Databases Reference
In-Depth Information
Bp
(
θ
i
)=
θ
i
∈A
m
(
A
)
|
.
(6)
A
|
When the information is provided by two independent BoEs
{
Θ,
F
1
,m
1
(
)
}
and
that span the
same
FoD
Θ
, they can be combined or 'fused'
to create a single BoE
{
Θ,
F
2
,m
2
(
)
}
{
Θ,
F
,m
(
)
}
by using
Dempster's rule of combination
(DRC)
[24]:
⎧
⎨
0
,
for
C
=
∅
;
m
(
C
)=
(7)
A
∩
B
=
C
m
1
(
A
)
m
2
(
B
)
K
⎩
,
for
C
=
∅
,
−
A∩B
=
∅
where
K
= 0. DRC is one of the most widely
used belief theoretic evidence combination functions. This fusion operation is
denoted as
m
1
⊕
≡
1
m
1
(
A
)
m
2
(
B
)
m
2
and referred to as the
orthogonal sum of m
1
(
)
and m
2
(
)
.
2.2 Data Model
The tr
aining data set (database) is denoted by
D
TR
=
,where
T
i
,i
=
1
,N
TR
, is a data instance in the training data set;
N
TR
is the cardinality of
D
TR
. Assume that there are
N
F
features in the database. An instance can
then be represented as follows:
{
T
i
}
T
i
=
<F
i
,C
i
>,
where
F
i
=
<f
1
i
,f
2
i
,...,f
N
F
i
>.
(8)
The
i
-th
featu
re vector is represented by
F
i
; its features are denoted by
f
ji
,j
= 1
,N
F
. The class label assigned to the
i
-th data instance is denoted
by
C
i
.
With this notation in place, for all
i
= 1
,N
TR
, the FoD of the class label
is taken to be identical, finite and equal to
θ
(1
C
,θ
(2
C
,...,θ
(
N
C
)
Θ
C
=
{
}
,
(9)
C
where
N
C
is the number of discernible class labels. Taking the example
given in Sect. 1, we would have
Θ
C
=
{
NotDangerous
,
OfConcern
,
Dangerous
,
ExtremelyDangerous
}
. We refer to the class label
C
i
as a
partially ambigu-
ous
class label if it can be represented as a single composite proposition and
C
i
=
Θ
C
;if
C
i
=
Θ
C
,
we refer to it as a
completely ambiguous
class label.
For all
i
= 1
,N
TR
, the FoD of each feature
f
ji
is also considered to be
identical, finite and equal to
,...,Θ
(
n
f
ji
)
f
ji
Θ
(1)
f
ji
,Θ
(2)
f
ji
Θ
f
ji
=
{
}
,
(10)
where
n
f
ji
represents the number of possible values that the
j
-th feature may
take. Thus the possible values that each feature vector
F
i
may take is a subset
of the
N
F
-fold cross product of 2
Θ
f
ji
,thatis,2
Θ
f
1
i
×
2
Θ
f
N
F
i
[7].
2
Θ
f
2
i
×···×
