Parametric Vine Copula Construction for Flood Analysis for Kelantan River Basin in Malaysia
Abstract
The multivariate approach of flood characteristics such as flood peak flow (P), volume (V), and duration (D) is much beneficial in recognizing the critical behaviour of flood episodes at a river basin scale. The incorporation of 2-dimensional copulas for establishing bivariate flood dependency frequently appears, but it could be more comprehensive if we focus all the three flood characteristic simultaneously. In such circumstances, incorporation of vine or Pair-Copula Construction (PCC) could produce a better approximation of joint probability density and much practical approach in the uncertainty analysis, in comparison with conventional trivariate copula distribution. This study demonstrated the efficacy of parametric vine copula in the modelling of trivariate flood characteristics for the Kelantan River basin in Malaysia. The D-vine tree structure is selected where the Gaussian and Frank copula is recognized for bivariate flood pairs (P-V) and (P-D) pairs in the first stage, using the maximum-pseudo-likelihood (MPL) estimation procedure. Similarly, the Gumbel copula is selected in the modelling of conditioned data obtained through the conditional distribution function of bivariate copulas selected in the previous stage based on the partial differentiation, also called h-function. Finally, the full density function of the 3-dimension structure is derived and compared with the observed flood characteristics. Furthermore, tail dependence properties and behaviour of D-vine copula are also investigated, which reveals for well capturing the general behaviour of Gaussian and Frank copula fitted to flood pair (P-V) and (V-D) and reproduces the overall flood correlation structure fairely well. Both the primary ‘OR’ and ‘AND’ joint return periods for trivariate flood events are estimated which pointing that ‘AND’ joint case produces higher return value than ‘OR’ case.
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References
Yue, Sheng. "The bivariate lognormal distribution to model a multivariate flood episode." Hydrological Processes 14, no. 14 (2000): 2575-2588. doi:10.1002/1099-1085(20001015)14:14<2575::AID-HYP115>3.0.CO;2-L.
Zhang, L., and Vijay P. Singh. "Trivariate flood frequency analysis using the Gumbel–Hougaard copula." Journal of Hydrologic Engineering 12, no. 4 (2007): 431-439. doi:10.1061/(asce)1084-0699(2007)12:4(431).
Gräler, B., M. J. van den Berg, S. Vandenberghe, A. Petroselli, S. Grimaldi, B. De Baets, and N. E. C. Verhoest. “Multivariate Return Periods in Hydrology: a Critical and Practical Review Focusing on Synthetic Design Hydrograph Estimation.” Hydrology and Earth System Sciences 17, no. 4 (April 2, 2013): 1281–1296. doi:10.5194/hess-17-1281-2013.
Cunnane, Conleth. “Review of Statistical Models for Flood Frequency Estimation.” Hydrologic Frequency Modeling (1987): 49–95. doi:10.1007/978-94-009-3953-0_4.
Nadarajah, S., and J. T. Shiau. “Analysis of Extreme Flood Events for the Pachang River, Taiwan.” Water Resources Management 19, no. 4 (August 2005): 363–374. doi:10.1007/s11269-005-2073-2.
Yue, S., and C. Y. Wang. “A Comparison of Two Bivariate Extreme Value Distributions.” Stochastic Environmental Research and Risk Assessment (SERRA) 18, no. 2 (April 1, 2004): 61–66. doi:10.1007/s00477-003-0124-x.
Sklar, M. "Fonctions de repartition an dimensions et leurs marges." Publ. inst. statist. univ. Paris 8 (1959): 229-231.
Nelsen, R. B. “An Introduction to Copulas.” Springer Series in Statistics (2006). doi:10.1007/0-387-28678-0.
Salvadori, G., and C. De Michele. “Frequency Analysis via Copulas: Theoretical Aspects and Applications to Hydrological Events.” Water Resources Research 40, no. 12 (December 2004). doi:10.1029/2004wr003133.
De Michele, C. “A Generalized Pareto Intensity-Duration Model of Storm Rainfall Exploiting 2-Copulas.” Journal of Geophysical Research 108, no. D2 (2003). doi:10.1029/2002jd002534.
Zhang, L. S. V. P., and V. P. Singh. "Bivariate flood frequency analysis using the copula method." Journal of hydrologic engineering 11, no. 2 (2006): 150-164. doi:10.1061/(asce)1084-0699(2006)11:2(150).
