Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
권호 표지
DOI QR코드

DOI QR Code

Derivation of Plotting Position Formulas Considering the Coefficients of Skewness for the GEV Distribution

왜곡도 계수를 고려한 GEV 분포의 도시위치공식 유도

  • Kim, Soo-Young (School of Civil and Environmental Engineering, Yonsei University) ;
  • Heo, Jun-Haeng (School of Civil and Environmental Engineering, Yonsei University) ;
  • Choi, Min-Young (School of Civil and Environmental Engineering, Yonsei University)
  • 김수영 (연세대학교 사회환경시스템공학부) ;
  • 허준행 (연세대학교 사회환경시스템공학부) ;
  • 최민영 (연세대학교 사회환경시스템공학부)
  • Received : 2010.11.10
  • Accepted : 2011.01.10
  • Published : 2011.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

Probability plotting position is generally used for the graphical analysis of the annual maximum quantile and the estimation of exceedance probability to display the fitness between sample and an appropriate probability distribution. In addition, it is used to apply a specific goodness of fit test. Plotting position formula to define the probability plotting position has been studied in many researches. Especially, the GEV distribution which is an important probability distribution to analyze the frequency of hydrologic data was popular. In this study, the theoretical reduced variates are derived using the mean value of order statistics to derived an appropriate plotting position formula for the GEV distribution. In addition, various forms of plotting position formula considering various sample sizes and coefficients of skewness related with shape parameters are applied. The parameters of plotting position formulas are estimated using the genetic algorithm. The accuracy of derived plotting position formula is estimated by the errors between the theoretical reduced variates and those by various plotting position formulas including the derived ones in this study. As a result, the errors by derived plotting position formula is the smallest at the range of shape parameter with -0.25~0.10.

연최대수문량의 도시적 분석에 주로 이용되어 온 확률도시위치는 표본자료와 적정 확률분포형의 적합도를 표시하여 초과확률을 산정할 수 있도록 하며, 일부 적합도 검정에도 사용되기도 한다. 확률도시위치를 결정하는 도시위치공식은 오래 전부터 꾸준히 연구되어 왔는데, 특히 빈도해석에 널리 사용되는 GEV 분포에 대한 연구는 다른 분포형보다 더욱 활발히 이루어져 왔다. 본 연구에서는 GEV 분포에 적합한 도시위치공식을 추정하고자 GEV 분포의 순서통계량의 평균 개념을 이용하여 이론적 축소변량을 유도하였다. 또한 다양한 표본크기와 형상 매개변수와 연관이 있는 왜곡도 계수를 고려한 다양한 형태의 도시위치공식을 적용하고, 유전자 알고리즘을 적용하여 도시위치공식의 매개변수를 추정하였다. 유도된 도시위치공식의 정확성을 알아보기 위해 이론적 축소변량과 금회 유도된 도시위치공식을 포함한 다양한 도시위치공식에 의해 계산되는 축소변량 사이의 오차를 비교하였다. 그 결과, 본 연구에서 제안한 도시위치공식은 GEV 분포의 형상 매개변수가 -0.25~0.10의 범위를 가질 때 이론적 축소변량과 가장 작은 오차를 보이는 것으로 나타났다.

