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In-Depth Information
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
.