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Def. 15. A detection area DetArea(d) of a detector d is a set of problem elements that
are detected by d .
We say that a detector d detects a problem element e (denoted by d m e ) iff. e belongs
to a detection area of d . A set of detectors D detects a problem element e (denoted
D m e ) iff. e is detected by a detector d that belongs to D . This can be stated using
mathematical notation as: d m e
e
DetArea(d) , D m e
d
D
d m e .
Def. 16. An immunological model's misfit function F IMM is a function defined as
follows:
(
)
1
iff.
D
m
e
IMM
IMM
F
M
,
e
=
df
0
iff.
¬
D
m
e
Def. 17. An i mmunological novelty detection system NDS IMM is an ordered triple
(F IMM , M IMM , 1).
By setting novelty threshold to 1 it is granted that the elements detected by D are
classified as novelties.
Sliding Window Procedure Parameters. For an NDinTS problem the source of data
is a system with an unknown dynamics. In the most known immunological
approaches the following systems were used: a cutting machine [10, 12, 16], a
refrigeration system [11] an aircraft system [19] and a computer network [14, 15]. In
the above mentioned works the parameters of sliding window procedure were
established in an arbitrary manner and in some of them the values were not reported.
In [10, 12] only 5 and 7 were used for window length. In [11] m=5,7,8,10 , but no
information about the delay and offset is given. In [19] there is no information on the
window length and in [14, 15] the window length m=1 and 3 . It seems then that these
parameters do not attract the attention of the authors as much as other parameters of
immunological novelty detection system.
The rest of this work is a discussion on the impact of these parameters and some
expectations following the Takens embedding theorem. This needs some introduction
into dynamical systems area.
3 Introduction to Dynamical Systems Analysis
Some basic concepts must be defined first.
Def. 18. A system's state space or a phase space is a k -dimensional space of
orthogonal coordinates, which represents every variables necessary to define the
momentary state of a system.
Def. 19. A dynamical system DS is an ordered pair (X, f) , where X is a subset of state
space and f:X
X is a mapping in this space. Usually X=R k .
Def. 20. A state vector or simply a state is a vector x =(x 1 , x 2 , …, x k )
X .
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