Keywords

References

  1. 김규태, 김태순, 김수영, 허준행 (2008). “누가분포함수를 활용한 강우강도식의 국내적용성 평가.” 대한토목학회논문집, 대한토목학회, 제28권, 제4B호, pp. 363-374.
  2. 김민규, 김재희, 김승 (2008). “유입량의 변동성을 고려한 저수지 연계 운영 모형의 가중치 산정.” 한국수자원학회논문집, 한국수자원학회, 제41권, 제1호, pp. 1-15.
  3. 김수영, 허준행, 신홍준, 고연우 (2009). “Gumbel 분포에 대한 도시위치공식의 비교.” 한국수자원학회논문집, 한국수자원학회, 제42권, 제5호, pp. 365-374. https://doi.org/10.3741/JKWRA.2009.42.5.365
  4. 김태순, 허준행, 배덕효 (2007a). “단일저수지 월간운영률의개발: 1. 다목적유전자알고리즘을적용한조각선형운영률.” 대한토목학회논문집, 대한토목학회, 제27권, 제4B호, pp. 387- 393.
  5. 김태순, 허준행, 배덕효, 김진훈(2007b). “단일저수지 월간 운영률의 개발 : 2. 확률론적 장기저류량 예측.” 대한토목학회논문집, 대한토목학회, 제27권, 제4B호, pp. 395-401.
  6. 김태순, 신주영, 김수영, 허준행 (2007). “유전자알고리즘을이용한강우강도식매개변수추정에관한연구(I): 기존 매개변수추정방법과의 비교.” 한국수자원학회논문집, 한국수자원학회, 제40권, 제10호, pp. 811-821.
  7. 신주영, 김태순, 김수영, 허준행 (2007). “유전자알고리즘을 이용한 강우강도식 매개변수 추정에 관한 연구(II): 장.단기간 구분 방법의 제시.” 한국수자원학회논문집, 한국수자원학회, 제40권, 제10호, pp. 823-832. https://doi.org/10.3741/JKWRA.2007.40.10.823
  8. Adamowski, K. (1981). “Plotting position formula forflood frequency.” Water Resource Bulletin, Vol. 17,No. 2, pp. 197-201. https://doi.org/10.1111/j.1752-1688.1981.tb03922.x
  9. Ahmed, J.A., and Sarma, A.K. (2005). “Genetic algorithmfor optimal operating policy of a multipurposereservoir.” Water Resources Management, Vol. 19,No. 2, pp. 145-161. https://doi.org/10.1007/s11269-005-2704-7
  10. Arnell, N.W., Beran, M., and Hosking, J.R.M. (1986).“Unbiased plotting positions for the general extremevalue distribution.” Journal of Hydrology, Vol. 86, pp.59-69. https://doi.org/10.1016/0022-1694(86)90006-5
  11. Beard, L.R. (1942). “Statistical analysis in hydrology.”Proceedings of the American Society of CivilEngineers, Vol. 68, pp. 1077-1088.
  12. Benson, M.A. (1975). “Plotting positions and economicsof engineering planning.” Proceedings of the AmericanSociety of Civil Engineering, Vol. 88, pp. 58-71.
  13. Beyer, H.-G., and Deb, K. (2001). “On self-adaptivefeatures in real-parameter evolutionary algorithms.”IEEE Transactions on Evolutionary Computation,Vol. 5, No. 3, pp. 250-270. https://doi.org/10.1109/4235.930314
  14. Blom, G. (1958). Statistical Estimates and TransformedBeta Variables. Wiley, New York, NY.
  15. Cai, X.M., McKinney, D.C., and Lasdon, L.S. (2001).“Solving nonlinear water management models usinga combined genetic algorithm and linear programmingapproach.” Advances inWater Resources, Vol. 24, No.6, pp. 667-676. https://doi.org/10.1016/S0309-1708(00)00069-5
  16. Celeste, A.B., Suzuki, K., and Kadota, A. (2004). “Geneticalgorithms for real-time operation of multipurposewater resource systems.” Journal of Hydroinformatics,Vol. 6, No. 1, pp. 19-38.
  17. Chang, F.J., Chen, L., and Chang, L.C. (2005).“Optimizing the reservoir operating rule curves bygenetic algorithms.” Hydrological Processes, Vol. 19,No. 11, pp. 2277-2289. https://doi.org/10.1002/hyp.5674
  18. Chang, F.J., Lai, J.S., and Kao, L.S. (2003). “Optimizationof operation rule curves and flushing schedule in areservoir.” Hydrological Processes, Vol. 17, No. 8, pp.1623-1640. https://doi.org/10.1002/hyp.1204
  19. Chang, L.C. (2008). “Guiding rational reservoir floodoperation using penalty-type genetic algorithm."Journal of Hydrology, Vol. 354, No. 1-4, pp. 65-74. https://doi.org/10.1016/j.jhydrol.2008.02.021
