Skip to main content
SpringerLink
Log in

A shape-based similarity measure for time series data with ensemble learning

  • Theoretical Advances
  • Published:
Pattern Analysis and Applications Aims and scope Submit manuscript

Abstract

This paper introduces a shape-based similarity measure, called the angular metric for shape similarity (AMSS), for time series data. Unlike most similarity or dissimilarity measures, AMSS is based not on individual data points of a time series but on vectors equivalently representing it. AMSS treats a time series as a vector sequence to focus on the shape of the data and compares data shapes by employing a variant of cosine similarity. AMSS is, by design, expected to be robust to time and amplitude shifting and scaling, but sensitive to short-term oscillations. To deal with the potential drawback, ensemble learning is adopted, which integrates data smoothing when AMSS is used for classification. Evaluative experiments reveal distinct properties of AMSS and its effectiveness when applied in the ensemble framework as compared to existing measures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. To be precise, C and C vec are not exactly the same because the latter does not retain the position of the first element c 1.

  2. Another shape-based similarity measure, DDTW, uses Euclidean distance. However, the distance is measured between the slopes of two vectors, which is similar to (inversed) cosine similarity in concept.

  3. This situation can arise when assessing AMSS(Q N−2, C M−1) or AMSS(Q N−1, C M−2).

  4. Ensemble approaches have been applied to time-series classification in the past [14, 26]. However, none has attempted to incorporate different smoothing techniques in an ensemble framework.

  5. EDR was chosen because it is reported to be the most accurate measure among the edit-distance family, including LCSS and ERP [9].

  6. DDTW(Q, Q) always becomes 0, whereas AMSS(Q, Q) linearly increases with the length of Q because AMSS is defined as a sum of similarities between matched subsequences.

  7. Note that these results are different from Table 1 due to the different treatment of the training data.

References

  1. Aßfalg J, Kriegel H-P, Kröger P, Kunath P, Pryakhin A, and Renz M (2006) Similarity search on time series based on threshold queries. In: Proceedings of the 10th international conference on extending database technology, pp 276–294

  2. Berndt DJ, Clifford J (1996) Finding patterns in time series: a dynamic programming approach. In: Fayyad UM, Piatetsky-Shapiro G, Smyth P, Uthurusamy R (eds) Advances in knowledge discovery and data mining. AAAI, Menlo Park, pp 229–248

  3. Chan K-P, Fu AW-C (1999) Efficient time series matching by wavelets. In: Proceedings of the 15th international conference on data engineering, pp 126–133

  4. Chen L, Ng R (2004) On the marriage of lp-norms and edit distance. In: Proceedings of the 30th international conference on very large data bases, pp 792–803

  5. Chen L, Øzsu MT, Oria V (2005) Robust and fast similarity search for moving object trajectories. In: Proceedings of the 2005 ACM SIGMOD international conference on management of data, pp 491–502

  6. Chen Y, Nascimento MA, Ooi BC, Tung AKH (2007) Spade: on shape-based pattern detection in streaming time series. In: Proceedings of the IEEE 23rd international conference on data engineering, pp 786–795

  7. Das G, Gunopulos D, Mannila H (1997) Finding similar time series. In: Proceedings of the first european symposium on principles of data mining and knowledge discovery, pp 88–100

  8. Dietterich TG (2000) Ensemble methods in machine learning. In: Proceedings of the 1st international workshop on multiple classifier systems, pp 1–15

  9. Ding H, Trajcevski G, Scheuermann P, Wang X, and Keogh E (2008) Querying and mining of time series data: experimental comparison of representations and distance measure. Proc VLDB Endowment 1:1542–1552

    Google Scholar 

  10. Faloutsos C, Ranganathan M, Manolopoulos Y (1994) Fast subsequence matching in time-series databases. In: Proceedings of the 1994 ACM SIGMOD international conference on management of data, pp 419–429

  11. Frentzos E, Gratsias K, Theodoridis Y (2007) Index-based most similar trajectory search. In: Proceedings of the IEEE 23rd international conference on data engineering, pp 816–825

  12. Freund Y, Schapire RE (1995) A decision-theoretic generalization of on-line learning and an application to boosting. In: Proceedings of the second european conference on computational learning theory, pp 23–37

  13. Freund Y, and Schapire RE (1997) A decision theoretic generalization of on-line learning and an application to boosting. J Comput Syst Sci 55(1):119–139

    Article  MathSciNet  MATH  Google Scholar 

  14. Geurts P, Wehenkel L (2005) Segment and combine approach for non-parametric time-series classification. In: Proceedings of the 9th European conference on principles and practice of knowledge discovery in databases, pp 478–485

