Geoscience Reference
In-Depth Information
In addition to evaluating options based on simplicity, we would like to compare precision of
the different options. Unfortunately, such an evaluation would be difficult, requiring either com-
plicated theoretical analysis or extensive simulation studies based on acquiring reasonably good
approximations to spatial patterns of classification error. A key point of this discussion of design
alternatives for two-stage cluster sampling is that while the problem can be simply stated and the
objectives for what needs to be achieved are clear, determining an optimal solution is elusive.
Simple changes in sampling protocol may lead to complications in the analysis, whereas maintaining
a simple analysis may require a complex sampling protocol.
2.2.4
Stratification and Local Spatial Control
Clustering to achieve local spatial control also conflicts with the effort to stratify by cover types.
Several design alternatives may be considered to remedy this problem. An easily implemented
approach is the following. A stratified random sample of pixels is obtained using the mapped LC
classes as strata. To incorporate local spatial control and increase the sample size, the eight pixels
touching each sampled pixel are also included in the sample. That is, a cluster consisting of a 3
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3 block of pixels is created, but the selection protocol is based on the center pixel of the cluster.
Two potential drawbacks exist for this protocol. First, the sample size control feature of stratified
random sampling is diminished because the eight pixels surrounding an originally selected sample
pixel could be any LC type, not necessarily the same type as the center pixel of the block. Sample
size planning becomes trickier because we do not know which LC classes will be represented by
the surrounding eight pixels or how many pixels will be obtained for each LC class present. This
will not be a problem if we have abundant resources because we could specify the desired minimum
sample size for each LC class based on the identity of the center pixels. However, having an
overabundance of accuracy assessment resources is unlikely, so the loss of control over sample
allocation is a legitimate concern.
Second, and more importantly, this protocol creates a complex inclusion probability structure
because a pixel may be selected into the sample via two conditions: it is an originally selected
center pixel of the 3
3 cluster or it is one of the eight pixels surrounding the initially sampled
center pixel. To use the data within a rigorous probability-sampling framework, the inclusion
probability determined for each pixel must account for this joint possibility of selection. We require
the probability of being selected as a center pixel, the probability of being selected as an accom-
panying pixel in the 3
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3 block, and the probability of being selected by both avenues in the same
sample (i.e., the intersection event). The first probability is readily available because it is the
inclusion probability of a stratified random sample, n
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are the sample and
population numbers of pixels for stratum h. The other two probabilities are much more complicated.
The probability of a pixel's being selected because it is adjacent to a pixel selected in the initial
sample depends on the map LC labels of the eight pixels surrounding the pixel in question, and
this probability differs among different LC types. Although it is conceptually possible to enumerate
the necessary information to obtain these probabilities, it is practically difficult. Finding the inter-
section probability would be equally complex. Rather than derive the actual inclusion probabilities,
we could use the stratified random sampling inclusion probabilities as an easily implemented, but
crude, approximation. This would violate the principle of consistent estimation and raise the
question of how well such an approximation worked.
A second general alternative is to change the way the stratification is implemented. The problem
arises because the strata are defined at the pixel level while the selection procedure is applied to
the cluster level. Stratifying at the cluster level, for example a 3
/N
, where n
and N
h
h
h
h
3 block of pixels, resolves this
problem but creates another. The nonhomogeneous character of the clusters creates a challenge
when deciding to which stratum a block should be assigned if it consists of two or more cover
types. Rules to determine the assignment must be specified. For example, assigning the block to
the most common class found in the 3
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3 block is one possibility, with a tie-breaking provision
