Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
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Nonlinear Autoregressive Modeling of Southern Oscillation Index

비선형 자기회귀모형을 이용한 남방진동지수 시계열 분석

  • Published : 2006.12.31
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Abstract

We have presented a nonparametric stochastic approach for the SOI(Southern Oscillation Index) series that used nonlinear methodology called Nonlinear AutoRegressive(NAR) based on conditional kernel density function and CAFPE(Corrected Asymptotic Final Prediction Error) lag selection. The fitted linear AR model represents heteroscedasticity, and besides, a BDS(Brock - Dechert - Sheinkman) statistics is rejected. Hence, we applied NAR model to the SOI series. We can identify the lags 1, 2 and 4 are appropriate one, and estimated conditional mean function. There is no autocorrelation of residuals in the Portmanteau Test. However, the null hypothesis of normality and no heteroscedasticity is rejected in the Jarque-Bera Test and ARCH-LM Test, respectively. Moreover, the lag selection for conditional standard deviation function with CAFPE provides lags 3, 8 and 9. As the results of conditional standard deviation analysis, all I.I.D assumptions of the residuals are accepted. Particularly, the BDS statistics is accepted at the 95% and 99% significance level. Finally, we split the SOI set into a sample for estimating themodel and a sample for out-of-sample prediction, that is, we conduct the one-step ahead forecasts for the last 97 values (15%). The NAR model shows a MSEP of 0.5464 that is 7% lower than those of the linear model. Hence, the relevance of the NAR model may be proved in these results, and the nonparametric NAR model is encouraging rather than a linear one to reflect the nonlinearity of SOI series.

본 연구에서는 조건부 핵밀도함수와 CAFPE(Corrected Asymptotic Final Prediction Error) 차수결정 방법에 근거한 비매개변수적 비선형 자기회귀 (Nonlinear AutoRegressive, NAR) 모형을 소개하고 이를 SOI(Southern Oscillation Index)에 적용하였다. SOI 자료에 대해서 선형 AR 모형을 적용하였으나 잔차에 대한 검정결과 이분산성(heteroscedasticity)을 나타내었다. 또한 BDS(Brock-Dechert-Sheinkman) 검정에서 비선형성이 존재함을 확인하였다. 따라서 NAR 모형에 SOI 자료를 적용시켰다. CAFPE를 이용하여 가장 적합한 모형으로 지체 1, 2와 4가 선택되었으며 조건부 평균함수를 추정하여 SOI 자료를 모의한 결과 잔차에 대해서 정규성과 이분산성 가정이 Jarque-Bera 검정과 ARCH-LM 검정에서 각각 기각되었으며 또한 조건부 표준편차함수의 최적 차수로 3, 8과 9가 CAPFE를 통해 선택되었다. 조건부 평균함수와 표준편차함수를 모두 고려한 모형에 대한 잔차 검정 결과 잔차의 I.I.D 가정을 만족하였으며 특히, BDS 검정에서 신뢰구간 95%와 99%에서 모두 만족한 결과를 나타내었다. 마지막으로 전체의 15%에 해당하는 SOI 자료에 대해서 One-Step 예측을 수행하였으며 선형 모형에 비해 평균제곱예측오차가 7% 적게 나타났다. 따라서, NAR 모형은 여타의 매개변수적 방법과 달리 모형 선택에 있어 자유로우며 비선형성을 고려할 수 있는 모형으로서 SOI 자료와 같은 비선형 자료를 위한 모의방법으로 선형 모형에 비해 많은 장점을 가지고 있다.

