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Markov-chain Monte Carlo methods for flood data analysis

Markov-chain Monte Carlo methods for flood data analysis. Anita Ivett Szabó, András Zempléni ELTE TTK 2004. Introduction. Generalized extrem e value distribution (GEV) Bayesian approach MCMC algorithm. Bayesian approach. Assume we have some apriori information on the river level

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Markov-chain Monte Carlo methods for flood data analysis

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  1. Markov-chain Monte Carlomethods for flood dataanalysis Anita Ivett Szabó, András Zempléni ELTE TTK 2004.

  2. Introduction • Generalized extreme value distribution (GEV) • Bayesian approach • MCMC algorithm

  3. Bayesian approach • Assume we have some apriori information on the river level •  Letbe the parameter of the GEV distribution • apriori information: we have an apriori distribution on the parameterset with continuous density function . • Let the sample X1,…,Xn be independent identically distributed random variables (the annual river level maxima). The joint distribution of the sample is

  4. Bayesian approach • According to the Bayes-theorem the aposterioridistribution is (the aposteriori distribution considers both the known apriori distribution and the sample) • The aposteriori distribution can be used for prediction: Let Z be an observation in the future The density function of the random variable Z is . Then is the predictive density function of Z given a samplex. (1) (2)

  5. MCMC method • Unfortunately to compute the integrals (1),(2) in closed formulaeare impossible. • The method: MCMC • We generate a Markov-chain such that the stationary distribution of this Markov chain is the needed aposteriori distribution. We give the draft of the Metropolis-Hastings algorithm(Gibbs-sampler).

  6. MCMC method • We generate a sequence : Let be arbitrary : let the distribution of be , where as the function of x is a density function, forming a family of distributions in in each step let and The generated sequence is a Markov-chain, for which its stationary distribution is the aposteriori distribution.

  7. MCMC method is the distribution of thefuture maxima given the sample and the apriori information. (3)

  8. Diagnostics Measurement of convergence (CODA package, add-on routine to R): • Geweke diagnostics: Geweke (1992) proposed a convergence diagnostic for Markov chainsbased on a test for equality of the means of the first and last partof a Markov chain. If thesamples are drawn from the stationary distribution of the chain, the twomeans are equal and Geweke's statistic has anasymptotically standardnormal distribution. • Heidelberger and Welch diagnostics: the convergence test uses the Cramer-von-Mises statistic to test the nullhypothesis that the sampled values come from a stationary distribution.

  9. Data Settlement time period level maximum Tivadar1901-2000 Vásárosnamény1990-2000 Záhony1901-1998 Polgár1991-2000 Szolnok1991-1999 Szeged1991-2000 Runoff maximum Csenger1920-2002 Garbolc1950-2002 Felsőberecki1939-2001 Tiszabecs1938-2002

  10. Application I Consider the water level data from Vásárosnamény. The parameters of the MCMC algorithm: Initial value: Apriori distribution (Gaussian): Distribution of the iterative step:

  11. Geweke-diagnostic

  12. Geweke-diagnostic

  13. Geweke-diagnostic

  14. Heidelberger-Welch Parameters Stationarity testStart iterationp-value Passed10.735 Halfwidth testMeanHalfwidth Passed6031.17 Stationarity testStart iterationp-value Passed 10.662 Halfwidth testMeanHalfwidth Passed1740.987 Stationarity testStart iterationp-value Passed 10.943 Halfwidth testMeanHalfwidth Passed -0.4930.00415

  15. Empirical density functions

  16. Parameter estimation Bayesian: =(602.82; 173.71; -0.49) ML: =(606.87; 171.74; -0.52) Method of moments =(606.34; 173.8; -0.52).

  17. Confidence intervals 95% empirical confidence interval for the parameters (563.932; 638.251) (149.99; 205.628) (-0.604; -0.378)

  18. Return level Return level (30 years) 891; 95% confidence interval (871, 916) Return level (50 years) 906; 95% confidence interval (886, 933) Return level (100 years) 922; 95% confidence interval (901, 953)

  19. Application II Consider runoff data from Felsőberecki (river Bodrog). The parameters of the MCMC algorithm: Initial value: Apriori distribution (Gaussian): Distribution of the iterative step:

  20. Geweke diagnostic

  21. Geweke diagnostic

  22. Geweke diagnostic

  23. Heidelberger-Welch Parameters Stationarity testStart iterationp-value Passed10.0954 Halfwidth testMeanHalfwidth Passed 437 1.88 Stationarity testStart iterationp-value Passed 10.927 Halfwidth testMeanHalfwidth Passed 204 1.42 Stationarity testStart iterationp-value Passed 10.899 Halfwidth testMeanHalfwidth Passed -0.0194 0.00995

  24. Empirical density functions

  25. Parameter estimation and the confidence intervals =(436.7372; 204.4207; -0.0194) 95% confidence interval for the parameters (389.8745, 487.212) (170.4324, 242.1865) (-0.1987, 0.2075)

  26. Return level Return level (30 years) 1115; 95% confidence interval (947, 1405) Return level (50 years) 1227; 95% confidence interval (1008, 1630) Return level (100 years)1351; 95% confidence interval (1078, 1956)

  27. Application III We consider data at Vásárosnamény and at Tivadar parallel (2-dimensional approach) The parameters of the MCMC algorithm: Initial value: Apriori distribution (Gaussian): ~N(500,200)*N(log 200, 2)*N(0,1)*N(500,200)*N(log 200, 2)*N(0,1) Distribution of the iterative step:

