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the reconstructed state space, and is less visible in higher-dimensional spaces because
the distance between trajectories increases with the dimension of reconstruction.
The bottom two graphs on the
Fig. 3
depicts the effect of
m
on the size of resulting
detectors set (DC) and average detection rate (DC/TC). For both of them there is a
clear optima for
m=5,6,7
. For these values of
m
the resulting set of detectors is
smallest, and the generated detectors have the biggest average detection rate, defined
as a ratio of DC to TC. It seems that the average detection rate is optimal for
m=6
rather than for
m=5
. This may be due to the fact, that the series used for calculating
DR was generated with
τ
MG
=20
. For this value the dimensionality of the underlying
attractor is greater than
2
and the estimated optimal reconstruction dimension is
6
.
5 Summary
The formal basis for Novelty Detection in Time Series problem and the sliding
window procedure in particular indicates the close connection with a state space
reconstruction method, known as Method of Delays. This encourages taking advance
of the wide spectrum of solutions presented in the dynamical systems analysis
literature. Especially the methods for estimation of optimal reconstruction parameters
can be used to fix the parameters of the sliding window procedure.
The experiments conducted for an chaotic time series showed that the estimated
optimal reconstruction dimension coincides with the optima of several detection
system's characteristics. More experiments are needed to check also the effect of
reconstruction lag.
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