Keywords

References

  1. 권현한, 문영일 (2006). '상태-공간 모형과 Nearest Neighbor 방법을 통한 수문시계열 예측에 관한 연구.', 대한토목학회 논문집, 대한토목학회, Vol.25 No.4B, pp. 275-283
  2. 문영일, 이정규 (1995). '변동핵 추정법에 의한 비매개 변수적 홍수빈도 분석.', 대한토목학회 논문집, 대한토목학회, Vol.15 No.2, pp. 413-424
  3. 문영일, 권현한 (2004). '수리.수문학적 댐 위험도 분석 (I) -비매개변수적 LHS Monte Carlo Simulation 위험도 해석 기법 개발.', 대한토목학회 논문집, 대한토목학회, Vol.24 No.5B, pp. 477-487
  4. 문영일, 권현한 (2004). '수리.수문학적 댐 위험도 분석 (II) - 수리.수문학적 댐 위험도 분석(II)-댐 위험도 해석 예시.', 대한토목학회 논문집, 대한토목학회, Vol.24 No.5B, pp. 489-500
  5. Akaike, H. (1969). 'Fitting autoregressive models for prediction.', Annals of the Institute of Statistical Mathematics, Vol. 21, pp. 243-247 https://doi.org/10.1007/BF02532251
  6. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, in B. Petrov & F. Csaki (eds.), 2nd International Symposium on Information Theory, Academiai Kiado, Budapest, pp. 267-281
  7. Ahn, J.H. and Kim, H. S. (2005). 'Nonlinear Modeling of El Nino/Southern Oscillation Index.', Journal of Hydrologic Engineering, Vol. 10, No. 1. pp. 8-15 https://doi.org/10.1061/(ASCE)1084-0699(2005)10:1(8)
  8. Brock, W. A, Hsieh, D. A., and LeBaron, B. (1991). Nonlinear dynamics, chaos, and instability: Statistical theory and economic evidence, MIT Press, Cambridge, USA
  9. Brock, W. A, Dechert, W. D., LeBaron, B., and Scheinkman, J.A. (1997). 'A Test for Independence Based on the Correlation Dimension.', Econometric Reviews, Vol. 15, pp. 197-235 https://doi.org/10.1080/07474939608800353
  10. Chu, P. S., and Katz, R. W. (1985). 'Modeling and forecasting the southern oscillation: A time-domain approach.', Mon. Weather Rev., Vol. 113, pp. 1876-1888 https://doi.org/10.1175/1520-0493(1985)113<1876:MAFTSO>2.0.CO;2
  11. Elshorbagy, A., Simonovic, S. P. and Panu, U. S. (2002). 'Estimation of missing streamflow data using principles of chaos theory.', J. Hydrol., Vol. 255, pp. 123- 133 https://doi.org/10.1016/S0022-1694(01)00513-3
  12. Engel, R. F. (1982). 'Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.', J. Econometr., Vol. 50, pp. 987 - 1007 https://doi.org/10.2307/1912773
  13. Engel, R. F. (1983). 'Estimates of the variance of U.S. inflation based upon the ARCH model.', J. Money, Credit, Banking, Vol. 15, pp. 287 - 301
  14. Hannan, E. J. and Quinn, B. G. (1979). 'The determination of the order of an autoregression.', Journal of the Royal Statistical Society, pp. 190-195
  15. Hense, A. (1987). 'On the possible existence of a strange attractor for the southern oscillation.', Beitr. Phys. Atmos., Vol. 60, pp. 34-47
  16. 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
  17. Jarque, C. M. and Bera, A. K. (1987). 'A test for normality of observations and regression residuals.', International Statistical Review, Vol. 41, pp. 1781-1816
  18. Jayawardena, A. W., Li, W. K. and Xu P. (2002). 'Neighborhood selection for local modeling and prediction of hydrological time series.' J. Hydrol., Vol. 258, pp. 40-57 https://doi.org/10.1016/S0022-1694(01)00557-1
  19. Kim, H. S., Kang, D. S., and Kim, J. H. (2003). 'The BDS statistic and residual test.' Stoch. Environ. Res. Risk Assess., Vol. 17, pp. 104 -115 https://doi.org/10.1007/s00477-002-0118-0
  20. Kwon, H.-H. and Y.-I. (2000). Moon, 'Improvement of Overtopping Risk Evaluations Using Probabilistic Concepts for Existing Dams.', Stochastic Environmental Research and Risk Assessment, Vol. 20, pp. 223-237 https://doi.org/10.1007/s00477-005-0017-2
  21. 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, pp. 975-985 https://doi.org/10.1029/95WR03402
  22. Lall, U. and Mann, M. E. (1995). 'The Great Salt Lake: A barometer of low-frequency climatic variability.' Water Resour. Res., Vol. 31, pp. 2503-2515 https://doi.org/10.1029/95WR01950
  23. 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
  24. Lomnicki, Z.A. (1961). 'Tests for departure from normality in the case of linear stochastic processes.', Metrika, Vol. 4, pp. 37-62 https://doi.org/10.1007/BF02613866
  25. Rissanen, J. (1978). 'Modeling by shortest data description.', Automatica, Vol. 14, pp. 465 - 471 https://doi.org/10.1016/0005-1098(78)90005-5
  26. Rodriguez-Iturbe I., De Power, F. B., Sharifi, M. B. and Georgakakos, K. P. (1989). 'Chaos in rainfall.', Water Resour. Res., Vol. 25, pp. 1667-1675 https://doi.org/10.1029/WR025i007p01667
  27. Schwarz, G. (1978). 'Estimating the dimension of a model.', Annals of Statistics, Vol. 6, pp. 461 - 464 https://doi.org/10.1214/aos/1176344136
  28. Silverman, B. W. (1996). 'Density Estimation for Statistics and Data Analysis.', Chapman & Hall, London
  29. Terrence C. and Compo GP. (1998). 'A practical guide to wavelets analysis.', Bulletin of the American Meteorological Society, Vol. 79, No.1, pp. 61-78 https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2
  30. Trenberth, K. E., and T. J. Hoar. (1996). 'The 1990-1995 El Nino - Southern Oscillation event: Longest on record.', Geophys. Res. Lett., 23, 57 - 60 https://doi.org/10.1029/95GL03602
  31. Tschernig, R. and Yang, L. (2000a). Nonparametric estimation of generalized impulse response functions, SFB 373 Discussion Paper 89, 2000, Humboldt-University, Berlin
  32. Tschernig, R. and Yang, L. (2000b). 'Nonparametric lag selection for time series.', Journal of Time Series Analysis, Vol. 21, pp. 456-487
  33. Yang, L. and Tschernig, R. (2000b). 'Multivariate bandwidth selection for local linear regression.', Journal of the Royal Statistical Society, Series B, Vol. 61, pp. 793-815