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Towards Symbolic Time Series Representation Improved by Kernel Density Estimators

Part of the Lecture Notes in Computer Science book series (TLDKS,volume 12930)


This paper deals with symbolic time series representation. It builds up on the popular mapping technique Symbolic Aggregate approXimation algorithm (SAX), which is extensively utilized in sequence classification, pattern mining, anomaly detection, time series indexing and other data mining tasks. However, the disadvantage of this method is, that it works reliably only for time series with Gaussian-like distribution. In our previous work (Kloska and Rozinajova, dwSAX, 2020) we have proposed an improvement of SAX, called dwSAX, which can deal with Gaussian as well as non-Gaussian data distribution. Recently we have made further progress in our solution - edwSAX. Our goal was to optimally cover the information space by means of sufficient alphabet utilization; and to satisfy lower bounding criterion as tight as possible. We describe here our approach, including evaluation on commonly employed tasks such as time series reconstruction error and Euclidean distance lower bounding with promising improvements over SAX.


  • Time series
  • Kernel density estimator
  • SAX
  • Tightness of lower bound

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  1. Bagnall, A., Lines, J., Keogh, E.: UEA & UCR time series classification repository (2021).

  2. Bountrogiannis, K., Tzagkarakis, G., Tsakalides, P.: Data-driven kernel-based probabilistic SAX for time series dimensionality reduction. In: 2020 28th European Signal Processing Conference (EUSIPCO), pp. 2343–2347. IEEE (2021)

    Google Scholar 

  3. Eghan, R.E., Amoako-Yirenkyi, P., Omari-Sasu, A.Y., Frimpong, N.K.: Time-frequency coherence and forecast analysis of selected stock returns in Ghana using Haar wavelet. J. Adv. Math. Comput. Sci. 1–12 (2019)

    Google Scholar 

  4. Fournier-Viger, P., Lin, J.C.W., Kiran, R.U., Koh, Y.S., Thomas, R.: A survey of sequential pattern mining. Data Sci. Pattern Recogn. 1(1), 54–77 (2017)

    Google Scholar 

  5. Hwang, J.N., Lay, S.R., Lippman, A.: Nonparametric multivariate density estimation: a comparative study. IEEE Trans. Sig. Process. 42(10), 2795–2810 (1994)

    CrossRef  Google Scholar 

  6. Jones, M.C., Marron, J.S., Sheather, S.J.: A brief survey of bandwidth selection for density estimation. J. Am. Stat. Assoc. 91(433), 401–407 (1996)

    CrossRef  MathSciNet  Google Scholar 

  7. Jones, M.C.: The performance of kernel density functions in kernel distribution function estimation. Stat. Prob. Lett. 9(2), 129–132 (1990)

    CrossRef  MathSciNet  Google Scholar 

  8. Keogh, E., Chakrabarti, K., Pazzani, M., Mehrotra, S.: Dimensionality reduction for fast similarity search in large time series databases. Knowl. Inf. Syst. 3(3), 263–286 (2001).

    CrossRef  MATH  Google Scholar 

  9. Keogh, E., Lin, J., Fu, A.: HOT SAX: efficiently finding the most unusual time series subsequence. In: Fifth IEEE International Conference on Data Mining (ICDM 2005), p. 8 IEEE (2005)

    Google Scholar 

  10. Kloska, M., Rozinajova, V.: Distribution-wise symbolic aggregate ApproXimation (dwSAX). In: Analide, C., Novais, P., Camacho, D., Yin, H. (eds.) IDEAL 2020. LNCS, vol. 12489, pp. 304–315. Springer, Cham (2020).

    CrossRef  Google Scholar 

  11. Lin, J., Keogh, E., Lonardi, S., Chiu, B.: 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 (2003)

    Google Scholar 

  12. Lin, J., Keogh, E., Wei, L., Lonardi, S.: Experiencing SAX: a novel symbolic representation of time series. Data Min. Knowl. Disc. 15(2), 107–144 (2007).

    CrossRef  MathSciNet  Google Scholar 

  13. Mahmoudi, M.R., Heydari, M.H., Roohi, R.: A new method to compare the spectral densities of two independent periodically correlated time series. Math. Comput. Simul. 160, 103–110 (2019)

    CrossRef  MathSciNet  Google Scholar 

  14. Sato, T., Takano, Y., Miyashiro, R.: Piecewise-linear approximation for feature subset selection in a sequential logit model. J. Oper. Res. Soc. Jpn. 60(1), 1–14 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Scott, D.W.: On optimal and data-based histograms. Biometrika 66(3), 605–610 (1979)

    CrossRef  MathSciNet  Google Scholar 

  16. Scott, D.W.: Scott’s rule. Wiley Interdisc. Rev. Comput. Stat. 2(4), 497–502 (2010)

    CrossRef  Google Scholar 

  17. Senin, P., Malinchik, S.: SAX-VSM: interpretable time series classification using SAX and vector space model. In: 2013 IEEE 13th International Conference on Data Mining, pp. 1175–1180. IEEE (2013)

    Google Scholar 

  18. Sheather, S.J., Jones, M.C.: A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Stat. Soc.: Ser. B (Methodol.) 53(3), 683–690 (1991)

    MathSciNet  MATH  Google Scholar 

  19. Shieh, J., Keogh, E.: iSAX: indexing and mining terabyte sized time series. In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 623–631 (2008)

    Google Scholar 

  20. Silverman, B.W.: Density Estimation for Statistics and Data Analysis. Routledge, Boca Raton (2018)

    CrossRef  Google Scholar 

  21. Tamura, K., Ichimura, T.: Clustering of time series using hybrid symbolic aggregate approximation. In: 2017 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 1–8. IEEE (2017)

    Google Scholar 

  22. Wand, M.: Data-based choice of histogram bin width. Am. Stat. 51(1), 59–64 (1997)

    Google Scholar 

  23. Wang, X., Mueen, A., Ding, H., Trajcevski, G., Scheuermann, P., Keogh, E.: Experimental comparison of representation methods and distance measures for time series data. Data Min. Knowl. Disc. 26(2), 275–309 (2013).

    CrossRef  MathSciNet  Google Scholar 

  24. Yang, S., Liu, J.: Time-series forecasting based on high-order fuzzy cognitive maps and wavelet transform. IEEE Trans. Fuzzy Syst. 26(6), 3391–3402 (2018)

    CrossRef  Google Scholar 

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This research was supported by TAILOR, a project funded by Horizon 2020 research and innovation programme under GA no 952215 and “Knowledge-based Approach to Intelligent Big Data Analysis” - Slovak Research and Development Agency under the contract No. APVV-16-0213.

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Correspondence to Matej Kloska .

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Kloska, M., Rozinajova, V. (2021). Towards Symbolic Time Series Representation Improved by Kernel Density Estimators. In: Hameurlain, A., Tjoa, A.M. (eds) Transactions on Large-Scale Data- and Knowledge-Centered Systems L. Lecture Notes in Computer Science(), vol 12930. Springer, Berlin, Heidelberg.

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