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can make to improve the agreement, given the quantity of the comparison map. Therefore, the
disagreement of location is 2/16. The distinction between information of quantity and information
of location is the foundation of this chapter's philosophy of map comparison.
It is worthwhile to consider in greater detail this concept of separation of information of quantity
vs. information of location in map comparison before introducing the technical methodology of
the analysis. The remainder of this introduction uses the puzzle example of Figure 17.1 to illustrate
the concepts that the Methods section then formalizes in mathematical detail.
The following analogy is helpful to grasp the fundamental concept. Imagine that the reference
map of Figure 17.1 is an original masterpiece that has been painted with two colors: light and dark.
A forger would like to forge the masterpiece, but the only information that she knows for certain
is that the masterpiece has exactly two colors: light and dark. Armed with partial information about
the masterpiece (reference map), the forger must create a forgery (comparison map).
To create the forgery, the forger must answer two basic questions: What proportion of each
color of paint should be used? Where should each color of paint be placed? The first question
requires information of quantity and the second question requires information of location.
If the forger were to have perfect information about the quantity of each color of paint in the
masterpiece, then she would use 4/16 light paint and 12/16 dark paint for the forgery, so that the
proportion of each color in the forgery would match the proportion of each color in the masterpiece.
The quantity of each color in the forgery must match the quantity of each color in the masterpiece
in order to allow the potential agreement between the forgery and the masterpiece to be perfect.
At the other extreme, if the forger were to have no information on the quantity of each color in
the masterpiece, then she would select half light paint and half dark paint, since she would have
no basis on which to treat either category differently from the other category. In the most likely
case, the forger has a medium level of information, which is a level of information somewhere
between no information and perfect information. Perhaps the forger would apply 6/16 light paint
and 10/16 dark paint to the forgery, as in Figure 17.1.
Now, let us turn our attention to information of location. If the forger were to have perfect
information about the location of each type of paint in the masterpiece, then she would place the
paint of the forgery in the correct location as best as possible, such that the only disagreement
between the forgery and the masterpiece would derive from error (if any) in the quantity of paint.
If the forger were to have no information about the location of each color of paint in the masterpiece,
then the she would spread each color of paint evenly across the canvas, such that each grid cell
would be covered smoothly with light paint and dark paint. In the most likely case, the forger has
a medium level of information of location about the masterpiece, so perhaps the forgery would
have a pair of grid cells that are incorrect in terms of location, as in Figure 17.1. However, in the
case of Figure 17.1, the error of location is not severe, since the error could be corrected by a swap
of neighboring grid cells.
After the forger completes the forgery, we compare the forgery directly to the masterpiece in
order to find the types and magnitudes of agreement between the two. There are two basic types
of comparison, one based on information of quantity and another based on information of location.
Each of the two types of comparisons leads to a different follow-up question.
First, we could ask, Given its medium level of information of quantity, how would the forgery
appear if the forger would have had perfect information on location during the production of the
forgery? For the example, in Figure 17.1, the answer is that the forger would have adjusted the
forgery by swapping the location of cell #9 with cell #3, #10, or #13. As a result, the agreement
between the adjusted forgery and the masterpiece would be 14/16, because perfect information on
location would imply that the only error would be an error of quantity, which is 2/16.
Second, we could ask, Given its medium level of information of location, how would the forgery
appear if the forger would have had perfect information of quantity during the production of the
forgery? In this case, the answer is that the forger would have adjusted the forgery by using more
dark paint and less light paint, but each type of paint would be in the same location as in Figure
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