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shown that the sliding window procedure and MOD are equivalent, and also the well
known methods of state space reconstruction parameters estimation are discussed. In
section 4 the results of V-detector novelty detection system on Mackey-Glass time
series are presented and discussed. The summary is presented in section 5.
2 Problem Definition
To be able to define the NDinTS, the general ND problem must be defined first.
Def. 1.
The
problem space
P
is a space containing all elements subject to
classification by novelty detection system.
Def. 2.
A
problem's element
is any element
e
that belongs to the problem space
P.
Def 3.
The
classification mapping
is a mapping
classify:P
{normal, novel}
, that
assigns each element of the problem space to with one of two classes:
normal, novel
.
1
→
Def. 4.
The
normal subspace
P
-
is a set of elements classified as
normal
P
-
=
df
{e
∈
P| classify(e) = normal}
.
Def. 5.
The
novel subspace
P
+
is set of elements classified as
novel
P
+
=
df
{e
∈
P| classify(p) = novel}
.
The problem can be formulated as follows: given a subset
S
of a normal subspace
P
-
, estimate the classification mapping. As it was stated in Section 1, the common
approach is based on a model of normal data. It can be informally defined as follows:
Def. 6.
A
model
M
X
is a finite mathematical representation of systems behavior given
by a set of problem's elements
X
.
M
X
∈
M
.
Def. 7.
A
misfit function
F(M,e)
is a function
F:
M
×
P
→
R
that determines how much
the element
e
does not fit into the model
M
.
Def. 8.
A
novelty detection system
NDS
is an ordered triple
(F, M
S
, p)
, such that:
F
is a misfit function,
M
S
is a model of input data set
S
,
p
is a misfit threshold value.
Def. 9.
An
estimated classification mapping
classify_est(NDS, e)
is a mapping defined
as follows
2
:
(
)
normal
iff
F
M
,
e
<
p
⎧
(
)
S
classify
_
est
NDS
,
e
=
⎨
(
)
df
novel
iff
F
M
,
e
≥
p
⎩
S
1
In the Artificial Immune Systems nomenclature, the classes and the following subspaces
P
-
and
P
+
sets are usually named
Self
and
Non-Self
.
2
There are also other definitions of classification mapping that allows more then one level on
novelty or even a non-crisp discrimination.
