Geoscience Reference
In-Depth Information
belongs to a particular category; therefore, the sum of the membership values over all categories
is 1. In addition, each grid cell has a weight to denote its membership in any particular stratum,
where the stratum weight can also range from 0 to 1. The weights do not necessarily need to sum
to 1. For example, if a cell's weights are 0 for all strata, then that cell is eliminated from the
analysis. These ideas are expressed mathematically in Equation 17.1 through Equation 17.4, where
j
is the category index,
J
is the number of categories,
R
is the membership of category
j
in cell
dnj
n
of stratum
d
of the reference map,
S
is the membership of category
j
in cell
n
of stratum
d
of
dnj
the comparison map, and
W
is the weight for the membership of cell
n
in stratum
d
:
dn
0
££
R
dnj
1
(17.1)
0
££
S
dnj
1
(17.2)
J
J
Â
1
Â
R
=
S
=
1
(17.3)
dnj
dnj
j
=
j
=
1
0
££
W
dn
1
(17.4)
Just as each cell has some proportional membership to each category, each stratum has some
proportional membership to each category. We define the membership of each stratum to each
category as the proportion of the stratum that is covered by that category. For each stratum, we
compute this membership to each category as the weighted proportion of the cells that belong to
that category. Similarly, the entire landscape has membership to each particular category, where
the membership is the proportion of the landscape that is covered by that category. We compute
the landscape-level membership by taking the weighted proportion over all grid cells. Equation
17.5 through Equation 17.9 show how to compute these levels of membership for every category
at both the stratum scale and the landscape scale. These equations utilize standard dot notation to
denote summations, where
N
denotes the number of cells that have some positive membership in
d
stratum
d
of the map and
D
denotes the number of strata. Equation 17.5 shows that
W
denotes
d·
the sum of the cell weights for stratum
d
. Equation 17.6 shows that
R
denotes the proportion of
d·j
category
j
in stratum
d
of the reference map. Equation 17.7 shows that
R
denotes the proportion
··j
of category
j
in the entire reference map. Equation 17.8 shows that
S
denotes the proportion of
d·j
category
j
in stratum
d
of the comparison map. Equation 17.9 shows that
S
denotes the proportion
··j
of category
j
in the entire comparison map:
Nd
=
Â
1
WW
d
(17.5)
◊
dn
n
=
N
d
(
)
Â
1
WR
*
dn
dnj
n
=
R
(17.6)
dj
◊ =
W
d
◊
D
N
d
(
)
Â
Â
WR
*
dn
dnj
d
=
1
n
=
1
R
(17.7)
◊◊ =
j
D
W
d
◊
d
=
1
