Geoscience Reference
In-Depth Information
N
d
(
)
Â
1
WS
*
dn
dnj
n
=
S
(17.8)
dj
◊ =
W
d
◊
D
N
d
(
)
Â
Â
WS
*
dn
dnj
d
=
1
n
=
1
S
(17.9)
◊◊ =
j
D
W
d
◊
d
=
1
17.2.3
Minimum Function
The Minimum function gives the agreement between a cell of the reference map and a cell of
the comparison map. Specifically, Equation 17.10 gives the agreement in terms of proportion correct
between the reference map and the comparison map for cell
. Equation 17.11 gives
the landscape-scale agreement weighted appropriately with grid cell weights, where M(
n
of stratum
d
m
) denotes
the proportion correct between the reference map and the comparison map:
J
Â
(
)
agreement in cell
n
of stratum
d
=
MIN R
S
(17.10)
dnj
,
dnj
j
=
1
È
˘
D
N
J
d
Â
Â
Â
Í
Í
˙
˙
W
MIN R
(
,
S
)
dn
dnj
dnj
Î
˚
d
=
1
n
=
1
j
=
1
M
()
m
=
(17.11)
D
N
d
Â
Â
W
dn
d
=
1
n
=
1
The Minimum function expresses agreement between two cells in a generalized way because
it works for both hard and soft classifications. In the case of hard classification, the agreement is
either 0 or 1, which is consistent with the conventional definition of agreement for hard classifica-
tion. In the case of soft classification, the agreement is the sum over all categories of the minimum
membership in each category. The minimum operator makes sense because the agreement for each
category is the smaller of the membership in the reference map and the membership in the
comparison map for the given category. If the two cells are identical, then the agreement is 1.
17.2.4
Agreement Expressions and Information Components
Figure 17.3 gives the 15 mathematical expressions that lay the foundation of our philosophy
of map comparison. The central expression, denoted M(
), is the agreement between the reference
map and the comparison map, given by Equation 17.11. The other 14 mathematical expressions
show the agreement between the reference map and an “other” map that has a specific combination
of information. The first argument in each Minimum function (e.g.,
m
) denotes the cells of the
reference map and the second argument in each Minimum function (e.g.,
R
dnj
) denotes the cells of
the other map. The components of information in the other maps are grouped into two orthogonal
concepts: (1) information of quantity and (2) information of location.
There are three levels of information of quantity no, medium, and perfect, denoted, respectively,
S
dnj
as
n
,
m
, and
p
. For the five mathematical expressions in the “no information of quantity” column,
