theory, the potential observations over each unit of the
population cannot be considered dependent, if we assume that there are no mea-
surement errors. However, an inherent and fully recognized feature of spatial data is
that they are dependent, as expressed in Tobler
π
s First Law of Geography (see Sect.
'
It is then clear that sampling schemes for spatial units can be constructed by
introducing a suitable model of spatial dependence within a model-based or model-
assisted framework. This approach suggests a rationale for the intuitive procedure
of spreading the selected units over space (Benedetti and Palma
1995
; Dunn and
Harrison
1993
; Rogerson and Delmelle
2004
; Salehi
2004
). Close observations will
provide overlapping information as an immediate consequence of the dependence.
Under such an assumption, we must find the sample configuration that is the best
representation of the whole population. This leads us to define our selection as a
combinatorial optimization problem. In fact, provided that the sample size is fixed,
our aim is to minimize an objective function defined over the whole set of possible
samples, which represents a measure of the loss of information due to dependence.
An optimal sample
selected with certainty
is, of course, undesirable if we
assume the randomization hypothesis, which is the background for design-based
inference. Thus, we should move from the concept of dependence to that of spatial
homogeneity measured in terms of the local variance of the variable of interest.
Another interesting approach, based on a particular algorithm of random num-
bers generation, is the BAS method (Robertson et al.
2013
) that is very easy to
implement, but difficult to adapt to a spatial finite population sampling framework.
The layout of this chapter is as follows. A motivation for developing specific
designs for spatial units that take advantage of their particular nature is contained in
Sect.
7.2
. Then, we briefly summarize the main features of some selection criteria
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