T Cell–Inspired Algorithms - Immunological Computation: Theory and Applications - page 86

1.0 L ML M MH H

0.166 0.333 0.5 0.666 0.833 1.0

Universe of discourse

Figure 4.9 Partition of the interval [0, 1] in basic fuzzy sets.

An example of fuzzy detection rules in the self/nonself space with dimension n = 3

=

and linguistic values S {L, M, H} is as follows:

∈∧ ∈ ∪ ∧ ∈ ∪
Here, the basic fuzzy sets correspond to a fuzzy division of the real interval [0.0, 1.0]
using triangular and trapezoidal fuzzy membership functions. Figure 4.9 shows an
example of such a division using fi ve basic fuzzy sets representing the linguistic
values “low,” “medium-low,” “medium,” “medium-high,” and “high.”
Given a set of rules { R 1 , …, R k }, each one with a condition part Cond i , the
degree of abnormality of a sample x is defi ned by

If xLx LMx MH

1

( ) ( ) then non_self

2 3

1 …
where Cond i (x) represents the fuzzy true value produced by the evaluation of Cond i
in x and µ non_self ( x ) represents the degree of membership of x to the nonself set;
thus, a value close to 0 means that x is normal and a value close to 1 indicates that
x is abnormal.
To generate the fuzzy rule detectors, the same evolutionary algorithm described
in Section 6.1 (NSDR with DC) was used. However, the use of fuzzy rules does
not require the generation of rules for diff erent levels of deviation. h us, all the
detection rules are generated in a single run. However, the use of fuzzy rules
requires changes to the chromosome representation, fi tness evaluation, and dis-
tance calculation, which are discussed in the following text.
Each individual (chromosome) in the GA represents the condition part of a
rule because the consequent part is same for all rules (i.e., the sample belongs to
nonself ). As described earlier, a condition is a conjunction of atomic conditions.

non_self ()

x max { Cond ( )} x

i

i , , k