Environmental Engineering Reference
In-Depth Information
In addition to appropriateness measures a
second related measure can be defined for the
fuzzy labels in LA. For value x
the mass functions derived from the trapezoidal
appropriateness measures in Figure 8.1.
2 V
the mass
function m
x
:
Þ!½
0; 1
represents the uncer-
tainty about which set of labels is appropriate to
describe x. For example, m
x
({low, medium}) is the
probability that low and medium are both appro-
priate to describe x and all other labels are inap-
propriate. This mass value would tend to be high
for values of x where both low and medium are
highly appropriate and hence where the two ap-
propriateness measures overlap. The mass func-
tion m
x
is a probability distribution on the power
set of labels P(LA) so that:
X
F
LA
m
x
ð
P
ð
LA
Learning Algorithms
This section gives a brief description of two learn-
ing algorithms, Linguistic Decision Trees and
Fuzzy Bayes, based on the Label Semantics frame-
work introduced in the previous section. These
will be the core AI technologies applied in the
hydrological case studies given in the following
two sections.
Þ¼
1
ð
8
:
1
Þ
F
Linguistic decision tree
Linguistic Decision Trees (LDTs) have been pro-
posed by Lawry (2006) as a tree-structured repre-
sentation for conditional rules involving fuzzy
labels. LDTs consist of nodes corresponding to
input variables (attributes) and branches corre-
sponding to linguistic expressions describing vari-
ables in terms of a set of predefined fuzzy labels.
Associated with each complete branch there is a
probability distribution. In the case of a classifi-
cation model this will be the probabilities for the
different classes. For a predictionmodel where the
output is a real value then the probabilities define a
mass function on the label sets describing that
value.
The strong relationship between mass functions
and appropriateness measures is based on the fact
that the probability that label L is appropriate to
describe x is equivalent to the probability that
label L is included in the set of labels appropriate
to describe x; consequently we have the relation-
ship:
m
L
ð
x
Þ¼
X
F
:
L
2
F
m
x
ð
F
Þ
ð
8
:
2
Þ
Furthermore, Lawry (2006) shows that, assuming
labels can be ranked in terms of their appropriate-
ness, mass functions can be determined directly
from appropriateness measures. Figure 8.2 shows
... ...
{
L
1
}{
L
1
,L
2
}
{
L
2
}{
L
2
,L
3
}{
L
3
} {
L
3
,L
4
} {
L
4
} {
L
4
,L
5
} {
L
5
}
{
L
n
− 1
,L
n
}
L
n
}
Fig. 8.2
Mass function for the possible sets of appropriate labels derived from the appropriateness measures in
Figure 8.1.
