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12.9 Evaluation
The goal of our evaluation is to assess how well the RRR distance is suited for: (i)
calculating the similarity between time series with order-invariance (in Sect.
12.9.1
),
(ii) clustering time series that contain similar trajectories at arbitrary positions (in
Sect.
12.9.2
), and (ii) identifying prototypical time series that cover as much as pos-
sible patterns which co-occur in other sequences of the dataset (in Sect.
12.9.3
).
12.9.1 Order-Invariance
In this section, we demonstrate the practicality of our proposed RRR distance on a
sample dataset of synthetic time series. As illustrated in Fig.
12.2
a, we consider four
different normally distributed pseudorandom time series with artificially implanted
sinus patterns. The first two time series comprise the same subsequences in reverse
order, whereas the last two time series contain a subset of the artificially implanted
signals.
Figure
12.2
b illustrates the cross recurrence plot (CRP) of time series ABCD and
DCBA as well as ABCD and A**D introduced in Fig.
12.2
a. Lines parallel to the
main diagonal (from upper left to bottom right corner) indicate similar subsequences
in both time series. The percentage of recurrence points that form diagonal lines is
much higher in the CRP of the time series ABCD and DCBA than in the CRP of
the pair ABCD and A**D. As discussed in Sect.
12.6
, we quantify the local small-
scale structures in the recurrence plots by means of the determinism DET (refer to
Eq.
12.5
).
Figure
12.2
c shows a direct comparison of
Dynamic Time Warping
and our intro-
duced RRR distance measure. As expected, the hierarchical cluster tree generated by
means of DTW indicates a relatively small distance between the time series ABCD,
A**D and *BC*, because they exhibit similar subsequences at the same positions.
However, DTW treats the time series DCBA as an outlier, because the artificially
implanted patterns occur in reverse order and cross-alignment is prevented. In con-
trast, the RRR measure considers the time series ABCD and DCBA as most similar,
as the order of the matched patterns is disregarded. Furthermore, the dendrogram
generated by means of RRR reveals that the time series A**D and *BC* are dissim-
ilar to ABCD and DCBA, which is due to the fact that the overlap of same or similar
subsequences is relatively small (
50%).
The results presented in Fig.
12.2
serve to demonstrate that the proposed RRR
distance measure is able to handle time series with order-invariance. In the following,
we investigate the capability of our RRRmeasure to cluster time series which exhibit
same or similar subsequences at arbitrary positions in time.
≤
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