  20. Chen, L. (2003). “Real coded genetic algorithm optimizationof long term reservoir operation.” Journal of theAmerican Water Resources Association, Vol. 39, No.5, pp. 1157-1165. https://doi.org/10.1111/j.1752-1688.2003.tb03699.x
  21. Chen, L., J. McPhee, J., and Yeh, W. W. G. (2007). “Adiversified multiobjective GA for optimizing reservoirrule curves.” Advances in Water Resources, Vol. 30,No. 5, pp. 1082-1093. https://doi.org/10.1016/j.advwatres.2006.10.001
  22. Cunnane, C. (1978). “Unbiased plotting positions - Areview.” Journal of Hydrology, Vol. 37, No. 3/4, pp.205-222. https://doi.org/10.1016/0022-1694(78)90017-3
  23. De, M. (2000). “A new unbiased plotting position formulafor Gumbel distribution.” Stochastic EnvironmentalResearch and Risk Assessment, Vol. 14, pp. 1-7. https://doi.org/10.1007/s004770050001
  24. Deb, K., and Beyer, H.-G. (2001). “Self-adaptive geneticalgorithms with simulated binary crossover.” EvolutionaryComputation Journal, Vol. 9, No. 2, pp. 197-221. https://doi.org/10.1162/106365601750190406
  25. Filliben, J.J. (1969). Simple and robust linear estimationof the location parameter of a symmetric distribution.Unpublished Ph.D. dissertation, Princeton university,Princeton, New Jersey.
  26. Goel, N.K., and De, M. (1993). “Development of unbiasedplotting position formula for General ExtremeValue distribution.” Stochastic Environmental Researchand Risk Assessment, Vol. 7, pp. 1-13.
  27. Goldberg, D.E. (1989). Genetic algorithms in search,optimization & machine learning. Addison Wesley,Massachusetts.
  28. Greenwood, J.A., Landwehr, J.M., Matalas, N.C., andWallis, J.R. (1979). “Probability weighted moments :definition and relation to parameters of severaldistributions expressible in inverse Form.” WaterResources Research, Vol. 15, No. 5, pp. 1049-1054. https://doi.org/10.1029/WR015i005p01049
  29. Gringorten, I.I. (1963). “A plotting rule for extremeprobability paper.” Journal of Geophysical Research,Vol. 68, No. 3, pp. 813-814. https://doi.org/10.1029/JZ068i003p00813
  30. Gumbel, E.J. (1958). Statistics of Extremes. ColumbiaUniversity Press, New York, N.Y., pp. 28-34.
  31. Guo, S.L. (1990a). “A discussion on unbiased plottingpositions for the general extreme value distribution.”Journal of Hydrology, Vol. 121, pp. 33-44. https://doi.org/10.1016/0022-1694(90)90223-K
  32. Guo, S.L. (1990b). “Unbiased plotting position formulaefor historical floods.” Journal of Hydrology, Vol. 121,pp. 45-61. https://doi.org/10.1016/0022-1694(90)90224-L
  33. Haktanir, T., and Bozduman, A. (1995). “A study onsensitivity of the probability-weighted momentsmethod on the choice of the plotting position formula.”Journal of Hydrology, Vol. 168, pp. 265-281. https://doi.org/10.1016/0022-1694(94)02642-O
  34. Hazen A. (1914). “Storage to be provided in impoundingreservoirs for municipal water supply.” TransactionsAmerican Society of Civil Engineers, Vol. 1308, No.77, pp. 1547-1550.
  35. Herrera, F., Lozano, M., and Verdegay, J.L. (1998).“Tackling real-coded genetic algorithms: operatorsand tools for behavioural analysis.” Journal ArtificialIntelligence Review, Vol. 12, No. 4, doi>10.1023. https://doi.org/10.1023
  36. Holland, J.H. (1975). Adaptation in natural and artificialsystems. University of Michigan Press.