Salvadori, G., and C. De Michele. “Statistical Characterization of Temporal Structure of Storms.” Advances in Water Resources 29, no. 6 (June 2006): 827–842. doi:10.1016/j.advwatres.2005.07.013.
Wong, G., M. F. Lambert, M. Leonard, and A. V. Metcalfe. “Drought Analysis Using Trivariate Copulas Conditional on Climatic States.” Journal of Hydrologic Engineering 15, no. 2 (February 2010): 129–141. doi:10.1061/(asce)he.1943-5584.0000169.
Genest, Christian, and Anne-Catherine Favre. "Everything you always wanted to know about copula modeling but were afraid to ask." Journal of hydrologic engineering 12, no. 4 (2007): 347-368. doi:10.1061/(asce)1084-0699(2007)12:4(347).
Reddy, M. Janga, and Poulomi Ganguli. “Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas.” Water Resources Management 26, no. 14 (September 11, 2012): 3995–4018. doi:10.1007/s11269-012-0124-z.
Renard, B., and M. Lang. “Use of a Gaussian Copula for Multivariate Extreme Value Analysis: Some Case Studies in Hydrology.” Advances in Water Resources 30, no. 4 (April 2007): 897–912. doi:10.1016/j.advwatres.2006.08.001.
Salvadori, G., C. De Michele, and F. Durante. “On the Return Period and Design in a Multivariate Framework.” Hydrology and Earth System Sciences 15, no. 11 (November 4, 2011): 3293–3305. doi:10.5194/hess-15-3293-2011.
Ganguli, Poulomi, and M. Janga Reddy. “Probabilistic Assessment of Flood Risks Using Trivariate Copulas.” Theoretical and Applied Climatology 111, no. 1–2 (May 25, 2012): 341–360. doi:10.1007/s00704-012-0664-4.
Daneshkhah, Alireza, Renji Remesan, Omid Chatrabgoun, and Ian P. Holman. “Probabilistic Modeling of Flood Characterizations with Parametric and Minimum Information Pair-Copula Model.” Journal of Hydrology 540 (September 2016): 469–487. doi:10.1016/j.jhydrol.2016.06.044.
Madadgar, Shahrbanou, and Hamid Moradkhani. “Drought Analysis Under Climate Change Using Copula.” Journal of Hydrologic Engineering 18, no. 7 (July 2013): 746–759. doi:10.1061/(asce)he.1943-5584.0000532.
Li, Fan, and Qian Zheng. “Probabilistic Modelling of Flood Events Using the Entropy Copula.” Advances in Water Resources 97 (November 2016): 233–240. doi:10.1016/j.advwatres.2016.09.016.
Joe, Harry. “Multivariate Models and Multivariate Dependence Concepts.” C&H/CRC Monographs on Statistics & Applied Probability (May 1997). doi:10.1201/b13150.
Aas, Kjersti, Claudia Czado, Arnoldo Frigessi, and Henrik Bakken. “Pair-Copula Constructions of Multiple Dependence.” Insurance: Mathematics and Economics 44, no. 2 (April 2009): 182–198. doi:10.1016/j.insmatheco.2007.02.001.
Grimaldi, Salvatore, and Francesco Serinaldi. “Asymmetric Copula in Multivariate Flood Frequency Analysis.” Advances in Water Resources 29, no. 8 (August 2006): 1155–1167. doi:10.1016/j.advwatres.2005.09.005.
Serinaldi, Francesco, and Salvatore Grimaldi. "Fully nested 3-copula: procedure and application on hydrological data." Journal of Hydrologic Engineering 12, no. 4 (2007): 420-430. doi:10.1061/(asce)1084-0699(2007)12:4(420)
Genest, C., A.-C. Favre, J. Béliveau, and C. Jacques. “Metaelliptical Copulas and Their Use in Frequency Analysis of Multivariate Hydrological Data.” Water Resources Research 43, no. 9 (September 2007). doi:10.1029/2006wr005275.
Kao, Shih-Chieh, and Rao S. Govindaraju. “Trivariate Statistical Analysis of Extreme Rainfall Events via the Plackett Family of Copulas.” Water Resources Research 44, no. 2 (February 2008). doi:10.1029/2007wr006261.
