Skip to main content
SpringerLink
Log in

Long memory time series forecasting by using genetic programming

  • Published:
Genetic Programming and Evolvable Machines Aims and scope Submit manuscript

Abstract

Real-world time series have certain properties, such as stationarity, seasonality, linearity, among others, which determine their underlying behaviour. There is a particular class of time series called long-memory processes, characterized by a persistent temporal dependence between distant observations, that is, the time series values depend not only on recent past values but also on observations of much prior time periods. The main purpose of this research is the development, application, and evaluation of a computational intelligence method specifically tailored for long memory time series forecasting, with emphasis on many-step-ahead prediction. The method proposed here is a hybrid combining genetic programming and the fractionally integrated (long-memory) component of autoregressive fractionally integrated moving average (ARFIMA) models. Another objective of this study is the discovery of useful comprehensible novel knowledge, represented as time series predictive models. In this respect, a new evolutionary multi-objective search method is proposed to limit complexity of evolved solutions and to improve predictive quality. Using these methods allows for obtaining lower complexity (and possibly more comprehensible) models with high predictive quality, keeping run time and memory requirements low, and avoiding bloat and over-fitting. The methods are assessed on five real-world long memory time series and their performance is compared to that of statistical models reported in the literature. Experimental results show the proposed methods’ advantages in long memory time series forecasting.

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

Similar content being viewed by others

Notes

  1. As a consequence of the increase in the number of input lagged variables.

  2. The gamma function, denoted by \(\Upgamma\), is an extension of the factorial function to real and complex numbers.

  3. Defined by: B i  y t  = y t−i.

  4. In this case, in order to improve run time, a CI method that generates one or more \(\hat{d}\) solution values, is used.

  5. This approach is only used as a reference.

  6. Here, the best solution from the previous population is included in the current population.

References

  1. B. Ray, Modeling long-memory processes for optimal long-range prediction. J. Time Ser. Anal. 14(5), 511–525 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Barkoulas, C. Baum, Long-memory forecasting of US monetary indices. J. Forecast. 25(4) (2006)

  3. J. Geweke, S. Porter-Hudak, The estimation and application of long memory time series models. J. Time Ser. Anal. 4(4), 221–238 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Reisinger, K. Stanley, R. Miikkulainen, Evolving reusable neural modules. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2004), pp. 69–81

  5. P. Evangelista, M. Embrechts, B. Szymanski, Taming the curse of dimensionality in kernels and novelty detection. in Applied Soft Computing Technologies: The Challenge of Complexity, ed. by A. Abraham, B.D. Baets, M. Koppen, B. Nickolay (2006), pp. 431–444

  6. M. Verleysen, D. Francois, The Curse of Dimensionality in Data Mining and Time Series Prediction. Lecture Notes Computer Science (Springer, Berlin/Heidelberg, 2005), pp. 758–770

  7. A. Sorjamaa, J. Hao, N. Reyhani, Y. Ji, A. Lendasse, Methodology for long-term prediction of time series. Neurocomputing 70(16–18), 2861–2869 (2007)

    Article  Google Scholar 

  8. J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, Cambridge, MA, 1992)

    MATH  Google Scholar 

  9. S. Luke, L. Panait, A comparison of bloat control methods for genetic programming. Evol. Comput. 14(3), 309–344 (2006)

    Article  Google Scholar 

  10. P. Monsieurs, E. Flerackers, Reducing Bloat in Genetic Programming. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2001), pp. 471–478

  11. W. Langdon, R. Poli, Genetic programming bloat with dynamic fitness. in Proceedings of the First European Workshop on Genetic Programming (Springer, London, 1998), pp. 97–112

  12. P. Nordin, W. Banzhaf, Complexity compression and evolution. in Proceedings of the 6th International Conference on Genetic Algorithms (Morgan Kaufmann Publishers Inc., San Francisco, CA, 1995), pp. 310–317

  13. T. Blickle, L. Thiele, Genetic programming and redundancy. in Genetic Algorithms within the Framework of Evolutionary Computation (Workshop at KI-94, Saarbrucken), (Max-Planck-Institut für Informatik, 1994), pp. 33–38

  14. W. Langdon, Evolution of size in variable length representations. in The 1998 IEEE International Conference on Evolutionary Computation, ICEC’98 (1998), pp. 633–638

