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data set
Obs
A,m,
τ
consist of a states that do not belong to the original attractor.
Therefore a type (1) novelties detected by a novelty detection system can be defined
as follows:
Def. 25.
A
type (1) novelty
is a state that does not belong to the original attractor.
The type (1) novelties are then caused by the change in the underlying system's
dynamics that causes the trajectory to diverge from the attractor.
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3.2 Reconstruction Parameters
The parameters of reconstruction may be mapped directly to the sliding window
procedure parameters. While the proper reconstruction ensures the detection of type
(1) novelties, the Takens theorem itself, which underlies theorem 2, does not give any
guidance on how those parameters should be fixed. Only the minimal sufficient value
of
m
is given. What's more, the assumptions for Takens theorem, which is an infinite
series of noise free data, are unrealistic [20]. In the real problems only a finite series is
given. This may lead to another type of novelties.
Def. 26.
A
type (2) novelty
is a state that does belong to the original attractor but is
not observed in the exemplary time series
A
.
While type (1) novelties are caused by change in the dynamics of the system, type (2)
novelties are the results of not having the full information. In general to represent a
whole attractor an infinite time series
A*
is needed, from which only a subseries
A
is
known. From the condition 1 it follows that the perfect novelty detection in time
series system a following must be true:
Cond. 3.
P
NDS-
= Obs
A*
The generalization of input data should then reconstruct a whole attractor basing on
an observed finite series
A
, so that only the type (1) novelties are detected.
Having only a finite set of imperfect data makes estimation of reconstruction
dimension more difficult, and also makes the reconstruction quality dependant on the
value of delay [20, 32]. Nevertheless many methods of estimating the proper
reconstruction parameters have been proposed. A small survey of them is presented
in the next few paragraphs. This methods may be used to estimate the values for
sliding window procedure parameters.
Reconstruction delay.
Commonly two limits are given for the value of
[20]: the
lower - so that the reconstructed attractor is expanded from the diagonal; and the
upper - so that the attractor does not fold on itself. The most popular methods are
based on a decorrelation (linear or general) of successive element of series [33],
the geometrical expansion from the diagonal [28] or a mean time between peeks [20].
For references to works presenting other approaches like higher-order correlations,
fill-factor, wavering products, small-window solution, see [28]. Many authors [20, 21,
27, 28] suggest that the independent parameter that should be estimated is not the lag
τ
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Assuming that the observed system already converged to the attractor.