Song, Songbai, and Vijay P. Singh. “Meta-Elliptical Copulas for Drought Frequency Analysis of Periodic Hydrologic Data.” Stochastic Environmental Research and Risk Assessment 24, no. 3 (July 17, 2009): 425–444. doi:10.1007/s00477-009-0331-1.
Bedford, Tim, and Roger M. Cooke. “Vines--a New Graphical Model for Dependent Random Variables.” The Annals of Statistics 30, no. 4 (August 2002): 1031–1068. doi:10.1214/aos/1031689016.
Czado, Claudia, and Aleksey Min. “Bayesian Inference for D-Vines: Estimation and Model Selection.” Dependence Modeling (December 2010): 249–264. doi:10.1142/9789814299886_0012.
Abdulrazzaq, Zaidoon T., Raghad H. Hasan, and Nadia A. Aziz. “Integrated TRMM Data and Standardized Precipitation Index to Monitor the Meteorological Drought.” Civil Engineering Journal 5, no. 7 (July 21, 2019): 1590–1598. doi:10.28991/cej-2019-03091355.
Gyasi-Agyei, Yeboah, and Charles S. Melching. “Modelling the Dependence and Internal Structure of Storm Events for Continuous Rainfall Simulation.” Journal of Hydrology 464–465 (September 2012): 249–261. doi:10.1016/j.jhydrol.2012.07.014.
Vernieuwe, H., S. Vandenberghe, B. De Baets, and N. E. C. Verhoest. “A Continuous Rainfall Model Based on Vine Copulas.” Hydrology and Earth System Sciences 19, no. 6 (June 12, 2015): 2685–2699. doi:10.5194/hess-19-2685-2015.
Czado, Claudia, Stephan Jeske, and Mathias Hofmann. "Selection strategies for regular vine copulae." Journal de la Société Française de Statistique 154, no. 1 (2013): 174-191.
Hobæk Haff, Ingrid, Kjersti Aas, and Arnoldo Frigessi. “On the Simplified Pair-Copula Construction — Simply Useful or Too Simplistic?” Journal of Multivariate Analysis 101, no. 5 (May 2010): 1296–1310. doi:10.1016/j.jmva.2009.12.001.
Kurowicka, Dorota, and Roger Cooke. “Uncertainty Analysis with High Dimensional Dependence Modelling.” Wiley Series in Probability and Statistics (February 24, 2006). doi:10.1002/0470863072.
Aas, Kjersti, and Daniel Berg. “Models for Construction of Multivariate Dependence – a Comparison Study.” The European Journal of Finance 15, no. 7–8 (December 2009): 639–659. doi:10.1080/13518470802588767.
Drainage, D. I. D. "Irrigation Department. Annual flood report of DID for Peninsular Malaysia." Unpublished report. Kuala Lumpur: DID (2004).
Adnan, Nor Aizam, and Peter M. Atkinson. “Exploring the Impact of Climate and Land Use Changes on Streamflow Trends in a Monsoon Catchment.” International Journal of Climatology 31, no. 6 (April 15, 2011): 815–831. doi:10.1002/joc.2112.
Nashwan, Mohamed Salem, Tarmizi Ismail, and Kamal Ahmed. “Flood Susceptibility Assessment in Kelantan River Basin Using Copula.” International Journal of Engineering & Technology 7, no. 2 (April 18, 2018): 584. doi:10.14419/ijet.v7i2.10447.
Shiau, J. T. “Fitting Drought Duration and Severity with Two-Dimensional Copulas.” Water Resources Management 20, no. 5 (June 27, 2006): 795–815. doi:10.1007/s11269-005-9008-9.
Favre, Anne-Catherine, Salaheddine El Adlouni, Luc Perreault, Nathalie Thiémonge, and Bernard Bobée. “Multivariate Hydrological Frequency Analysis Using Copulas.” Water Resources Research 40, no. 1 (January 2004). doi:10.1029/2003wr002456.
Zhang, Runrun, Xi Chen, Qinbo Cheng, Zhicai Zhang, and Peng Shi. “Joint Probability of Precipitation and Reservoir Storage for Drought Estimation in the Headwater Basin of the Huaihe River, China.” Stochastic Environmental Research and Risk Assessment 30, no. 6 (April 11, 2016): 1641–1657. doi:10.1007/s00477-016-1249-z.
