Wind and Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Alternative robust estimation methods for parameters of Gumbel distribution: an application to wind speed data with outliers

  • Aydin, Demet (Department of Statistics, Faculty of Science and Letters, Sinop University)
  • Received : 2017.05.19
  • Accepted : 2018.02.16
  • Published : 2018.06.25
⟨ Previous Next ⟩

Abstract

An accurate determination of wind speed distribution is the basis for an evaluation of the wind energy potential required to design a wind turbine, so it is important to estimate unknown parameters of wind speed distribution. In this paper, Gumbel distribution is used in modelling wind speed data, and alternative robust estimation methods to estimate its parameters are considered. The methodologies used to obtain the estimators of the parameters are least absolute deviation, weighted least absolute deviation, median/MAD and least median of squares. The performances of the estimators are compared with traditional estimation methods (i.e., maximum likelihood and least squares) according to bias, mean square deviation and total mean square deviation criteria using a Monte-Carlo simulation study for the data with and without outliers. The simulation results show that least median of squares and median/MAD estimators are more efficient than others for data with outliers in many cases. However, median/MAD estimator is not consistent for location parameter of Gumbel distribution in all cases. In real data application, it is firstly demonstrated that Gumbel distribution fits the daily mean wind speed data well and is also better one to model the data than Weibull distribution with respect to the root mean square error and coefficient of determination criteria. Next, the wind data modified by outliers is analysed to show the performance of the proposed estimators by using numerical and graphical methods.