  15. K. Kinnear Jr, Generality and Difficulty in Genetic Programming: Evolving a Sort. in Proceedings of the 5th International Conference on Genetic Algorithms (Morgan Kaufmann Publishers Inc. San Francisco, CA, 1993), pp. 287–294

  16. J. Rosca, Generality versus size in genetic programming. in Genetic Programming 1996: Proceedings of the First Annual Conference (The MIT Press, Cambridge, MA, 1996), pp. 381–387

  17. L. Becker, M. Seshadri, Comprehensibility and Overfitting Avoidance in Genetic Programming for Technical Trading Rules. Worcester Polytechnic Institute, Computer Science Technical Report (2003)

  18. D. Robilliard, C. Fonlupt, Backwarding: An Overfitting Control for Genetic Programming in a Remote Sensing Application. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2002), pp. 245–254

  19. N. Nikolaev, L. de Menezes, H. Iba, Overfitting avoidance in genetic programming of polynomials. in Proceedings of the Evolutionary Computation on 2002. CEC’02. Proceedings of the 2002 Congress-Volume 02 (IEEE Computer Society, Washington, DC, 2002), pp. 1209–1214

  20. W. Langdon, R. Poli, Fitness Causes Bloat: Mutation. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 1998), pp. 37–48

  21. W. Banzhaf, W. Langdon, Some considerations on the reason for bloat. Gene. Prog. Evol. Mach. 3(1), 81–91 (2002)

    Article  MATH  Google Scholar 

  22. K. Man, Long memory time series and short term forecasts. Int. J. Forecast. 19(3), 477–491 (2003)

    Article  Google Scholar 

  23. H. Hurst, Long-term storage capacity of reservoirs. in Transactions of the American Society of Civil Engineers, vol. 116 (American Society of Civil Engineering, 1951), pp. 770–808

  24. Z. Ding, C. Granger, R. Engle, A long memory property of stock market returns and a new model. Econ. Soc. Monogr. 33, 349–372 (2001)

    Google Scholar 

  25. N. Crato, de P. Lima, Long-range dependence in the conditional variance of stock returns. Econ. Lett. 45(3), 281–285 (1994)

    Article  MATH  Google Scholar 

  26. A. Lo, Long-term memory in stock market prices. Econom. J. Econ. Soc. 1279–1313 (1991)

  27. T. Bollerslev, H. Ole Mikkelsen, Modeling and pricing long memory in stock market volatility. J. Econom. 73(1), 151–184 (1996)

    Article  MATH  Google Scholar 

  28. U. Hassler, J. Wolters, Long memory in inflation rates: International evidence. J. Bus. Econ. Stat. 37–45 (1995)

  29. R. Baillie, C. Chung, M. Tieslau, Analysing inflation by the fractionally integrated ARFIMA-GARCH model. J. Appl. Econom. 11(1), 23–40 (1996)

    Article  Google Scholar 

  30. J. Doornik, M. Ooms, Inference and forecasting for ARFIMA models with an application to US and UK inflation. Stud. Nonlinear Dyn. Econom. 8(2), 1208–1218 (2004)

    Google Scholar 

  31. R. Baillie, S. Chung, Modeling and forecasting from trend-stationary long memory models with applications to climatology. Int. J. Forecast. 18(2), 215–226 (2002)

    Article  Google Scholar 

  32. R. Caballero, S. Jewson, A. Brix, Long memory in surface air temperature detection, modeling, and application to weather derivative valuation. Climat. Res. 21(2), 127–140 (2002)

    Article  Google Scholar 

  33. H. Hamisultane, Pricing the Weather Derivatives in the Presence of Long Memory in Temperatures. Technical report, Working paper, EconomiX, Nanterre (2006)

  34. J. Hosking, Modeling persistence in hydrological time series using fractional differencing. Water Resour. Res. 20(12), 1898–1908 (1984)

    Article  Google Scholar 

  35. M. Ooms, P. Franses, A seasonal periodic long memory model for monthly river flows. Environ. Modell. Softw. 16(6), 559–569 (2001)