  37. Huang, W.C., Yuan, L.C., and Lee, C.M. (2002). “Linkinggenetic algorithms with stochastic dynamic programmingto the long-term operation of a multireservoirsystem.” Water Resources Research, Vol. 38, No. 12,pp. 1304. https://doi.org/10.1029/2001WR001122
  38. In-na, N., and Nguyen, V-T-V. (1989). “An unbiasedplotting position formula for the generalized extremevalue distribution.” Journal of Hydrology, Vol. 106, p.193-209. https://doi.org/10.1016/0022-1694(89)90072-3
  39. Jenkinson, A.F. (1955). “The frequency distribution ofthe annual maximum(or minimum) values of meteorologicalelements.” Quarterly Journal of the RoyalMeteorological Society, Vol. 87, pp. 158-171.
  40. Jothiprakash, V., and Shanthi, G. (2006). “Single reservoiroperating policies using genetic algorithm.” WaterResources Management, Vol. 20, No. 6, pp. 917-929. https://doi.org/10.1007/s11269-005-9014-y
  41. Kimball, B.F. (1946). “Assignment of frequencies to acompletely ordered set of sample data.” Transactionon the American Geophysical Union, Vol. 27, pp. 843-846. https://doi.org/10.1029/TR027i006p00843
  42. Kimball, B.F. (1960). “On the choice of plotting positionson probability paper.” Journal of American StatisticalAssociation, Vol. 55, pp. 546-560. https://doi.org/10.2307/2281914
  43. Lee, Y.D., Kim, S.K., and Ko, I.H. (2007). Geneticalgorithm to determine weighting factors in multipleobjective reservoir operation model under inflowuncertainty. Working Paper, Korea University.
  44. Nguyen, V-T-V., In-na, N., and Bobee, B. (1989). “Newplotting-position formula for Pearson type III distribution.”Journal of Hydraulic Engineering, Vol. 115,No. 6, pp. 709-730. https://doi.org/10.1061/(ASCE)0733-9429(1989)115:6(709)
  45. Nguyen, V-T-V., and In-na, N. (1992). “Plottingformula for Pearson type III distribution consideringhistorical information.” Environmental Monitoringand Assessment, Vol. 23, pp. 137-152. https://doi.org/10.1007/BF00406958
  46. Reddy, M.J., and Kumar, D.N. (2006). “Optimal reservoiroperation using multi-objective evolutionary algorithm.”Water Resources Management, Vol. 20, No. 6, pp.861-878. https://doi.org/10.1007/s11269-005-9011-1
  47. Tung, C.P., Hsu, S.Y., Liu, C.M., and Li, J.S. (2003).“Application of the genetic algorithm for optimizingoperation rules of the LiYuTan Reservoir in Taiwan.”Journal of the American Water Resources Association,Vol. 39, No. 3, pp. 649-657. https://doi.org/10.1111/j.1752-1688.2003.tb03682.x
  48. Wardlaw, R., and Sharif, M. (1999). “Evaluation ofgenetic algorithms for optimal reservoir systemoperation.” Journal ofWater Resources Planning andManagement, Vol. 125, No. 1, pp. 25-33. https://doi.org/10.1061/(ASCE)0733-9496(1999)125:1(25)
  49. Weibull, W. (1939). “A statistical theory of strength ofmaterials.” Ing. Vetenskaps Akad. Handl, No. 151,Generalstabens Litografiska Anstals Forlag, Stockholm.
  50. Xuewu, J., Jing, D., Shen, H.W., and Salas, J.D. (1984).“Probability plots for Pearson type III distribution.”Journal of Hydrology, Vol. 74, pp. 1-29. https://doi.org/10.1016/0022-1694(84)90137-9

Cited by

  1. Prediction of Local Scour Around Bridge Piers Using GEP Model vol.34, pp.6, 2014, https://doi.org/10.12652/Ksce.2014.34.6.1779