Information Technology Reference
In-Depth Information
Def. 23.
A
delay vector
x
(t)
of an univariate time series
X
is a vector:
x
(t) = (x(t-(d-1)
- dimension and delay of reconstruction
respectively - are the reconstruction's parameters.
In the
d
-dimensional reconstructed state space delay vectors form a reconstructed
attractor. In [30] Takens formulated a theorem that if the dimension of reconstruction
is big enough, namely bigger than twice the dimension of underlying attractor, then
the delay vectors form an embedding of the original system space
5
. This means that
the mapping from the original attractor to the reconstructed one is one-to-one and
reversible, so every element of the original attractor is mapped onto one element of
the reconstructed attractor and vice-versa.
This theorem, known as Takens embedding theorem, applies also to the attractor's
neighborhood. Therefore it can be said that at least in the vicinity of the attractor, the
states that do not belong to the attractor in the original state space are mapped onto
states that do not belong to the attractor in the reconstructed space. This is a very
important property of an embedding as it is very closely related to the NDinTS
problem. The connection is due to the fact that the sliding window procedure is no
more than a state space reconstruction using MOD. To see this we must introduce the
definition of a delay vector for a discrete time series:
τ
), …, x(t-
τ
), x(t))
, where:
d
,
τ
Def. 24.
A
delay vector
x
t,m,
τ
for a discrete univariate time series is a vector:
x
t,m,
τ
=
df
(x
t
, x
t+
τ
,
…, x
t+(m-1)
τ
)
It is an equivalent to the definition 11, which defined on observation. It may be then
said that:
Theorem 1.
If the source of a time series
A
is a dynamical system, which state already
converged to an attractor, then the observations set
Obs
A,m,
τ
form a reconstruction of
the underlying attractor in
m-
dimensional reconstructed state space.
From theorem 1 it follows that for the observation set
Obs
A,m,
τ
applies all the
implications of Takens theorem and its generalizations. Therefore:
Theorem 2.
If the window length
m
is big enough, so that observations set
Obs
A,m,
τ
forms an embedding of the original attractor, then:
(a) the states that belongs to the original attractor are mapped onto
Obs
A,m,
τ
(b) the states from the original attractor's vicinity that do not belong to this attractor
are mapped onto the supplement of
Obs
A,m,
τ
The immunological novelty detection system detects only the elements that does not
belong to the input data set. From Theorem 2 it follows that the supplement of input
5
Precisely: if the dynamical system and the observed quantity are generic, then the delay-
coordinate map from a
d
-dimensional compact manifold
M
to
R
2d+1
is a diffeomorphism on
M
. Generalized in [25] for a compact-invariant subset of
R
k
, and furthermore in [22] for a
finite-dimensional subset of infinite-dimensional state space.
