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Fig. 6.
An example time sequence.
Although correct, this simple signature extraction technique is not par-
ticularly precise. The signature extraction methods introduced in the fol-
lowing sections take into account more information about the full sequence
shape, and therefore lead to fewer false alarms.
Figure 6 shows a time series containing measurements of atmospheric
pressure. In the following three sections, the methods described will be
applied to this sequence, and the resulting simplified sequence (recon-
structed from the extracted signature) will be shown superimposed on the
original.
3.2.
Spectral Signatures
Some of the methods presented in this section are not very recent, but
introduce some of the main concepts used by newer approaches.
Agrawal
et al
. (1993) introduce a method called the
-index in which a
signature is extracted from the frequency domain of a sequence. Underlying
their approach are two key observations:
F
•
Most real-world time sequences can be faithfully represented by their
strongest Fourier coecients.
•
Euclidean distance is preserved in the frequency domain (Parseval's
Theorem [Shatkay (1995)]).
Based on this, they suggest performing the Discrete Fourier Transform
on each sequence, and using a vector consisting of the sequence's
first
amplitude coecients as its signature. Euclidean distance in the signa-
ture space will then underestimate the real Euclidean distance between
the sequences, as required.
Figure 7 shows an approximated time sequence, reconstructed from a
signature consisting of the original sequence's ten first Fourier components.
This basic method allows only for whole-sequence matching. In 1994,
Faloutsos
et al.
introduce the
k
ST
-index, an improvement on the
F
-index
