2.1 Extensions of Apriori Algorithm
The basic Apriori algorithms(Meshram, 2004) do
not consider any domain knowledge and as a result
they can generate many “irrelevant" (i.e.,
uninteresting) rules. The above limitations and
generalization is achieved by incorporating two
modifications to the basic Apriori algorithm. They
are: Axis attributes, Reference attributes.
Interestingness Measures based on Attributes:
The basic algorithms implicitly measure the
interestingness (i.e., relevancy) of patterns by their
support and confidence values, without regard to any
available prior domain knowledge. That is, assume I
is the interestingness measure of a pattern p, then
I(p) = f(support(p); confidence(p)) Where f is some
ranking function. We attempt to utilize the schema
level information about audit records to direct the
pattern mining process. Assume IA is a measure on
whether a pattern p contains the specified important
(i.e. “interesting") attributes, our extended
interestingness measure is
Ie(p) = fe(IA(p); f(support(p); confidence(p)))
= fe(IA(p); I(p))
Where fe is a ranking function that first considers
the attributes in the pattern, then the support and
confidence values.
Using the Axis Attributes (IA): We describe
several schema-level characteristics of audit data, in
the forms of “what attributes must be considered",
that can be used to guide the mining of relevant
features. We call the essential attribute(s) as axis
attribute(s) when they are used as a form of item
constraints in the association rules algorithm. During
candidate generation, an item set must contain
value(s) of the axis attribute(s). We consider the
correlation among non-axis attributes as not
interesting. In other words,
IAx (p) = 1 if p contains axis attribute(s)
= 0 otherwise (not interesting attribute).
Using the Reference Attributes (IR): Another
interesting characteristic of system audit data is that
some attributes can be the references of other
attributes. These reference attributes normally carry
information about some “subject", and other
attributes describe the “actions" that refer to the
same “subject". It is important to use the “subject" as
a reference when finding such frequent sequential
“action" patterns because the “actions" from
different “subjects" are normally irrelevant. This
kind of sequential pattern can be represented as:
(Subject = X; action = a);
(Subject = X; action = b) → (subject = X; action = c)
Thus subject is simply reference (or a variable).In
other words,
IAr (p) = 1 if the item sets of p refer to the same
reference attribute value otherwise 0.
Level-wise Approximate Mining: In daily network
traffic, some services, for example, gopher, account
for very low occurrences. Yet we still need to
include their patterns into the network traffic profile,
we will then get unnecessarily a very large number
of patterns related to the high frequency services.
Here the idea is to first find the episodes related
to high frequency axis attribute values. We then
iteratively lower the support threshold to find the
episodes related to the low frequency axis values by
restricting the participation of the “old" axis values
that already have output episodes. More specifically,
when an episode is generated, it must contain at least
one “new" (low frequency) axis value. Then we can
infer the more information required in the
classification of the network data.
2.2 Integration of Fuzzy Logic with
Modified Apriori
Although association rules can be mined from audit
data for anomaly intrusion detection, the mined rules
are at the data level. Integrating fuzzy logic with
association rules allows one to extract more abstract
patterns at a higher level.
2.2.1 Probability of Fuzzy Logic Association
Rules
Let T = {t
1
, t
2
, ..., t
n
} be the database and ti
represents the i
th
tuple in T. Moreover, we use I =
{i
1
, i
2
, ..., i
m
} to represent all attributes appeared in T
and ij represents the j
th
attribute. Since I contain set
of items, we call I an itemset. We can retrieve the
value of attribute i
k
in the j
th
record simply by t
j
[ik].
Besides, each attribute ik will associate with several
fuzzy sets. We use F
jk
= {f1
jk
, f2
ik
, ..., fl
ik
} to
represent set of fuzzy sets associated with ik and fj
ik
represents the j
th
fuzzy set in F
ik
.
Given a database T with attributes I and those
fuzzy sets associated with attributes in I, we want to
find out some interesting, potentially useful
regularities in a guided way. Our proposed fuzzy
association rule is in the form: If X is A then Y is B.
In the above rule, X = {x
1
, x
2
, ..., x
p
} and Y = {y
1
,
y
2
, ..., y
q
} are item sets. X and Y are subsets of I and
they are disjoint which means that they share no
common attributes. A = {fx
1
, fx
2
, ..., fx
p
} and B =
{fy
1
, fy
2
, ..., fy
q
} contain the fuzzy sets associated
with the corresponding attributes in X and Y. For
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