Journal of Korean Society of Disaster and Security (한국방재안전학회논문집)

Korean Society of Disaster & Security (한국방재안전학회)
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Derivation of Probability Plot Correlation Coefficient Test Statistics and Regression Equation for the GEV Model based on L-moments

L-모멘트 법 기반의 GEV 모형을 위한 확률도시 상관계수 검정 통계량 유도 및 회귀식 산정

  • 안현준 (연세대학교 대학원 토목환경공학과 통합과정) ;
  • 정창삼 (인덕대학교 토목환경공학과) ;
  • 허준행 (연세대학교 사회환경공학부 토목환경공학과)
  • Received : 2020.02.07
  • Accepted : 2020.02.28
  • Published : 2020.03.31
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Abstract

One of the important problem in statistical hydrology is to estimate the appropriated probability distribution for a given sample data. For the problem, a goodness-of-fit test is conducted based on the similarity between estimated probability distribution and assumed theoretical probability distribution. Probability plot correlation coefficient test (PPCC) is one of the goodness-of-fit test method. PPCC has high rejection power and its application is simple. In this study, test statistics of PPCC were derived for generalized extreme value distribution (GEV) models based on L-moments and these statistics were suggested by the multiple and nonlinear regression equations for its usability. To review the rejection power of the newly proposed method in this study, Monte Carlo simulation was performed with other goodness-of-fit tests including the existing PPCC test. The results showed that PPCC-A test which is proposed in this study demonstrated better rejection power than other methods, including the existing PPCC test. It is expected that the new method will be helpful to estimate the appropriate probability distribution model.

수문 통계 분야에서 관측된 자료를 대표할 수 있는 확률분포 모형을 추정하는 일은 매우 중요한 문제이다. 이를 위해 표본 자료로부터 추정되는 확률분포 모형과 가정된 이론적 확률분포 모형의 일치 정도를 통해 적합도 검정을 수행한다. 확률 도시 상관계수 검정(PPCC)은 적합도 검정 방법 중 하나로 적용 방법이 간편하면서도 높은 기각력을 가지고 있다. 본 연구에서는 L-모멘트 법 기반의 generalized extreme value(GEV) 분포 모형을 위한 PPCC의 검정 통계량을 유도하고 이를 다변량 비선형 형태의 회귀식으로 제시하였다. 새롭게 제시된 방법의 기각력을 검토하고자 기존의 적합도 검정 방법들과 모의실험을 수행하였으며 그 결과 본 연구에서 제시된 PPCC-A 검정 방법이 기존의 PPCC 검정을 비롯한 다른 적합도 검정 방법보다 우수한 기각력을 보이는 것으로 나타났다. 이를 통해 표본 자료를 좀 더 정확하게 대표할 수 있는 확률분포 모형을 구축하는 데 도움이 될 것으로 기대된다.