De Michele, C., G. Salvadori, M. Canossi, A. Petaccia, and R. Rosso. "Bivariate statistical approach to check adequacy of dam spillway." Journal of Hydrologic Engineering 10, no. 1 (2005): 50-57. doi:10.1061/(asce)1084-0699(2005)10:1(50).
Kojadinovic, Ivan, and Jun Yan. “Modeling Multivariate Distributions with Continuous Margins Using thecopulaRPackage.” Journal of Statistical Software 34, no. 9 (2010). doi:10.18637/jss.v034.i09.
Genest, Christian, and Bruno Rémillard. “Validity of the Parametric Bootstrap for Goodness-of-Fit Testing in Semiparametric Models.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 44, no. 6 (December 2008): 1096–1127. doi:10.1214/07-aihp148.
Genest, Christian, Bruno Rémillard, and David Beaudoin. “Goodness-of-Fit Tests for Copulas: A Review and a Power Study.” Insurance: Mathematics and Economics 44, no. 2 (April 2009): 199–213. doi:10.1016/j.insmatheco.2007.10.005.
Alamgir, Mahiuddin, Tarmizi Ismail, and Muhammad Noor. “Bivariate Frequency Analysis of Flood Variables Using Copula in Kelantan River Basin.” Malaysian Journal of Civil Engineering 30, no. 3 (November 19, 2018). doi:10.11113/mjce.v30n3.515.
Hosking, J. R. M., J. R. Wallis, and E. F. Wood. “Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments.” Technometrics 27, no. 3 (August 1985): 251–261. doi:10.1080/00401706.1985.10488049.
Ljung, G. M., and G. E. P. Box. “On a Measure of Lack of Fit in Time Series Models.” Biometrika 65, no. 2 (August 1, 1978): 297–303. doi:10.1093/biomet/65.2.297.
Benth, Fred Espen, and Jūratė Šaltytė‐Benth. “Stochastic Modelling of Temperature Variations with a View Towards Weather Derivatives.” Applied Mathematical Finance 12, no. 1 (March 2005): 53–85. doi:10.1080/1350486042000271638.
Fisher, N. I, and P Switzer. “Graphical Assessment of Dependence.” The American Statistician 55, no. 3 (August 2001): 233–239. doi:10.1198/000313001317098248.
Genest, Christian, and Jean-Claude Boies. “Detecting Dependence With Kendall Plots.” The American Statistician 57, no. 4 (November 2003): 275–284. doi:10.1198/0003130032431.
Gringorten, Irving I. “A Plotting Rule for Extreme Probability Paper.” Journal of Geophysical Research 68, no. 3 (February 1, 1963): 813–814. doi:10.1029/jz068i003p00813.
Anderson, T. W., and D. A. Darling. “A Test of Goodness of Fit.” Journal of the American Statistical Association 49, no. 268 (December 1954): 765–769. doi:10.1080/01621459.1954.10501232.
Akaike, H. “A New Look at the Statistical Model Identification.” IEEE Transactions on Automatic Control 19, no. 6 (December 1974): 716–723. doi:10.1109/tac.1974.1100705.
Schwarz, Gideon. “Estimating the Dimension of a Model.” The Annals of Statistics 6, no. 2 (March 1978): 461–464. doi:10.1214/aos/1176344136.
Hannan, E. J., and B. G. Quinn. “The Determination of the Order of an Autoregression.” Journal of the Royal Statistical Society: Series B (Methodological) 41, no. 2 (January 1979): 190–195. doi:10.1111/j.2517-6161.1979.tb01072.x.
Salvadori, G. “Bivariate Return Periods via 2-Copulas.” Statistical Methodology 1, no. 1–2 (December 2004): 129–144. doi:10.1016/j.stamet.2004.07.002.
Caperaa, P. “A Nonparametric Estimation Procedure for Bivariate Extreme Value Copulas.” Biometrika 84, no. 3 (September 1, 1997): 567–577. doi:10.1093/biomet/84.3.567.
Bedford, Tim, Alireza Daneshkhah, and Kevin J. Wilson. “Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas.” Risk Analysis 36, no. 4 (September 2, 2015): 792–815. doi:10.1111/risa.12471.
DOI: 10.28991/cej-2020-03091561
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