Environmental Engineering Reference
In-Depth Information
Y
t
not
low
low
U
t
− 1
{
l
}
:
1
low
med.
{
&
not
low
high
{
&
not
med.
Y
t
− 1
m,l
}
:
0
.
8
,
m,h
}
:
0
.
85
,
{
m
}
:
0
.
2
{
m
}
:
0
.
15
low
{
&
not
med.
med
{
&
not
high
high
{
m,l
}
:
1
m,h
}
:
0
.
7
,
m,h
}
:
0
.
1
,
{
h
}
:
0
.
3
{
h
}
:
0
.
9
Fig. 8.3
Hypothetical linguistic decision tree (LDT) mapping current and historical flow and rainfall measurements to
predicted future flow measurements. See text for definitions of labels.
Consider a time series consisting of flow Y
t
and
rainfall measurements U
t
. Figure 8.3 shows a
hypothetical LDT relating current and historical
flow and rainfall measurements to predicted
future flow Y
t
þd
. Here it is assumed that both
flow and rainfall measurements are described by
labels low (l), medium (m) and high (h) although
these may be defined according to differently
scaled appropriateness measures. The mass func-
tion associatedwith each branch refers to the set of
labels appropriate to describe Y
t
þd
given the con-
ditions on the other variables identified in that
branch. Hence each branch represents a probabi-
listic rule based on fuzzy labels. For example, the
branch labelled B in Figure 8.3 represents the
following rule:
.
IF Y
t
is not low AND U
t
1
is low AND Y
t
1
is
medium but not high THEN Y
t
þd
is either be-
tween medium and high (with probability 0.7) or
only high (with probability 0.3).
Qin (2005) andQin and Lawry (2005) introduced
the Linguistic ID3 LID3 algorithm to learn LDTs
fromdata. LID3 is an extension of the well-known
ID3 algorithm (Quinlan 1986) to generate proba-
bilistic decision trees involving fuzzy labels.
Shannon's entropy measure is used as a search
heuristic in the generation of the LDT so that
branches are expanded using attributes that min-
imize the uncertainty (entropy) associated with
the resulting branch probability distributions.
LID3 has been applied to a wide range of problem
areas including radar image classification and
