Contrast of frequency analysis between beta-kappa and beta-Pareto distributions with three of widespread application
DOI:
https://doi.org/10.24850/j-tyca-15-02-09Keywords:
Beta-kappa distribution, beta-Pareto distribution, maximum likelihood fit, standard error of fit, mean absolute error, Q-Q graphics, predictionsAbstract
The hydrological design of several hydraulic works or the revision of the constructed ones is based on the design floods, which are maximum flows of the river, associated with low probabilities of exceedance or predictions. Its most reliable estimate is made through frequency analysis, statistical process that consists of representing the record of maximum annual flows, with a probability distribution function (PDF) or probabilistic model, used to make the desired predictions. In this contrast study, the beta-kappa and beta-Pareto FDPs are proposed, and the following three were considered to be widely used FDPs: Log-Pearson type III, general extreme values, and generalized logistics. Therefore, it is exposed, for the first two FDP, a summary of his theory and his method of fit for maximum likelihood is presented. Eleven annual extreme hydrological data records are processed and the fits are contrasted with two indices: The standard error of fit and the mean absolute error. The selection of the predictions in the seven return periods (Tr) studied was based on the lower values of the fit errors and on the search for representative predictions in the Tr ≥ 500 years. The conclusions suggest the inclusion of the beta-kappa and beta-Pareto distributions in the frequency analysis due to their versatility and fit facility.
References
Aldama, A. A., Ramírez, A. I., Aparicio, J., Mejía-Zermeño, R., & Ortega-Gil, G. E. (2006). Seguridad hidrológica de las presas en México. Jiutepec, México: Instituto Mexicano de Tecnología del Agua.
Benson, M. A. (1962). Plotting positions and economics of engineering planning. Journal of Hydraulics Division, 88(6), 57-71. DOI: 10.1061/jYCEAj.0001293
Bobée, B., & Ashkar, F. (1991). Chapter 1. Data requirements for hydrologic frequency analysis. In: The Gamma Family and derived distributions applied in Hydrology (pp. 1-12). Littleton, USA: Water Resources Publications.
Campos-Aranda, D. F. (1998). Distribución de probabilidades β-ρ: descripción y aplicación en hidrología superficial. En: XV Congreso Nacional de Hidráulica (AMH) (pp. 965-971), del 13 al 16 de octubre, Oaxaca, Oaxaca, México.
Campos-Aranda, D. F. (2002). Contraste de seis métodos de ajuste de la distribución Log-Pearson tipo III en 31 registros históricos de eventos máximos anuales. Ingeniería Hidráulica en México, 17(2), 77-97.
Campos-Aranda, D. F. (2019). Mejores FDP en 19 series amplias de PMD anual del estado de San Luis Potosí, México. Tecnología y ciencias del agua, 10(5), 34-74. DOI: 10.24850/j-tyca-2019-05-02
Coles, S. (2001). Theme 2.6.7: Model diagnostics. In: An introduction to statistical modeling of extreme values (pp. 36-44). London, UK: Springer-Verlag.
Chai, T., & Draxler, R. R. (2014). Root mean square error (RMSE) or mean absolute error (MAE)? - Arguments against avoiding RMSE in the literature. Geoscientific Model Development, 7(3), 1247-1250. DOI: 10.5194/gmd-7-1247-2014
Domínguez, M., R., & Arganis, M. L. (2012). Validation of method to estimate design discharge flow for dam spillways with large regulating capacity. Hydrological Sciences Journal, 57(3), 460-478. DOI: 10.1080/02626667.2012.665993
Hosking, J. R. M., & Wallis, J. R. (1997). Appendix: L-moments for some specific distributions. In: Regional frequency analysis. An approach based on L-moments (pp. 191-209). Cambridge, UK: Cambridge University Press.
Kite, G. W. (1977). Chapter 12. Comparison of frequency distributions. In: Frequency and risk analyses in hydrology (pp. 156-168). Fort Collins, USA: Water Resources Publications.
Mason, S. J., Waylen, P. R., Mimmack, G. M., Rajaratnam, B., & Harrison, J. M. (1999). Changes in extreme rainfall events in South Africa. Climatic Change, 41(2), 249-257. DOI: 10.1023/A:1005450924499
Meylan, P., Favre, A. C., & Musy, A. (2012). Chapter 3. Selecting and checking data series. In: Predictive hydrology. A frequency analysis approach (pp. 29-70). Boca Raton, USA: CRC Press.
Mielke, P. W. (1973). Another family of distributions for describing and analyzing precipitation data. Journal of Applied Meteorology, 12(2), 275-280. DOI: 10.1175/1520-0450(1973)012<0275:AFODFD>2.0.CO;2
Mielke, P. W., & Johnson, E. S. (1974). Some generalized Beta distributions of the second kind having desirable application features in hydrology and meteorology. Water Resources Research, 10(2), 223-226. DOI: 10.1029/WR010i002p00223
Murshed, M. S., Kim, S., & Pak, J. S. (2011). Beta-κ distribution and its application to hydrologic events. Stochastic Environmental Research and Risk Assessment, 25(7), 897-911. DOI: 10.1007/s00477-011-0494-4
Nguyen, T. H., El Outayek, S., Lim, S. H., & Nguyen, T. V. T. (2017). A systematic approach to selecting the best probability models for annual maximum rainfalls - A case study using data in Ontario (Canada). Journal of Hydrology, 553, 49-58. DOI: 10.1016/j.jhydrol.2017.07.052
Oberhettinger, F. (1972). Chapter 15. Hypergeometric functions. In: Abramowitz, M., & Stegun, I. A. (eds.). Handbook of mathematical functions (pp. 555-566). New York, USA: Dover Publications.
Öztekin, T. (2007). Wakeby distribution for representing annual extreme and partial duration rainfall series. Meteorological Applications, 14(4), 381-387. DOI: 10.1002/met.37
Rao, A. R., & Hamed, K. H. (2000). Theme 1.8. Tests on hydrologic data. Chapters 7, 8 and 9. In: Flood frequency analysis (pp. 12-21, 207-321). Boca Raton, USA: CRC Press.
Stedinger, J. R. (2017). Chapter 76. Flood frequency analysis. In: Singh, V. P. (ed.). Handbook of applied hydrology (2nd ed.). (pp. 76.1-76.8). New York, USA: McGraw-Hill Education.
Strupczewski, W. G., Markiewicz, I., Kochanek, K., & Singh, V. P. (2008). Short walk into two-shape-parameter flood frequency distributions. In: Singh, V. P. (ed.). Hydrology and hydraulics (pp. 669-716). Highlands Ranch, USA: Water Resources Publications.
Teegavarapu, R. S. V., Salas, J. D., & Stedinger, J. R. (2019). Chapter 1. Introduction. In: Statistical analysis of hydrologic variables (pp. 1-4). Reston, USA: American Society of Civil Engineers.
WRC, Water Resources Council. (1977). Guidelines for determining flood flow frequency (Bulletin # 17A of the Hydrology Committee). Washington, DC, USA: Water Resources Council.
Willmott, C. J., & Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30(1), 79-82. DOI: 10.3354/cr030079
Wilks, D. S. (1993). Comparison of three-parameter probability distributions for representing annual extreme and partial duration precipitation series. Water Resources Research, 29(10), 3543-3549. DOI: 10.1029/93WR01710
Wilks, D. S. (2011). Theme 4.5. Qualitative assessments of the goodness fit. In: Statistical methods in the atmospheric sciences (3rd ed.) (pp. 112-116). San Diego, USA: Academic Press (Elsevier).
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