    Article  Google Scholar 

  36. W. Wang, P. Van Gelder, J. Vrijling, Long-memory in streamflow processes of the Yellow river. in IWA International Conference on Water Economics, Statistics, and Finance (2005), pp. 8–10

  37. B. Mandelbrot, J. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 422–437 (1968)

  38. B. Mandelbrot, A fast fractional gaussian noise generator. Water Resour. Res. 7(3), 543–553 (1971)

    Article  Google Scholar 

  39. C. Granger, R. Joyeux, An Introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1(1), 15–29 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Hosking, Fractional differencing. Biometrika 68(1), 165–176 (1981)

    Article  MathSciNet  Google Scholar 

  41. P. Bloomfield, An exponential model for the spectrum of a scalar time series. Biometrika 60(2), 217–226 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Beran, Statistics for long-memory processes (Chapman & Hall/CRC, London, 1994)

    MATH  Google Scholar 

  43. Y. Yajima, On estimation of long-memory time series models. Aust. NZ. J. Stat. 27(3), 303–320 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Fox, M. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. Annal. Stat. 14(2), 517–532 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  45. W. Li, A. McLeod, Fractional time series modelling (Biometrika Trust, Great Britain, 1986)

  46. P. Robinson, Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econ. 47(1), 67–84 (1991)

    MATH  Google Scholar 

  47. P. Robinson, Efficient tests of nonstationary hypotheses. J. Am. Stat. Assoc. (1994), 1420–1437

  48. F. Sowell, Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J. Econom. 53(1), 2 (1992)

    Article  MathSciNet  Google Scholar 

  49. G. Bhardwaj, N. Swanson, An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series. J. Econom. 131(1-2), 539–578 (2006)

    Article  MathSciNet  Google Scholar 

  50. J. Barkoulas, C. Baum, Long memory and forecasting in Euroyen deposit rates. Asia-Pac Financ. Mark. 4(3), 189–201 (1997)

    MATH  Google Scholar 

  51. R. Baillie, Long memory processes and fractional integration in econometrics. J. Econom. 73(1), 5–59 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. J. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, Ann Arbor, MI, 1975)

    Google Scholar 

  53. L. Fogel, A. Owens, M. Walsh, Artificial Intelligence Through Simulated Evolution (Wiley, New York, 1966)

    MATH  Google Scholar 

  54. R. Storn, K. Price, Differential Evolution-a Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. International Computer Science Institute-Publications-TR (1995)

  55. P. Smith, Controlling Code Growth in Genetic Programming. Advances in Soft Computing (2000), pp. 166–171

  56. R. Olsson, Inductive functional programming using incremental program transformation. Artif. Intell. 74(1), 55–81 (1995)

    Article  Google Scholar 

  57. T. Soule, J. Foster, Effects of code growth and parsimony pressure on populations in genetic programming. Evol. Comput. 6(4), 293–309 (1998)

    Article  Google Scholar 

  58. B. Zhang, H. Mühlenbein, Balancing accuracy and parsimony in genetic programming. Evol. Comput. 3(1), 17–38 (1995)

    Article  Google Scholar 

  59. S. Luke, L. Panait, Fighting Bloat with Nonparametric Parsimony Pressure. Lecture Notes in Computer Science (2003), pp. 411–421

  60. W. Langdon, Data structures and genetic programming. Adv. Genet. Prog. 2, 395–414 (1996)

    Google Scholar 

  61. E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)

    Article  Google Scholar 

  62. E. De Jong, R. Watson, J. Pollack, Reducing bloat and promoting diversity using multi-objective methods. in Proceedings of the Genetic and Evolutionary Computation Conference (2001), pp. 11–18

  63. W. Langdon, J. Nordin, Seeding Genetic Programming Populations. Lecture Notes in Computer Science (2000), pp. 304–315

  64. A. Ekárt, S. Nemeth, Selection based on the pareto nondomination criterion for controlling code growth in genetic programming. Gene. Prog. Evol. Mach. 2(1), 61–73 (2001)

    Article  MATH  Google Scholar 

  65. E. De Jong, J. Pollack, Multi-objective methods for tree size control. Gene. Prog. Evol. Mach. 4(3), 211–233 (2003)

    Article  Google Scholar 

  66. A. Guven, Linear genetic programming for time-series modelling of daily flow rate. J. Earth Syst. Sci. 118(2), 137–146 (2009)

