KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
권호 표지
DOI QR코드

DOI QR Code

Dynamic Nonlinear Prediction Model of Univariate Hydrologic Time Series Using the Support Vector Machine and State-Space Model

Support Vector Machine과 상태공간모형을 이용한 단변량 수문 시계열의 동역학적 비선형 예측모형

  • Received : 2005.11.25
  • Accepted : 2006.01.17
  • Published : 2006.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

The reconstruction of low dimension nonlinear behavior from the hydrologic time series has been an active area of research in the last decade. In this study, we present the applications of a powerful state space reconstruction methodology using the method of Support Vector Machines (SVM) to the Great Salt Lake (GSL) volume. SVMs are machine learning systems that use a hypothesis space of linear functions in a Kernel induced higher dimensional feature space. SVMs are optimized by minimizing a bound on a generalized error (risk) measure, rather than just the mean square error over a training set. The utility of this SVM regression approach is demonstrated through applications to the short term forecasts of the biweekly GSL volume. The SVM based reconstruction is used to develop time series forecasts for multiple lead times ranging from the period of two weeks to several months. The reliability of the algorithm in learning and forecasting the dynamics is tested using split sample sensitivity analyses, with a particular interest in forecasting extreme states. Unlike previously reported methodologies, SVMs are able to extract the dynamics using only a few past observed data points (Support Vectors, SV) out of the training examples. Considering statistical measures, the prediction model based on SVM demonstrated encouraging and promising results in a short-term prediction. Thus, the SVM method presented in this study suggests a competitive methodology for the forecast of hydrologic time series.

최근에 수문시계열로부터 저차원의 비선형 거동을 재구성하고자 하는 연구가 활발히 진행되고 있다. 이러한 관점에서 본 연구에서는 Support Vector Machine(SVM)을 이용하여 우수한 상태-공간 재구성 능력을 갖는 비선형 예측모형을 구성하여 Great Salt Lake(GSL) Volume에 적용하였다. SVM은 Kernel 함수로부터 유도된 고차원의 특성공간 안에서 선형함수의 가상공간을 이용하는 Machine Learning 방법론이다. 또한 SVM은 훈련자료로부터 얻어지는 평균제곱오차가 아닌 일반화된 오차를 최소화함으로써 상대적으로 기존 방법에 비해 적은 수의 매개변수와 과적합(over fitting)을 피하면서 비선형 함수의 최적화가 가능하다. 본 연구에서 제시한 SVM 회귀분석의 적용성은 미국의 GSL의 2주 간격 Volume을 대상으로 검토하였다. SVM을 이용한 비선형 예측모형은 GSL Volume의 2주(1-Step), 8주(4-Step)와 반복예측(Iterated Prediction, 121-Step)까지 적용되었다. 본 연구에서는 극치사상 즉, 급격한 감소 및 증가 구간을 예측하는데 있어서 훈련구간과 예측구간을 구분하여 모형의 신뢰성을 평가하였다. 예측결과SVM은 훈련자료로부터 적은 수의 관측치를 이용하여 동역학적 거동을 추출할 수 있었으며 실제 관측자료와 거의 유사한 예측이 가능함을 통계적 지표로 확인할 수 있었다. 따라서 비선형 수문시계열의 단기 예측을 위한 모형으로 적용이 가능할 것으로 판단된다.