  28. Geweke diagnostic

  29. Geweke diagnostic

  30. Geweke diagnostic

  31. Heidelberger-Welch Vásárosnamény: Stationarity testStart iterationp-value Passed1 0.184 (0.735) Halfwidth testMeanHalfwidth Passed 602 (603) 2.19 (1.17) Stationarity testStart iterationp-value Passed 1 0.821 (0.662) Halfwidth testMeanHalfwidth Passed 173 (174) 1.46 (0.987) Stationarity testStart iterationp-value Passed 1 0.922 (0.943) Halfwidth testMeanHalfwidth Passed -0.491 (-0.493) 0.00597 (0.00415)

  32. Empirical density functions

  33. Parameter estimation and the confidence intervals Vásárosnamény =(601.744; 173.37; -0.491) Tivadar =(502.041; 172.91; -0.293) 95% confidence interval for the parameters (563.321; 640.652) (460.537; 540.823) (149.532; 205.133)(149.15; 201.65) (-0.613; -0.369)(-0.408; -0.163)

  34. Return level Vásárosnamény Tivadar Vásárosnamény: Return level (30 years) 892; 95% confidence interval (871, 916) Return level (50 years) 907; 95% confidence interval (886, 934) Return level (100 years) 922; 95% confidence interval (901, 953) Tivadar Return level (30 years) 872; 95% confidence interval (829, 927) Return level (50 years) 904; 95% confidence interval (858, 969) Return level (100 years) 942; 95% confidence interval (888, 1018)

  35. Return level (30 years) River Value Confidence interval (95%) Garbolc Túr 242 (m3/s) (202, 306) Tiszabecs3225 (m3/s) (2837, 3868) Tivadar Tisza 872 (cm) (830, 927) Tivadar (Namény) Tisza 872 (cm) (829, 927) Namény (Tivadar) Tisza 892 (cm) (871, 916) Namény Tisza 891 (cm) (871, 916) Namény (Záhony) Tisza 887 (cm) (868, 913) Csenger Szamos 2297 (m3/s) (1925, 2920) Záhony (Namény) Tisza721 (cm) (701, 744) Záhony Tisza 721 (cm) (701, 744) Záhony (Polgár) Tisza 721 (cm) (702, 744) Berecki Bodrog 1115 (m3/s) (947, 1405) Polgár (Záhony) Tisza 742 (cm) (719, 770) Polgár Tisza 759 (cm) (735, 793) Polgár (Szolnok) Tisza 749 (cm) (724, 777) Szolnok (Polgár) Tisza 919 (cm) (892, 956) Szolnok Tisza 920 (cm) (892, 958) Szolnok (Szeged) Tisza 920 (cm) (890, 960) Szeged (Szolnok) Tisza 896 (cm) (863, 941) Szeged Tisza 903 (cm) (867, 948)

  36. Return level (50 years) River Value Confidence interval (95%) Garbolc Túr 271 (m3/s) (219, 353) Tiszabecs 3372 (m3/s) (2969, 4248) Tivadar Tisza 904 (cm) (857, 970) Tivadar (Namény) Tisza 904 (cm) (858, 969) Namény (Tivadar) Tisza 907 (cm) (886, 934) Namény Tisza 906 (cm) (886, 933) Namény (Záhony) Tisza 903 (cm) (883, 931) Csenger Szamos 2639 (m3/s) (2140, 3508) Záhony (Namény) Tisza 736 (cm) (716, 762) Záhony Tisza 736 (cm) (716, 762) Záhony (Polgár) Tisza 736 (cm) (716, 762) Berecki Bodrog 1227 (m3/s) (1008, 1630) Polgár (Záhony) Tisza 759 (cm) (734, 791) Polgár Tisza 778 (cm) (751, 818) Polgár (Szolnok) Tisza 766 (cm) (740, 798) Szolnok (Polgár) Tisza 942 (cm) (911, 983) Szolnok Tisza 942 (cm) (911, 985) Szolnok (Szeged) Tisza 942 (cm) (909, 991) Szeged (Szolnok) Tisza 922 (cm) (886, 974) Szeged Tisza 929 (cm) (890, 983)

  37. Return level (100 years) Value Confidence interval (95%) Garbolc Túr 314 (m3/s) (242, 431) Tiszabecs 3745 (m3/s) (3121, 4815) Tivadar Tisza 942 (cm) (888, 1017) Tivadar (Namény) Tisza 942 (cm) (888, 1018) Namény (Tivadar) Tisza 922 (cm) (901, 953) Namény Tisza 922 (cm) (901, 953) Namény (Záhony) Tisza 919 (cm) (898, 954) Csenger Szamos 3167 (m3/s) (2423, 4407) Záhony (Namény) Tisza 752 (cm) (731, 780) Záhony Tisza 752 (cm) (732, 781) Záhony (Polgár) Tisza 752 (cm) (731, 780) Berecki Bodrog 1351 (m3/s) (1078, 1956) Polgár (Záhony) Tisza 778 (cm) (751, 815) Polgár Tisza 800 (cm) (769, 844) Polgár (Szolnok) Tisza 785 (cm) (757, 821) Szolnok (Polgár) Tisza 968 (cm) (931, 1017) Szolnok Tisza 968 (cm) (931, 1020) Szolnok (Szeged) Tisza 967 (cm) (929, 1028) Szeged (Szolnok) Tisza 953 (cm) (910, 1014) Szeged Tisza 961 (cm) (913, 1023)

  38. Conclusions Convergence: OK most of the cases (with some problems in the bivariate case and for cases with shorter observation periods). Method: very similar results to the classical approach.

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