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does not. COM2 shows a better specification of location than does COM1. When the analysis is
stratified, the scientist can see whether errors of location exist at the stratum level, substratum level,
or grid cell level. For example, Figure 17.9 shows that COM1 has errors at the stratum level and
not the substratum level, while COM2 has errors at the substratum level and not at the stratum
level. In another application from remote sensing, we could examine the influence of a hardening
rule, since the techniques work for both hard and soft classification. For example, COM1 could
show the soft category membership before a hardening rule is applied, and COM2 could show the
hard category membership after the hardening rule is applied. The format of Figure 17.9 would
then summarize the influence of the hardening rule.
In simulation modeling, a scientist commonly builds a model to predict how land changes over
time (Veldkamp and Lambin, 2001). The scientist performs validation to see how the model
performs and to obtain ideas for how to improve the model. When the model is run from T 1 to T 2 ,
the scientist validates the model by comparing the simulated landscape of T 2 with a reference map
T 2 . A null model would predict no change between T 1 and T 2 . In other words, if the scientist had
no simulation model, then the best guess at the T 2 map would be the T 1 map. Therefore, to see
whether the simulation model is performing better than a null model, the scientist needs to compare
(1) the agreement between the T 2 simulation map and the T 2 reference map vs. (2) the agreement
between the T 1 map and the T 2 reference map. In this situation, the format of Figure 17.9 is perfectly
suited to address this question because the analogy is that COM1 is the T 1 map, COM2 is the T 2
simulation map, and REF is the T 2 reference map.
The methods described here are particularly helpful in this case since land-cover and land-use
(LCLU) change models are typically stratified according to political units because data are typically
available by political unit and because the process of land change often happens by political unit.
For example, land-use activities in Brazil are planned at the regional and household scales, where
the household stratification is nested within the regional stratification. Researchers are dedicating
substantial effort to collecting data at a relevant scale in order to calibrate and to improve change
models. Therefore, it is essential that statistical methods budget the components of agreement and
disagreement at relevant scales, because researchers want to collect new data at the scale at which
the most uncertainty exists.
In land-change analysis, the scientist wants to know the manner in which land categories change
and persist over time. For this application, the methods of this chapter would use COM1 as the T 1
map and REF as the T 2 map. Figure 17.7 would supply a multiple-resolution analysis of LCLU
change, where agreement means persistence and disagreement means change. A disagreement in
quantity indicates that a category has experienced either a net gain or a net loss. Disagreement at
the stratum level means that a loss of a category in one stratum is accompanied by gain in that
category in another stratum. Disagreement at the grid cell level means that a loss of a category at
one location is accompanied by a gain of that category at another location within the same stratum.
Therefore, Figure 17.7 would show at what scales LCLU change occurs.
17.4.2
Quantity Information
We focus primarily on the center column of mathematical expressions of Figure 17.3, because
those expressions give the components of agreement. However, the other two columns can be
particularly helpful depending on the purpose of map comparison. In the case of remote sensing,
guidance is needed to improve the classification rules. For simulation modeling, guidance is required
to improve the simulation model's rules. It would be helpful to know the expected improvement
if the rule's specification of quantity changes, given a specific level of information of location. The
mathematical expressions in the rightmost column of Figure 17.3 show the expected results when
the rule specifies the quantity of each category perfectly with respect to the reference quantities.
At the other end of the spectrum, the mathematical equations of the leftmost column show the
expected results when the rule uses random chance to specify the quantities of each category.
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