Keywords

References

  1. Akgul, F.G., Senoglu, B. and Arslan, T. (2016), "An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution", Energ. Convers. Manage., 114, 234-240. https://doi.org/10.1016/j.enconman.2016.02.026
  2. Apery, R. (1979), "Irrationalite de ${\zeta}$ (2) et ${\zeta}$ (3)", Asterisque, 61, 11-13.
  3. Arslan, T., Acitas, S. and Senoglu, B. (2017), "Generalized Lindley and power Lindley distributions for modeling the wind speed data", Energ. Convers. Manage., 152, 300-311. https://doi.org/10.1016/j.enconman.2017.08.017
  4. Aydin, D. and Senoglu, B. (2015), "Monte Carlo comparison of the parameter estimation methods for the two-parameter Gumbel distribution", J. Modern Appl. Stat. Method., 14(2), 123-140. https://doi.org/10.22237/jmasm/1446351060
  5. Aydin, D. (2018), "Estimation of the lower and upper quantiles of Gumbel distribution: an application to wind speed data", Appl. Ecology Environ. Res., 16(1), 1-15. https://doi.org/10.15666/aeer/1601_001015
  6. Bai, J. (1995), "Least absolute deviation estimation of a shift", Econometric Theory, 11(3), 403-436. https://doi.org/10.1017/S026646660000935X
  7. Barnett, V. and Lewis, T. (1994), "Outliers in statistical data", John Wiley, New York, USA.
  8. Bernard, A. and Bosi-Levenbach, E.C. (1953), "The plotting of observations on probability paper", Statistica Neerlandica, 7, 163-173. https://doi.org/10.1111/j.1467-9574.1953.tb00821.x
  9. Blom, G. (1958), Statistical estimates and transformed beta variables, John Wiley and Sons, New York, USA.
  10. Boudt, K., Caliskan, D. and Croux, C. (2011), "Robust explicit estimators of Weibull parameters", Metrika, 73, 187-209. https://doi.org/10.1007/s00184-009-0272-1
  11. Croux, C., Filzmoser, P., Pison, G. and Rousseeuw, P.J. (2003), "Fitting multiplicative models by robust alternating regressions", Stat. Comput., 13, 23-36. https://doi.org/10.1023/A:1021979409012
  12. Corsini, G., Gini, F., Greco, M.V. and Verrazzani, L. (1995), "Cramer-Rao bounds and estimation of the parameters of the Gumbel distribution", Aerosp. Electron. Syst., 31(3), 1202-1204. https://doi.org/10.1109/7.395217
  13. Cox, N.J. (2005), "Speaking Stata: density probability plots", Stata J., 5(2), 259-273.
  14. Davis, R.A. and Dunsmuir, W.T.M. (1997), "Least absolute deviation estimation for regression with ARMA errors", J. Theor. Probab., 10, 481-497. https://doi.org/10.1023/A:1022620818679
  15. Dixon, W.J. (1950), "Analysis of extreme values", Annal. Math. Stat., 21, 488-506. https://doi.org/10.1214/aoms/1177729747
  16. Dunsmuir, W.T.M. and Spencer, N.M. (1991), "Strong consistency and asymptotic normality of L1 estimates of the autoregressive moving-average model", J. Time Series Anal., 12, 95-104. https://doi.org/10.1111/j.1467-9892.1991.tb00071.x
  17. Ercelebi, S.G. and Toros, H. (2009), "Extreme value analysis of Istanbul air pollution data", Clean Soil Air Water, 37, 122-131. https://doi.org/10.1002/clen.200800041
  18. Fischer, S., Fried, R. and Schumann, A. (2015), "Examination for robustness of parametric estimators for flood statistics in the context of extraordinary extreme events", Hydrology Earth Syst. Sci., 12, 8553-8576. https://doi.org/10.5194/hessd-12-8553-2015
  19. Graybeal, D. and Leathers, D. (2006), "Snowmelt-related flood risk in Appalachia: first estimates from a historical snow climatology", J. Appl. Meteorol. Clim., 45(1), 178-193. https://doi.org/10.1175/JAM2330.1
  20. Gumbel, E.J. (1941), "The return period of flood flows", Annal. Math. Stat., 12(2), 163-190. https://doi.org/10.1214/aoms/1177731747
  21. Herd, G.R. (1960), "Estimation of reliability from incomplete data", Proceedings of the 6th National Symposium on Reliability and Quality Control, IEEE, New York.
  22. Hong, H.P., Li, S.H. and Mara, T.G. (2013), "Performance of the generalized least-squares method for the Gumbel distribution and its application to annual maximum wind speeds", J. Wind Eng. Ind. Aerod., 119, 121-132. https://doi.org/10.1016/j.jweia.2013.05.012
  23. Jenkinson, A.F. (1955), "The frequency distribution of the annual maximum (or minimum) values of meteorological elements", Q. J. Roy. Meteor. Soc., 81, 158-171. https://doi.org/10.1002/qj.49708134804
  24. Johnson, L.G. (1964), The statistical treatment of fatigue experiments, Elsevier, New York, USA.
  25. Kang, D., Ko, K. and Huh, J. (2015), "Determination of extreme wind values using the Gumbel distribution", Energy, 86, 51-58. https://doi.org/10.1016/j.energy.2015.03.126
  26. Kantar, Y.M. and Yildirim, V. (2015), "Robust estimation for parameters of the extended Burr type III distribution", Commun. Statistics-Simulation and Comput., 44, 1901-1930. https://doi.org/10.1080/03610918.2013.839032
  27. Kantar, Y.M.; Usta, I.; Arik, I. and Yenilmez, I. (2018), "Wind speed analysis using the Extended Generalized Lindley Distribution", Renew. Energy, 118, 1024-1030. https://doi.org/10.1016/j.renene.2017.09.053
  28. Kao, J.H. (1958), "Computer methods for estimating Weibull parameters in reliability studies", IRE T. Reliab. Quality Control, 13, 15-22.
