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Elastic distances for time-series classification: Itakura versus Sakoe-Chiba constraints

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Abstract

In the field of time series data mining, the accuracy of the simple, but very successful nearest neighbor (NN) classifier directly depends on the chosen similarity measure. To improve the efficiency of elastic measures introduced to overcome the shortcomings of Euclidean distance, the Sakoe-Chiba band is usually applied as a constraint. In this paper, we provide a detailed analysis of the influence of the alternative Itakura parallelogram constraint on the accuracy of the NN classifier in combination with four well-known elastic measures, compared to the Sakoe-Chiba constraint and the unconstrained variants of these measures. The findings suggest that, although the Sakoe-Chiba band generally produces better results, for certain types of datasets the Itakura parallelogram represents a better choice.

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Data availability

The results presented in this paper are based on the UCR time Series Classification Archive (https://www.cs.ucr.edu/~eamonn/time_series_data_2018/).

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Acknowledgements

V. Kurbalija, M. Ivanović, and M. Radovanović acknowledge financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022-14/200125). The authors would like to express special thanks to Eamonn Keogh for his valuable comments about the paper, and for collecting and making available the UCR time-series datasets, and also to everyone who contributed data to the collection, without which the presented work would not have been possible.

Funding

Partial financial support was received from the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022-14/200125).

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Authors and Affiliations

  1. Department of Media Studies, Faculty of Philosophy, University of Novi Sad, Dr Zorana Đinđića 2, 21000, Novi Sad, Serbia

    Zoltan Geler

  2. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg D. Obradovića 4, 21000, Novi Sad, Serbia

    Vladimir Kurbalija, Mirjana Ivanović & Miloš Radovanović

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  1. Zoltan Geler
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Contributions

Z. Geler and V. Kurbalija designed and performed the experiments. V. Kurbalija and M. Radovanović provided the expertise for interpreting the results and designing the experiments. Z. Geler was involved in data acquisition and pre-processing. V. Kurbalija, M. Radovanović and M. Ivanović provided valuable feedback and interpretation of the experimental results. Z. Geler drafted the manuscript, while V. Kurbalija, M. Radovanović and M. Ivanović gave comments and made modifications to the manuscript. All authors approved the manuscript. M. Ivanović coordinated the joint research.

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Correspondence to Zoltan Geler.

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Appendix: The intuition behind the Sakoe-Chiba band and the Itakura parallelogram

Appendix: The intuition behind the Sakoe-Chiba band and the Itakura parallelogram

The reader may wonder why both the Sakoe-Chiba band and Itakura parallelogram exist. It may be helpful to revisit DTW in the context of its invention as a speech processing tool. In addition, let us use text as an intuitive proxy for time series. For some speech processing problems, it is easy to precisely locate the beginning and end of an utterance (by the dramatic change in volume). Imagine we wish to compare the name of a Southern US state with a stuttered "plosive" consonant, Mississippppppi with a version that has significant sibilance Missssissssippi. The reader will appreciate that DTW is an ideal tool to recognize that these refer to the same word. Note that for such comparisons there is an interesting constraint. Both utterances start with the same letter “M”, and end with the same letter “i”. This means there is no warping at the beginning of the alignment, the warping can grow as we transverse the words, and as we move to the end of the word, the fact that we must end with the same sound again constrains the amount of warping that is possible. The reader will appreciate that this, in geometric terms, describes a parallelogram. The parallelogram shaped alignment is very common in natural and human processes where there is a mechanism to naturally align the beginning and end of a process, but the amount of lagging or leading is proportional to the distance to the nearer of, the beginning or the end. For example, amateur musicians typically start and end in time, but may differ in the middle of a performance.

In contrast, consider the problem of comparing braided and upbraid. While these two words have a lot of common structure, their differences are concentrated at both ends, exactly were the Itakura parallelogram would penalize them, by forcing them to an alignment that is almost Euclidean. Such comparisons need just as much flexibility at every point, which the reader will appreciate describes a band.

We believe that this consideration exactly explains when the Sakoe-Chiba band or Itakura parallelogram are most appropriate. For example, in the GunPoint dataset, the creators use a metronome that produced an audible bleep every five seconds; this provided a cue that aligned the ends of the signal snippet. This explains why Itakura parallelogram works well on this dataset. A good example is GunPointMaleVersusFemale dataset in combination with DTW measure. Here, the IT version is more than two times better than SC version (error rates are 0.0024 and 0.0058 respectively). The alignment on the beginning and on the end of the time series is clearly visible in Fig. 

Fig. 27
figure 27

Several time series from the GunPointMaleVersusFemale dataset

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27 where several time series from this dataset are plotted.

Another example of good IT performance could be datasets where the images are converted to time series by measuring and recording the local angles of each pixel of an image contour. This idea is nicely shown in Fig. 

Fig. 28
figure 28

Transformation of raw image shape to time series illustrated on the FaceFour dataset [26]

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28 on the FaceFour dataset. The same idea is applied in the Plane dataset where the contours of 7 different airplanes are used to create time-series dataset in the same manner. In both datasets, time series are aligned at the beginning and the end, while in the middle sector the variations are significant, as seen in Fig. 

Fig. 29
figure 29

Several time series from the FaceFour A, and Plane B datasets

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29. This fact makes them more appropriate for the IT constraint which is evident in the obtained classification error rates:

  • For the FaceFour dataset using the LCS measure: 0.0089 (IT), 0.0179 (SC)

  • For the Plane dataset applying the DTW measure: 0.0029 (IT), 0.0038 (SC)

In contrast to datasets suitable for the IT approach, in the datasets where SC performs better, time series do not necessarily need to be aligned at the beginning and at the end. The example datasets with this property are more frequent and can be easily found in the UCR repository. Two such datasets (only a few time series are plotted) are shown in Fig. 30.

Fig. 30
figure 30

Several time series from the GesturePebbleZ2 A, and BeetleFly B datasets

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In both datasets, SC produced notably lower error rates:

  • For the GesturePebbleZ2 dataset using DTW: 0.1384 (IT), 0.0671 (SC)

  • For the BeetleFly dataset applying DTW: 0.2225 (IT), 0.125 (SC)

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Geler, Z., Kurbalija, V., Ivanović, M. et al. Elastic distances for time-series classification: Itakura versus Sakoe-Chiba constraints. Knowl Inf Syst 64, 2797–2832 (2022). https://doi.org/10.1007/s10115-022-01725-1

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