Keywords

References

  1. 김형수, 최시중, 김중훈(1998) DVS 알고리즘을 이용한 일 유량 자료의 예측, 대한토목학회논문집, 대한토목학회, 제18권 제116호, pp. 563-570
  2. 권현한, 문영일(2005) 상태-공간 모형과 Nearest Neighbor 방법을 통한 수문시계열 예측에 관한 연구, 대한토목학회 논문집, 대한토목학회, 제25권 제4B호, pp. 275-283
  3. 문영일(1997) 시계열 수문자료의 비선형 상관관계. 한국수자원학회논문집, 한국수자원학회, 제30권, pp. 641-648
  4. 문영일(2000) 지역가중다항식을 이용한 예측모형, 한국수자원학회 논문집, 한국수자원학회, 제33권, pp. 31-38
  5. 박무종, 윤용남(1989) Multiplicative ARIMA 모형에 의한 월유량의 추계학적 모의 예측, 한국수자원학회논문집, 한국수자원학회, 제22권, pp. 331-339
  6. 안상진, 이재경(2000) 추계학적 모의발생기법을 이용한 월 유출 예측, 한국수자원학회논문집, 한국수자원학회, 제33권, pp. 159-167
  7. 윤강훈, 서봉철, 신현석(2004) 신경망을 이용한 낙동강 유역 홍수기 댐유입량예측, 한국수자원학회논문집, 한국수자원학회,제37권. pp. 67-75
  8. Abarbanel, H.D.I. and Lall, U. (I996) Nonlinear dynamics and the Great Salt Lake: system identification and prediction. Climate Dynamics, Vol. 12. pp. 287-297 https://doi.org/10.1007/BF00219502
  9. Abarbanel, H.D.I., Lall, U., Moon, Y.I, Mann, M., and Sangoyomi, T. (1996) Nonlinear dynamic of the great salt lake: a predictable indicator of regional climate, Energy, Vol. 21 (7/8), pp. 655-665 https://doi.org/10.1016/0360-5442(96)00018-7
  10. Boser, B.E., Guyon, I., and Vapnik, V. (1992) A training algorithm for optimal margin classifiers, Proceedings Fifth ACM Workshop on Computational Learning Theory, pp. 144-152
  11. Cortes, C. and Vapnik, V. (1995). Support vector networks. Machine Learning, 20, pp. 273-297
  12. Cristianini, N. and Shawe-Taylor, J. (2000) An introduction to support vector machines and other kernel based learning methods. Cambridge University Press
  13. Dibike, B.Y., Velickov, S., Solomatine, D. and Abbot, B.M. (2001) Model induction with support vector machines: Introduction and applications, J Computing in Civil Eng., Vol. 15, No.3, pp. 208-216 https://doi.org/10.1061/(ASCE)0887-3801(2001)15:3(208)
  14. Drucker, H., Burges, C.J.C., Kaufman, L., Smola, A. and Vapnik, V. (1997) Support vector regression machines. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems, Vol. 9, pp. 155-161, Cambridge, MA, MIT Press
  15. Girosi, F. (1998) An equivalence between sparse approximation and support vector machines, Neural Computation, Vol. 10, No.6, pp.1455-1480 https://doi.org/10.1162/089976698300017269
  16. Guyon, I., Boser, B., and Vapnik, V, (1993) Automatic capacity tuning of very large VC-dimension classifiers. In Stephen Jose Hanson, Jack D. Cowan, and C. Lee Giles, editors, Advances in Neural Information Processing Systems, Vol. 5, pp. 147-155. Morgan Kaufmann, San Mateo, CA
  17. Hense, A. (1987) On the possible existence of a strange attractor for the southern oscillation. Beitr Phys. Atmos. Vol. 60, No. 1, pp. 34-47
  18. Hilborn, R.C. (1994) Chaos and Nonlinear Dynamics, Oxford University Press
  19. Huber, R, (1964) Robust estimation of a location parameter, Annals of Mathematical Statistics, Vol. 35, No. I, pp. 73-101 https://doi.org/10.1214/aoms/1177703732
  20. Jayawardena, A. W., and Lai, F. (1994) Analysis and prediction of chaos in rainfall and stream flow time series, J. Hydrol., Vol. 153, pp. 23-52 https://doi.org/10.1016/0022-1694(94)90185-6
  21. Kember, G, Flower, A.C., and Holubeshen, J. (1993) Forecasting river flow using nonlinear dynamics, Sthoch. Hydrol. Hydraul., Vol. 7, pp. 205-212 https://doi.org/10.1007/BF01585599
  22. Lall, U. and Mann, M.E. (1995) The great salt lake: a barometer of low-frequency climatic variability. Water Resour. Res., Vol. 31, No. 10, pp. 2503-2515 https://doi.org/10.1029/95WR01950
  23. Lall, U., Sangoyomi, T, and Abarbanel, H.D.I. (1996) nonlinear dynamics of the great salt lake: nonparametric short-term forecasting. Water Resour. Res., Vol. 32, No.4, pp. 975-985 https://doi.org/10.1029/95WR03402
  24. Liong S.Y. and Sivapragasam, C. (2002) flood stage forecasting with SVM, J. AWRA, Vol. 38, No.1, pp. 173-186