  29. Kao, J.H. (1959), "A graphical estimation of mixed Weibull parameters in life-testing of electron tubes", Technometrics, 1(4), 389-407. https://doi.org/10.1080/00401706.1959.10489870
  30. Kollu, R., Rayapudi, S.R., Narasimham, S.V.L. and Pakkurthi, K.M. (2012), "Mixture probability distribution functions to model wind speed distributions", Int. J. Energ. Environ. Eng., 3, 1-10. https://doi.org/10.1186/2251-6832-3-1
  31. Koutsoyiannis, D. (2004), "Statistics of extremes and estimation of extreme rainfall: I. theoretical investigation", Hydrol. Sci. J., 49(4), 575-590. https://doi.org/10.1623/hysj.49.4.575.54430
  32. Landwehr, J.M., Matalas, N.C. and Wallis, J.R. (1979), "Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles", Water Resour. Res., 15(5), 1055-1064. https://doi.org/10.1029/WR015i005p01055
  33. Lee, B.H., Ahn, D.J., Kim, H.G. and Ha, Y.C. (2012), "An estimation of the extreme wind speed using the Korea wind map", Renew Energ., 42, 4-10. https://doi.org/10.1016/j.renene.2011.09.033
  34. Ling, S. (2005), "Self-weighted LAD estimation for infinite variance autoregressive models", J. Roy. Stat. Soc.: Series B, 67, 381-393. https://doi.org/10.1111/j.1467-9868.2005.00507.x
  35. Mohammadi, K., Alavi, O. and McGowan, J.G. (2017), "Use of Birnbaum-Saunders distribution for estimating wind speed and wind power probability distributions: A review", Energ. Convers. Manage., 143, 109-122 https://doi.org/10.1016/j.enconman.2017.03.083
  36. Morgan, E.C., Lackner, M., Vogel, R.M. and Baise, L.G. (2011), "Probability distribution for offshore wind speeds", Energ. Convers. Manage., 52, 15-26. https://doi.org/10.1016/j.enconman.2010.06.015
  37. Mousa, M.A.M., Jaheen, Z.F. and Ahmad, A.A. (2002), "Bayesian estimation, prediction and characterization for the Gumbel model based on records", J. Theor. Appl. Stat., 36(1), 65-74.
  38. Olive, D. (2006), "Robust estimators for transformed location scale families", http://lagrange.math.siu.edu/Olive/pprloc3.pdf.
  39. Pan, J., Wang, H. and Yao, Q. (2007), "Weighted least absolute deviations estimation for ARMA models with infinite variance", Econometric Theory, 23(5), 852-879. https://doi.org/10.1017/S0266466607070363
  40. Pishgar-Komleh, S.H., Keyhani, A. and Sefeedpari, P. (2015), "Wind speed and power density analysis based on Weibull and Rayleigh distributions (a case study: Firouzkooh county of Iran)", Renew. Sust. Energ. Rev., 42, 313-322. https://doi.org/10.1016/j.rser.2014.10.028
  41. Pollard, D. (1991), "Asymptotics for least absolute deviation regression estimators", Econometric Theory, 7, 186-199. https://doi.org/10.1017/S0266466600004394
  42. Phien, H.N. (1987), "A review of methods of parameter estimation for the extreme-value type I distribution", J. Hydrol., 86, 391-398.
  43. Rasmussen, P.F. and Gautam, N. (2003), "Alternative PWMestimators of the Gumbel distribution", J. Hydrol., 280, 265-271. https://doi.org/10.1016/S0022-1694(03)00241-5
  44. Raynal, J.A. and Salas, J.D. (1986), "Estimation procedures for the type-1 extreme value distribution", J. Hydrol., 87, 315-336. https://doi.org/10.1016/0022-1694(86)90022-3
  45. Rousseeuw, P.J. (1984), "Least median of squares regression", J. Am. Stat. Association, 79, 871-880. https://doi.org/10.1080/01621459.1984.10477105
  46. Rousseeuw, P.J. and Leroy, A. (1987), Robust regression and outlier detection, Wiley, New York, USA.
  47. Simiua, E.I., Heckertb, N.A., Filliben, J.J. and Johnson, S.K. (2001), "Extreme wind load estimates based on Gumbel distribution of dynamic pressures: an assessment", Struct. Saf., 23, 221-229. https://doi.org/10.1016/S0167-4730(01)00016-9
  48. Sohoni, V., Gupta, S. and Nema, R. (2016), "A comparative analysis of wind speed probability distributions for wind power assessment of four sites", Turkish J. Elec. Eng. Comput. Sci., 24, 4724-4735. https://doi.org/10.3906/elk-1412-207
  49. Soukissian, T. (2013) "Use of multi-parameter distributions for offshore wind speed modeling: The Johnson SB distribution", Appl. Energ., 111, 982-1000. https://doi.org/10.1016/j.apenergy.2013.06.050
  50. Tiryakioglu, M. and Hudak, D. (2007), "On estimating Weibull modulus by the linear regression method", J. Mater. Sci., 42, 10173-10179. https://doi.org/10.1007/s10853-007-2060-5
  51. Trzpiot, G. (2009), "Extreme value distributions and robust estimation", Folia Oeconomica, 228, 85-92.
  52. Wolstenholme, L.C. (1999), Reliability modelling: a statistical approach, Chacman and Hall Press, New York, USA.
  53. Yavuz, A.A. (2013), "Estimation of the shape parameter of the Weibull distribution using linear regression methods: noncensored samples", Qual. Reliab. Eng. Int., 29, 1207-1219. https://doi.org/10.1002/qre.1472
  54. Zyl, J.M.V. and Schall, R. (2012), "Parameter estimation through weighted least-squares rank regression with specific reference to the Weibull and Gumbel distributions", Commun. Stat. - Simul. C., 41, 1654-1666. https://doi.org/10.1080/03610918.2011.611315