Databases Reference
In-Depth Information
Table2.3. Fuzzylogicoperators.
Logic Operator
Fuzzy Operator
p
AND
q
min
{µ
p
(
x
),
µ
q
(
y
)
}
p
OR
q
max
{µ
p
(
x
),
µ
q
(
y
)
}
NOT
p
1
− µ
p
(
x
)
phase. To this end, the mapping between a logic operator and a fuzzy
operator has to be defined, as shown in Table 2.3,
26,27,41
where
p
and
q
are fuzzy sets,
x
and
y
are continuous values of attributes and
µ
is
the membership function associated with a fuzzy set. According to this
table, the firing strength of a fuzzy rule can be calculated for a given
data instance. Since an input value can belong to more than one fuzzy
set, it is possible that a data sample will trigger more than one fuzzy rule
with different class labels. Winner-takes-all
16
or majority vote
37,38
are two
commonly used techniques to resolve the conflict. “Winner” refers to the
rule with maximum
CF
j
.
2.3.3.
Regression
Regression is a process of estimating the value of a continuous target
attribute as a function of one or more predictor attributes, a set of
parameters, and a set of arithmetic operators. In the context of intrusion
detection or other similar domains, regression can be viewed as a special
case of classification. That is, regression outputs an equation which
transforms data in a high dimensional space into a specific value or a range
of values in a low dimensional space according to different class labels, as
shown in Fig. 2.7.
The simplest regression function is a linear regression function with
the following format:
C
(
χ
)=
j
=1
χ
j
), where
n
is the number of
attributes,
w
j
is a weight
59
or coecient
60
of attribute
χ
j
. A GA usually
searches for the best set of weights or coecient that map any data from the
normal class to a value larger than
δ
(
C
(
χ
)
>δ
) and any data from the
anomaly class to a value less than
δ
(
C
(
χ
)
<δ
).
δ
is a user defined threshold.
Individuals in this case contain
n
genes, each standing for a weight or
coecient.
Compared with GAs, regression equations evolved by GP have
more complex structures, normally employing nonlinear functions. Only
arithmetic operators, such as “+”, “
(
w
j
×
−
”, “
×
”, “
÷
”, “log”, etc., and