  15. Gunopulos D, Das G (2000) Time series similarity measures. In: Tutorial notes of the sixth ACM SIGKDD international conference on knowledge discovery and data mining, pp 243–307

  16. Itakura F (1975) Minimum prediction residual principle applied to speech recognition. IEEE Trans Acoust Speech Signal Process 23(1):67–72

    Article  Google Scholar 

  17. Keogh E (2002) Exact indexing of dynamic time warping. In: Proceedings of the 28th international conference on very large data bases, pp 406–417

  18. Keogh E, Xi X, Wei L, Ratanamahatana CA (2006) The UCR time series classification/clustering homepage. http://www.cs.ucr.edu/˜eamonn/time_series_data/

  19. Keogh EJ, Pazzani MJ (2001) Derivative dynamic time warping. In: Proceedings of the 1st SIAM international conference on data mining, pp 1–11

  20. Lin J, Keogh E, Lonardi S, Chiu B (2003) A symbolic representation of time series, with implications for streaming algorithms. In: Proceedings of the 8th ACM SIGMOD workshop on research issues in data mining and knowledge discovery, pp 2–11

  21. Liu Y, Yao X (1999) Ensemble learning via negative correlation. Neural Netw 12(10):1399–1404

    Article  Google Scholar 

  22. Morse MD, Patel JM (2007) An efficient and accurate method for evaluating time series similarity. In: Proceedings of the 2007 ACM SIGMOD international conference on management of data, pp 569–580

  23. Nomiya H, Uehara K (2007) Multistrategical approach in visual learning. In: Proceedings of the 8th Asian conference on computer vision, pp 502–511

  24. Pavlidis T, Horowitz SL (1974) Segmentation of plane curves. IEEE Trans Comput 23:860–870

    Article  MathSciNet  MATH  Google Scholar 

  25. Rafiei D, Mendelzon A (1997) Similarity-based queries for time series data. In: Proceedings of the 1997 ACM SIGMOD international conference on management of data, pp 13–25

  26. Rodríguez JJ, Kuncheva LI (2007) Time series classification: Decision forests and SVM on interval and DTW features. In: Proceedings of the workshop and challenge on time series classification

  27. Sakoe H, Chiba S (1978) Dynamic programming algorithm optimization for spoken word recognition. IEEE Trans Acoust Speech Signal Process 26(1):46–49

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Takashi Okamura for his help with implementations and experiments.

Author information

Authors and Affiliations

  1. Kobe University, 1-1 Rokkodai, Nada, Kobe, 657-8501, Japan

    Tetsuya Nakamura, Keishi Taki, Kazuhiro Seki & Kuniaki Uehara

  2. Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585, Japan

    Hiroki Nomiya

Authors
  1. Tetsuya Nakamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keishi Taki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hiroki Nomiya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kazuhiro Seki
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kuniaki Uehara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuhiro Seki.

Appendix

Appendix

AMSS was compared with other (dis)similarity measures: Euclidean distance, DTW, DDTW, EDR, and SpADe as shown in Table 1. However, there is more comprehensive comparison reported by Ding et al. [9], who used k-fold cross validation for parameter tuning and evaluation for 13 different (dis)similarity measures. The value of k was individually set for each UCR data set (or class).

This paper did not take Ding et al.’s results for the comparison in Table 1 because of the significant amount of computation needed for the ensemble experiments with the way they tuned parameters through cross validation. For completeness, however, AMSS (without the ensemble framework) is compared with those reported by Ding et al. with their evaluation scheme. The results are summarized in Table 4, where boldface indicates the best performance (lowest error rates) across different measures.Footnote 7 “DTW (w)” and “LCSS (w)” indicate DTW and LCSS with warping constraint, respectively. All the results but DDTW were taken from Ding et al.’s paper [9], and DDTW’s results are based on our own implementation with no warping window. For this experiments, the same number of splits k as Ding et al. was used so that all the results are directly comparable to theirs. Among the 20 data sets, AMSS performed the best for 10 data sets including one tie.

Table 4 Classification performance in error rates of different (dis)similarity measures using 1NN
Full size table

In terms of the number of data sets for which error rates are lowest, AMSS was found to be the best (dis)similarity measure in this particular setting. As discussed in Sect. 5.2, however, no single measure works for every type of data. For example, simple Euclidean distance performed better than all the other measures, including AMSS, for the “Beef” data set. More work is needed to understand the interaction between the properties of similarity measures and the characteristics of the target data types.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakamura, T., Taki, K., Nomiya, H. et al. A shape-based similarity measure for time series data with ensemble learning. Pattern Anal Applic 16, 535–548 (2013). https://doi.org/10.1007/s10044-011-0262-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10044-011-0262-6

Keywords

Search

Navigation