Advertisement

Mathematical Programming for Piecewise Linear Representation of Discrete Time Series

  • Yang Xiyang
  • Zhang Jing
  • Yu Fusheng
  • Li ZhiweiEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1075)

Abstract

Piecewise linear representation (PLR) of a time series arises in variety of disciplines in data mining. Unlike most PLR methods who separate a discrete time series into a few discontinuous line segments, \( \ell 1 \) trend filtering method is one of the few PLR methods who generates continuous line segment representations. However, the approximation errors of \( \ell 1 \) trend filtering seldom reach its minimum. In this paper, we propose a binary integer programming model to produce a continuous PLR of time series with the least approximation error, and therefore it is well suitable to analyzing time series with an underlying piecewise linear trend. We describe the motives of the proposed method and give some illustrative examples. The improvement in approximation error is demonstrated by some experiments on some real-world time series datasets.

Keywords

Piecewise linear representation \( \ell 1 \) trend filtering Approximation error Binary integer programming 

Notes

Acknowledgements

Funding from Training Programs of Innovation and Entrepreneurship for Undergraduates (201810399037) are gratefully acknowledged.

References

  1. 1.
    Keogh, E., Pazzani, M.: An enhanced representation of time series which allows fast and accurate classification, clustering and relevance feedback. In: Proceedings of the 4th International Conference of Knowledge Discovery and Data Mining, pp. 239–241. AAAI Press (1998)Google Scholar
  2. 2.
    Yang, X., Yu, F., Pedrycz, W.: Long-term forecasting of time series based on linear fuzzy information granules and fuzzy inference system. Int. J. Approximate Reasoning 81, 1–27 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhu, Y., Wu, D., Li, S.: A piecewise linear representation method of time series based on feature points. In: International Conference on Knowledge-based and Intelligent Information and Engineering Systems (2007)Google Scholar
  4. 4.
    Thurlimann, C.M., Durrenmatt, D.J., Villez, K.: Soft-sensing with qualitative trend analysis for waste water treatment plant control. Control Eng. Pract. 70, 121–133 (2018)CrossRefGoogle Scholar
  5. 5.
    Sammaknejad, N., Huang, B., Fatehi, A., Miao, Y., Xu, F., Espejo, A.: Adaptive monitoring of the process operation based on symbolic episode representation and hidden Markov models with application toward an oil sand primary separation. Comput. Chem. Eng. 71, 281–297 (2014)CrossRefGoogle Scholar
  6. 6.
    Janusz, M.E., Venkatasubramanian, V.: Automatic generation of qualitative description of process trends for fault detection and diagnosis. Eng. Appl. Artif. Intell. 4(5), 329–339 (1991)CrossRefGoogle Scholar
  7. 7.
    Charbonnier, S., Gentil, S.: A trend-based alarm system to improve patient monitoring in intensive care units. Control Eng. Pract. 15, 1039–1050 (2007)CrossRefGoogle Scholar
  8. 8.
    Sundarraman, A., Srinivasan, R.: Monitoringtransitions in chemical plants using enhanced trend analysis. Comput. Chem. Eng. 27, 1455–1472 (2003)CrossRefGoogle Scholar
  9. 9.
    Luo, L., Xi, C.: Integrating piecewise linear representation and weighted support vector machine for stock trading signal prediction. Appl. Soft Comput. J. 13(2), 806–816 (2013)CrossRefGoogle Scholar
  10. 10.
    Chang, P.C., Fan, C.Y., Liu, C.H.: Integrating a piecewise linear representation method and a neural network model for stock trading points prediction. IEEE Trans. Syst. Man Cybern. Part C 39(1), 80–92 (2008)CrossRefGoogle Scholar
  11. 11.
    Fitzgerald, W., Lemire, D., Brooks, M.: Quasi-monotonic segmentation of state variable behavior for reactive control. In: Proceedings of the National Conference on Artificial Intelligence, Part 3, vol. 20, pp. 1145–1150 (2005)Google Scholar
  12. 12.
    Skelton, A., Willms, A.R.: An algorithm for continuous piecewise linear bounding of discrete time series data BIT. Numer. Math. 54(4), 1155–1169 (2014)CrossRefGoogle Scholar
  13. 13.
    Keogh, E., Chu, S., Hart, D., et al.: Segmenting time series: a survey and novel approach. In: Data Mining in Time Series Databases (2003)Google Scholar
  14. 14.
    Kim, S.J., Koh, K., Boyd, S., Gorinevsky, D.: l1 trend filtering. SIAM Rev. 51(2), 339–360 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yang, L., Liu, S., Tsoka, S., Papegeorgiou, L.G.: Mathematical programming for piecewise linear regression analysis. Expert Syst. Appl. 44, 156–167 (2016)CrossRefGoogle Scholar
  16. 16.
    Zhou, B., Ye, H., Zhang, H., Li, M.: A new qualitative trend analysis algorithm based on global polynomial fit. AIChE J. 63(8), 3374–3383 (2017)CrossRefGoogle Scholar
  17. 17.
    Wang, Y.X., Sharpnack, J., Smola, A.J., et al.: Trend filtering on graphs. J. Mach. Learn. Res. 17(1), 3651–3691 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Duan, L., Yu, F., Pedrycz, W., Wang, X., Yang, X.: Time-series clustering based on linear fuzzy information granules. Appl. Soft Comput. J. 73, 1053–1067 (2018)CrossRefGoogle Scholar
  19. 19.
    History whether data base homepage. http://www.tianqihoubao.com/aqi/beijing.html. Accessed 13 Jan 2019
  20. 20.
    Wang, J., Yan, Yu., Chen, K.: Determining the number of segments for piece-wise linear representation of discrete-time signals. Comput. Chem. Eng. 120, 46–53 (2019)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Yang Xiyang
    • 1
    • 2
  • Zhang Jing
    • 1
  • Yu Fusheng
    • 2
  • Li Zhiwei
    • 1
    Email author
  1. 1.Key Laboratory of Intelligent Computing and Information Processing of Fujian ProvinceQuanzhou Normal UniversityQuanzhouChina
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingChina

Personalised recommendations