    Article  Google Scholar 

  67. H. Akaike, A new look at the statistical identification model. IEEE Trans. Auto. Cont. 19(6), 716–723 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  68. N. Nikolaev, L.M. de Menezes, H. Iba, Overfitting avoidance in genetic programming of polynomials. in Proceedings of the 2002 Congress on Evolutionary Computation CEC2002 (IEEE Press, 2002), pp. 1209–1214

  69. H. Iba, Bagging, boosting, and bloating in genetic programming. in Proceedings of the Genetic and Evolutionary Computation Conference, Vol. 2. (1999), pp. 1053–1060

  70. B.S. Mulloy, R.L. Riolo, R.S. Savit, Dynamics of genetic programming and chaotic time series prediction. in Genetic Programming 1996: Proceedings of the First Annual Conference, Stanford University, CA, (MIT Press, 1996), pp. 166–174

  71. H. Iba, de H. Garis, T. Sato, A numerical approach to genetic programming for system identification. Evol. Comput. 3(4), 417–452 (1995)

    Article  Google Scholar 

  72. N. Wagner, Z. Michalewicz, M. Khouja, R.R. McGregor, Time series forecasting for dynamic environments: The DyFor genetic program model. IEEE Trans. Evol. Comput. 11(4), 433–452 (2007)

    Article  Google Scholar 

  73. R. Jagielski, Genetic programming prediction of solar activity. Intelligent Data Engineering and Automated LearningIDEAL 2000. Data Mining, Financial Engineering, and Intelligent Agents (2009), pp. 191–210

  74. N. Nikolaev, H. Iba, Genetic programming of polynomial harmonic models using the discrete Fourier transform. in Evolutionary Computation, 2001. Proceedings of the 2001 Congress on. Vol. 2. (IEEE, 2002), pp. 902–909

  75. N. Nikolaev, H. Iba, Regularization approach to inductive genetic programming. Evol. Comput. IEEE Trans. 5(4), 359–375 (2002)

    Article  Google Scholar 

  76. H. Iba, N. Nikolaev, Genetic programming polynomial models of financial data series. in Evolutionary Computation, 2000. Proceedings of the 2000 Congress on. Vol. 2. (IEEE, 2002), pp. 1459–1466

  77. R. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application (Wiley, New York, 1986)

    MATH  Google Scholar 

  78. K. Deb, Multi-objective Optimization Using Evolutionary Algorithms (Wiley, New York, 2001)

    MATH  Google Scholar 

  79. A. Konak, D. Coit, A. Smith, Multi-objective optimization using genetic algorithms: a tutorial. Reliab. Eng. Syst. Safe. 91(9), 992–1007 (2006)

    Article  Google Scholar 

  80. C. Coello, G. Lamont, D. Van Veldhuizen, Evolutionary Algorithms for Solving Multi-objective Problems (Springer, New York , 2007)

    MATH  Google Scholar 

  81. J. Doornik, Object-Oriented Matrix Programming using Ox 6.0 (2007) London: Timberlake Consultants Ltd. See http://www.doornik.com

  82. J. Doornik, M. Ooms, Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models. Comput. Stat. Data Anal. 42(3), 333–348 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  83. A. Fraser, Genetic Programming in C++ (Gpc++ Version 0.40). Public Domain Genetic Programming System (1994)

  84. R Development Core Team, R: A Language and Environment for Statistical Computing. (R Foundation for Statistical Computing, Vienna, Austria, 2010) ISBN 3-900051-07-0

Download references

Acknowledgments

I would like to thank the anonymous reviewers for their review, comments, and suggestions. The author acknowledges the financial support, offered through a doctoral fellowship, from CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina).

Author information

Authors and Affiliations

  1. Departamento de Informática, Universidad Nacional de San Luis, Ejército de los Andes 950, San Luis, D5700HHW, Argentina

    Emiliano Carreño Jara

Authors
  1. Emiliano Carreño Jara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emiliano Carreño Jara.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carreño Jara, E. Long memory time series forecasting by using genetic programming. Genet Program Evolvable Mach 12, 429–456 (2011). https://doi.org/10.1007/s10710-011-9140-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10710-011-9140-7

Keywords

Search

Navigation