Biomedical Engineering Reference
In-Depth Information
with
T
1
, …,
T
s
the linguistic terms of an attribute
(antecedent) with
m
objects. When there is overlap-
ping between linguistic terms (MFs) of an attribute
or between consequents, then ambiguity exists.
For all
u
∈
U
, the intersection
A
∩
B
of two
fuzzy sets is given by m
A
∩
B
= min[m
A
(
u
)
,
m
B
(
u
)]. The
fuzzy subsethood
S
(
A
,
B
) measures the degree to
which
A
is a subset of
B
, and is given by:
To summarize the underlying process for fuzzy
decision tree construction, the attribute associ-
ated with the root node of the tree is that which
has the lowest classification ambiguity (
G
(
E
)).
Attributes are assigned to nodes down the tree
based on which has the lowest level of classifica-
tion ambiguity (
G
(
P
|
F
)). A node becomes a leaf
node if the level of subsethood (
S
(
E
,
C
i
)) associated
with the evidence down the path, is higher than
some truth value b assigned across the whole of
the fuzzy decision tree. Alternatively a leaf node
is created when no augmentation of an attribute
improves the classification ambiguity associated
with that down the tree. The classification from
a leaf node is to the decision outcome with the
largest associated subsethood value.
The truth level threshold b controls the growth
of the tree; lower b may lead to a smaller tree (with
lower classification accuracy), higher b may lead to
a larger tree (with higher classification accuracy).
The construction process can also constrain the
effect of the level of overlapping between linguistic
terms that may lead to high classification ambigu-
ity. Moreover, evidence is strong if its membership
exceeds a certain significant level, based on the
notion of α-cuts (Yuan & Shaw, 1995).
min(
(
u
),
(
u
))
(
u
)
∑
∑
S
(
A
,
B
) =
.
A
B
A
u
∈
U
u
∈
U
Given fuzzy evidence
E
, the possibility of clas-
sifying an object to the consequent
C
i
can be
defined as:
p(
C
i
|E
) =
,
S
(
E
,
C
)
/
max
S
(
E
,
C
)
i
j
j
where the fuzzy subsethood
S
(
E
,
C
i
) represents
the degree of truth for the classification rule ('if
E
then
C
i
'). With a single piece of evidence (a
fuzzy number for an attribute), then the classifi-
cation ambiguity based on this fuzzy evidence is
defined as:
G
(
E
) =
g
(p(
C
|
E
)), which is measured
using the possibility distribution p(
C
|
E
) = (p(
C
1
|
E
), …, p(
C
L
|
E
)).
The classification ambiguity with fuzzy par-
titioning
P
= {
E
1
, …,
E
k
} on the fuzzy evidence
F
, denoted as
G
(
P
|
F
), is the weighted average
of classification ambiguity with each subset of
partition:
Tutorial FDT Analysis of Example
Data Set
Beyond the description of the FDT technique
presented in Yuan and Shaw (1995) and Beynon
et al
. (2004b), here the small example problem
previously described, with its data fuzzified, is
used to offer a tutorial on the calculations needed
to be made in a FDT analysis. Before the construc-
tion process commences, a threshold value of b
= 0.75 for the minimum required truth level was
used throughout.
The construction process starts with the con-
dition attribute that is the root node. Hence it is
necessary to calculate the classification ambiguity
G
(
E
) of each condition attribute. The evaluation
of a
G
(
E
) value is shown for the first attribute T1
k
G
(
P
|
F
) =
∑
,
w
(
E
|
F
)
G
(
E
∩
F
)
i
i
i
=
1
where
G
(
E
i
∩
F
) is the classification ambiguity
with fuzzy evidence
E
i
∩
F
, and where
w
(
E
i
|
F
)
is the weight which represents the relative size of
subset
E
i
∩
F
in
F
:
w
(
E
i
|
F
) =
k
∑
∑
min(
(
u
),
(
u
))
∑
min(
(
u
),
(
u
))
E
F
E
F
i
j
u
∈
U
j
=
1
u
∈
U
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