Geoscience Reference
In-Depth Information
expressions consider the other maps at the grid cell level, as indicated by the use of all three
subscripts:
) rows, the expressions consider the other maps at the
stratum level, as indicated by the use of two subscripts:
d
,
n
, and
j
. In the H(
x
) and K(
x
) rows, the
expressions consider the other maps at the study area level, as indicated by the use of one subscript:
d
and
j
. In the N(
x
) and P(
x
.
The concepts behind these combinations of components of information of location are as
follows. In row N(
j
), the categories of the other maps are spread evenly across the landscape, such
that every grid cell has an identical multinomial distribution of categories. In row H(
x
), the
categories of the other maps are spread evenly within each stratum, such that every grid cell in
each stratum has an identical multinomial distribution of categories. In row M(
x
), the grid cell level
information of location in the other maps is the same as in the comparison map. In row K(
x
), the
other maps derive from the comparison map, whereby the locations of the categories in the
comparison map are swapped within each stratum in order to match as best as possible the reference
map; however, this swapping of grid cell locations does not occur across stratum boundaries. In
row P(
x
), the other maps derive from the comparison map, whereby the locations of the categories
in the comparison map are swapped in order to match as best as possible the reference map, and
this swapping of grid cell locations can occur across stratum boundaries.
Each of the 15 mathematical expressions of Figure 17.3 is denoted by its location in the table.
The
x
denotes the level of information of quantity. For example, the overall agreement between
the reference map and the comparison map is denoted M(
x
), since the comparison map has a
medium level of information of quantity and a medium level of information of location, by
definition. The expression P(
m
p
) is in the upper right of Figure 17.3 and is always equal to 1, because
P(
) is the agreement between the reference map and the other map that has perfect information
of quantity and perfect information of location.
There are seven mathematical expressions that are especially interesting and helpful. They are
p
N(
n
), N(
m
), H(
m
), M(
m
), K(
m
), P(
m
), and P(
p
). For N(
n
), each cell of the other map is the same
and has a membership in each category equal to 1/
), each cell of the other map is the
same and has a membership in each category equal to the proportion of that category in the
comparison map. For H(
J
. For N(
m
), each cell within each stratum of the other map is the same and has a
membership in each category equal to the proportion of that category in each stratum of the
comparison map. For M(
m
), the other map is the
comparison map with the locations of the grid cells swapped within each stratum, so as to have
the maximum possible agreement with the reference map within each stratum. For P(
m
), the other map is the comparison map. For K(
m
), the other
map is the comparison map with the locations of the grid cells swapped anywhere within the map,
so as to have the maximum possible agreement with the reference map. For P(
m
p
), the other map
is the reference map, and therefore the agreement is perfect.
17.2.5
Agreement and Disagreement
) constitute
a sequence of measures of agreement between the reference map and other maps that have
increasingly accurate information. Therefore, usually 0 < N(
The seven mathematical expressions N(
n
), N(
m
), H(
m
), M(
m
), K(
m
), P(
m
), and P(
p
n
) < N(
m
) < H(
m
) < M(
m
) < K(
m
)
< P(
) = 1. This sequence partitions the interval [0,1] into components of the agreement
between the reference map and the comparison map. M(
m
) < P(
p
m
) is the total proportion correct, and 1
- M(
) is the total proportion error between the reference map and the comparison map. Hence,
the sequence of N(
m
m
) defines components of agreement, and the sequence
of M(
m
), K(
m
), P(
m
), and P(
p
) defines components of disagreement.
Table 17.2 defines these components mathematically. Beginning at the bottom of the table and
working up, the first component is agreement due to chance, which is usually N(
n
). However, if
the agreement between the reference map and the comparison map is less than would be expected
by chance, then the component of agreement due to chance may be less than N(
n
). Therefore, Table
17.2 defines the component of agreement due to chance as the minimum of N(
n
), N(
m
), H(
m
),
n
), N(
m
), H(
m
), and M(