Keywords

References

  1. Ahn, H., Shin, H., Kim, S., and Heo, J. -H. (2014). Comparison on Probability Plot Correlation Coefficient Test Considering Skewness of Sample for the GEV Distribution. Journal of Korea Water Resources Association. 47(2): 161-170. https://doi.org/10.3741/JKWRA.2014.47.2.161
  2. Arnell, N. W., Beran, M., and Hosking, J. R. M. (1986). Unbiased Plotting Positions for the General Extreme Value Distribution. Journal of Hydrology. 86: 59-69. https://doi.org/10.1016/0022-1694(86)90006-5
  3. Blom, G. (1958). Statistical Estimates and Transformed Beta Variables. John Wiley and Sons. New York.
  4. Chowdhury, J. D., Stedinger, J. R., and Lu, L. H. (1991). Goodness-of-fit Tests for Regional Generalized ExTreme Value Flood Distributions. Water Resources Research. 27(7): 1765-1776. https://doi.org/10.1029/91WR00077
  5. Cunnane, C. (1978). Unbiased Plotting Positions-A Review. Journal of Hydrology. 37(3/4): 205-222. https://doi.org/10.1016/0022-1694(78)90017-3
  6. Filliben, J. J. (1969). Simple and Robust Linear Estimation of the Location Parameter of a Symmetric Distribution. Unpublished Ph.D. Dissertation, Princeton University. Princeton. New Jersey.
  7. Filliben, J. J. (1975). The Probability Plot Correlation Coefficient Test for Normality. Technometrics. 17(1): 111-117. https://doi.org/10.1080/00401706.1975.10489279
  8. Goel, N. K. and De, M. (1993). Development of Unbiased Plotting Position Formula for General Extreme Value Distribution. Stochastic Environmental Research and Risk Assessment. 7: 1-13.
  9. Gringorten, I. I. (1963). A Plotting Rule for Extreme Probability Paper. Journal of Geophysical Research. 68(3): 813-814. https://doi.org/10.1029/JZ068i003p00813
  10. Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press. New York, NY.
  11. Hazen, A. (1914). Storage to Be Provided in Impounding Reservoirs for Municipal Water Supply. Transactions American society of Civil Engineers. 1308(77): 1547-1550.
  12. Heo, J. -H., Kim, K. -D., and Han, J. -H. (1999). Derivation of Rainfall Intensity-duration-frequency Equation Based on the Approproate Probability Distribution. Journal of Korea Water Resources Association. 32: 247-254.
  13. Heo, J. -H., Kho, Y., Shin, H., Kim, S., and Kim, T. (2008). Regression Equations of Probability Plot Correlation Coefficient Test Statistics from Several Probability Distributions. Journal of Hydrology. 355: 1-15. https://doi.org/10.1016/j.jhydrol.2008.01.027
  14. Heo, J. -H., Kho, Y. -W., and Kim, K. -D. (2001). Test Statistics Derivation and Power Test for Probability Plot Correlation Coefficient Goodness of Fit Test. Journal of Korean Society of Civil Engineering. 21(2-b): 85-92.
  15. In-na, N., and Nguyen, V-T-V. (1989). An Unbiased Plotting Position Formula for the Generalized Extreme Value Distribution. Journal of Hydrology. 106: 193-209. https://doi.org/10.1016/0022-1694(89)90072-3
  16. Jenkinson, A. F. (1955). The Frequency Distribution of the Annual Maximum (or Minimum) Values of Meteorological Elements. Quarterly Journal of the Royal Meteorological Society. 87: 158-171. https://doi.org/10.1002/qj.49708134804
  17. KICT (2000). Research Report on the Development of Water Management Techniques, Vol. I. Preparation of Korea Probability Rainfall Diagram. Gwacheon: KICT.
  18. Kim, S., Shin, H., Joo, K., and Heo, J.-H. (2012). Development of Plotting Position for the General Extreme Value Distribution. Journal of Hydrology. 475: 259-269. https://doi.org/10.1016/j.jhydrol.2012.09.055
  19. Kim, S., Shin, H., Ahn. H., and Heo, J.-H. (2015). Development of an Unbiased Plotting Position Formula Considering the Coefficient of Skewness for the Generalized Logistic Distribution. Journal of Hydrology. 527: 471-481. https://doi.org/10.1016/j.jhydrol.2015.05.002
  20. Kimball, B. F. (1946). Assignment of Frequencies to a Completely Ordered Set of Sample Data. Transaction on the American Geophysical Union. 27: 843- 846. https://doi.org/10.1029/TR027i006p00843
  21. Looney, S. W. and Gulledge, T. R. (1985). Use the CorRelation Coefficient with Normal Probability Plots. The American Statistician. 39(1): 75-79. https://doi.org/10.2307/2683917
  22. Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E. (1993). Frequency Analysis of Extreme Events. Handbook of Hydrology. D. R. Maidment, de., McGraw-Hill. New York, N.Y. pp. 18.24-18.26
  23. Vogel, R. M. (1986). The Probability Plot Correlation Coefficient Test for the Normal, Lognormal, and Gumbel Distributional Hypotheses. Water Resources Research. 22(4): 587-590. https://doi.org/10.1029/WR022i004p00587
  24. Vogel, R. M. and McMartin, D. E. (1991). Probability Plot Goodness-of-fit and Skewness Estimation Procedures for the Pearson Type III Distribution. Water Resources Research. 27(12): 3149-3158. https://doi.org/10.1029/91WR02116