Information Technology Reference
In-Depth Information
Let
m
i
represents the membership value of a positive data point
x
i
, while
m
i
represents the membership of a negative data point
x
i
in their own classes.
In the proposed FSVM-CIL method, these membership functions are defined as
follows:
m
i
=
f(x
i
)r
+
(5.17)
m
i
=
f(x
i
)r
−
(5.18)
where
f(x
i
)
generates a value between 0 and 1, which reflects the importance
of
x
i
in its own class. The values for
r
+
and
r
−
were assigned in order to reflect
the class imbalance, such that
r
+
=
1and
r
−
=
r
, where
r
is the minority-to-
majority-class ratio
(r
+
>r
−
)
(this was following the findings reported in [10],
where optimal results could be obtained from the DEC method when
C
−
/C
+
equals to the minority-to-majority-class ratio). According to this assignment of
values, a positive class data point is assigned a misclassification cost
m
i
C
, where
m
i
takes a value in the [0,1] interval, while a negative class data point is assigned
a misclassification cost
m
i
C
, where
m
i
takes value in the [0
,r
] interval, where
r<
1.
In order to define the function
f(x
i
)
introduced in Equations 5.17 and 5.18,
which gives the within-class importance of a training example, the following
methods have been considered in [40].
A. f(x
i
) is based on the distance from the own class center:
In this method,
f(x
i
)
is defined with respect to
d
ce
i
, which is the distance between
x
i
and its
own class center. The examples closer to the class center are treated as more
informative and assigned higher
f(x
i
)
values, while the examples far away from
the center are treated as outliers or noise and assigned lower
f(x
i
)
values. Here,
two separate decaying functions of
d
cen
i
have been used to define
f(x
i
)
,which
are represented by
f
cen
lin
(x
i
)
and
f
cen
exp
(x
i
)
as follows:
f
cen
(d
cen
i
/(
max
(d
cen
i
lin
(x
i
)
=
1
−
)
+
δ))
(5.19)
is a linearly decaying function.
δ
is a small positive value used to avoid the case
where
f(x
i
)
becomes zero.
f
cen
exp
(x
i
)
=
2
/(
1
+
exp
(d
cen
∗
β))
(5.20)
i
is an exponentially decaying fu
nc
ti
o
n, where
β
;
β
∈
[0
,
1] determines the steep-
2
ness of the d
e
cay.
d
cen
i
=
x
i
−
x
is the Euclidean distance to
x
i
from its own
class center
x
.
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