Geoscience Reference
In-Depth Information
is the number of arguments. We constructed four fuzzy rules for each cover type for four
different combinations of two arguments including (1) accuracy level (i.e., low and high) and (2)
majority (i.e., dominant or subordinate). Both of the arguments were available spatially; the first
was obtained from the accuracy maps constructed in previous steps and the second was derived
directly from the LC thematic map. The four fuzzy rules for cover type
where
k
x
are stated as follows:
•
Rule 1: if
x
is “dominant”
and the accuracy is “high,” then the degree of membership of
x
to
become
y
is:
Â
x
An Mx y
◊ ◊
(
Æ
)
i
xi
,
i
i
m
1
(
xy
Æ=
)
(13.7)
Â
x
An
◊
i
xi
,
i
•
Rule 2: if
x
is “subordinate
”
and the accuracy is “high,” then:
Â
x
A
◊ - ◊
(
9
n
)
Mx
(
Æ
y
)
i
xi
,
i
i
m
2
(
xy
Æ=
)
(13.8)
Â
x
A
◊ -
(
9
n
)
i
xi
,
i
•
Rule 3: if
x
is “dominant”
and the accuracy is “low,” then:
Â
x
(
1
- ◊ ◊
AnMx y
)
(
Æ
)
i
xi
,
i
i
m
3
(
xy
Æ=
)
(13.9)
Â
x
(
1
- ◊
An
)
i
xi
,
i
•
Rule 4: if
x
is “subordinate
”
and the accuracy is “low,” then:
Â
x
(
1
- ◊ - ◊
A
) (
9
n
)
Mx
(
Æ
y
)
i
xi
,
i
i
m
4
(
xy
Æ=
)
(13.10)
Â
x
(
1
- ◊ -
A
) (
9
n
)
i
xi
,
i
where is accuracy level for land-cover type
x
at point
i
with its values ranging from 0 to 1
and
n
x,i
is the number of pixels labeled
x
in the 3 ¥ 3 window surrounding the mapped pixel
i
. We
assigned values of based on the multilevel agreement for cover type
x
at that point.
is
equal to 1 if the agreement level is I and is equal to 0.8, 0.6, 0.4, 0.2, and 0 for agreement levels
II, III, IV, V, and VI, respectively. While Equations 13.7-13.10 are based on fuzzy set theory and
the error or confusion matrix is associated with probability theory, outcomes of Equations
13.7-13.10 are somewhat similar to information in a row of the error matrix. Note that while one
sampled point is used only once in computing the error matrix, it is employed four times at different
degrees in constructing the four fuzzy rules. For example, a sampled point in a high accuracy area
dominated by cover type
x
will contribute more to rule1 than to rules 2-4. In contrast, a sampled
point in a low accuracy area and subordinate cover type
x
will have a more significant contribution
to rule 4 above than to the other rules. Consequently, each rule represents the degrees of membership
of cover type conversion for specific conditions of accuracy and dominance that vary spatially on
A
i
x
A
i
x
A
i
x