  25. Lorenz, E.N. (1963) Deterministic nonperiodic flow. J. Atmos. Sci. Vol. 20, 130-141 https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
  26. Ljung, G.M and Box, G.E.P. (1978) On a measure of lack of fit in time-series models, Biometrika, Vol. 65. pp. 297-303 https://doi.org/10.1093/biomet/65.2.297
  27. Mackey, M.C. and Glass, L. (1977) Oscillation and chaos in physiological control systems. Science, Vol. 197, pp. 287-289 https://doi.org/10.1126/science.267326
  28. Mann M.E, Lall, U., and Saltzman, B. (1995) Decadal-to-centennial-scale climate variability: Insight into rise and fall of the Great Salt Lake. Geophysical Res. Let., Vol. 22, No.8, pp. 937-940 https://doi.org/10.1029/95GL00704
  29. Mattera, D. and Haykin, S. (1999) Support vector machines for dynamic reconstruction of a chaotic system. In B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods: Support Vector Learning, pp. 211-242, Cambridge, MA, MIT Press
  30. Moon, Y.I. and Lall, U. (1996) Large scale atmospheric indices and the great salt lake: interannual and interdecadal variability, ASCE, J. of Hydrologic Eng., Vol. 1, No.2, pp. 55-62 https://doi.org/10.1061/(ASCE)1084-0699(1996)1:2(55)
  31. Muller, K.R., Smola, A., Ratsch, G., Scholkopf, B., Kohlmorgen, J., and Vapnik, V. (1999) Predicting time series with support vector machines. In B. Scholkopf, C.J.C. Burges, and A.J. Smola, editors, Advances in Kernel Methods: Support Vector Learning, pp. 243-254, Cambridge, MA, MIT Press
  32. Rodriguez-Iturbe I., De Power, F.B., Sharifi, M.B., and Georgakakos, K.P. (1989) Chaos in rainfall, Water Resour. Res., Vol. 25, No.7, pp. 1667-1675 https://doi.org/10.1029/WR025i007p01667
  33. Sangoyomi, T. (1993) Climate variability and dynamics of Great Salt Lake hydrology, PhD dissertation. 247pp., Utah State Univ., Logan
  34. Sangoyomi, T, Lall, U., and Abarbanel, H.D.I. (1996) Nonlinear dynamics of the Great Salt Lake: Dimension estimation. Water Resour. Res., Vol. 32, No. 1, pp. 149-1599 https://doi.org/10.1029/95WR02872
  35. Sauer, Y., Yorke, J.A, and Casdagli, M. (1991) Embedology. Journal of Statistical Physics, Vol. 65, pp. 579-616 https://doi.org/10.1007/BF01053745
  36. Scholkopf, B., Burges, C., and Vapnik, V. (1995) Extracting support data for a given task. In U.M. Fayyad and R. Uthurusamy, editors, Proceedings, First International Conference on Knowledge Discovery & Data Mining, Menlo Park, AAAI Press
  37. Smith, J.A. (1991) Long-range streamflow forecasting using non-parametric regression, Water Resour, Bull., Vol. 27, No.1, pp. 39-46 https://doi.org/10.1111/j.1752-1688.1991.tb03111.x
  38. Smola, A.J. (1998) Learning with kernels. Ph.D. thesis. Technischen Universitat Berlin, Berlin, Germany
  39. Smola, A.J. and Scholkopf, B. (1998) A tutorial on support vector regression. NeuroCOLT2 Technical Report Series, NC2-TR-1998-030
  40. Takens, F. (1981) Detecting strange attractors in turbulence. In, Rand, D.A. and L.S. Young (eds.). Dynamical systems and Turbulence. Springer-Verlag. Berlin, pp. 366-381
  41. Tikhonov, A. and Arsenin, V. (1977) Solution of ill-posed problems, Washington, D.C. W.H. Winston
  42. Vapnik, V. (1995) The nature of statistical learning theory. Springer, New York
  43. Vapnik, V. (1998) Statistical learning theory. Wiley, New York
  44. Vapnik, V., Golowich, S., and Smola, A. (1997) Support vector method for function approximation, regression estimation, and signal processing. In M. Mozer, M. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 281-287, Cambridge, MA, MIT Press
  45. Yakowitz, S., and Karlsson, M. (1987) Nearest neighbor methods with application to rainfall/runoff prediction, Stochastic hydrology, Edited by Macneil, J.B., and Humphries, G.J., D. Reidel, Hingham, MA, pp. 149-160