Markov-chain Monte Carlo methods for flood data analysis Anita Ivett Szabó, András Zempléni ELTE TTK 2004.

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

Bayesian approach Assume we have some apriori information on the river level Let be the parameter of the GEV distribution apriori information: we have an apriori distribution on the parameterset with continuous density function. Let the sample X 1,…,X n be independent identically distributed random variables (the annual river level maxima). The joint distribution of the sample is

Bayesian approach According to the Bayes-theorem the aposteriori distribution 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 sample x. (1) (2)

MCMC method Unfortunately to compute the integrals (1), (2) in closed formulae are 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).

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.

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

Diagnostics Measurement of convergence (CODA package, add-on routine to R): -Geweke diagnostics: Geweke (1992) proposed a convergence diagnostic for Markov chains based on a test for equality of the means of the first and last part of a Markov chain. If the samples are drawn from the stationary distribution of the chain, the two means are equal and Geweke's statistic has an asymptotically standard normal distribution. -Heidelberger and Welch diagnostics: the convergence test uses the Cramer-von-Mises statistic to test the null hypothesis that the sampled values come from a stationary distribution.

Data Settlementtime period level maximum Tivadar Vásárosnamény Záhony Polgár Szolnok Szeged Runoff maximum Csenger Garbolc Felsőberecki Tiszabecs

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:

Geweke-diagnostic

Heidelberger-Welch Parameters Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed

Empirical density functions

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

Confidence intervals 95% empirical confidence interval for the parameters ( ; ) (149.99; ) (-0.604; )

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)

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:

Geweke diagnostic

Heidelberger-Welch Parameters Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed Stationarity test Start iteration p-value Passed Halfwidth test Mean Halfwidth Passed

Empirical density functions

Parameter estimation and the confidence intervals 95% confidence interval for the parameters ( , ) ( , ) ( , ) =( ; ; )

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)

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:

Geweke diagnostic

Heidelberger-Welch Vásárosnamény: Stationarity test Start iteration p-value Passed (0.735) Halfwidth test Mean Halfwidth Passed 602 (603) 2.19 (1.17) Stationarity test Start iteration p-value Passed (0.662) Halfwidth test Mean Halfwidth Passed 173 (174) 1.46 (0.987) Stationarity test Start iteration p-value Passed (0.943) Halfwidth test Mean Halfwidth Passed (-0.493) ( )

Empirical density functions

Parameter estimation and the confidence intervals 95% confidence interval for the parameters ( ; ) ( ; ) ( ; ) (149.15; ) (-0.613; ) (-0.408; ) Vásárosnamény =( ; ; ) Tivadar =( ; ; )

Return level 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) Vásárosnamény Tivadar

Return level (30 years) RiverValue Confidence interval (95%) Garbolc Túr 242(m 3 /s) (202, 306) Tiszabecs 3225(m 3 /s) (2837, 3868) Tivadar Tisza872(cm) (830, 927) Tivadar (Namény)Tisza872 (cm) (829, 927) Namény (Tivadar) Tisza892 (cm) (871, 916) Namény Tisza 891 (cm) (871, 916) Namény (Záhony) Tisza 887 (cm) (868, 913) Csenger Szamos 2297 (m 3 /s) (1925, 2920) Záhony (Namény) Tisza 721 (cm) (701, 744) Záhony Tisza 721 (cm) (701, 744) Záhony (Polgár)Tisza 721 (cm) (702, 744) BereckiBodrog1115 (m 3 /s) (947, 1405) Polgár (Záhony)Tisza 742 (cm) (719, 770) PolgárTisza 759 (cm) (735, 793) Polgár (Szolnok)Tisza 749 (cm) (724, 777) Szolnok (Polgár)Tisza 919 (cm) (892, 956) SzolnokTisza 920 (cm) (892, 958) Szolnok (Szeged)Tisza 920 (cm) (890, 960) Szeged (Szolnok)Tisza 896 (cm) (863, 941) SzegedTisza 903 (cm) (867, 948)

Return level (50 years) RiverValue Confidence interval (95%) GarbolcTúr271 (m 3 /s) (219, 353) Tiszabecs3372 (m 3 /s) (2969, 4248) TivadarTisza 904 (cm) (857, 970) Tivadar (Namény) Tisza 904 (cm) (858, 969) Namény (Tivadar) Tisza 907 (cm) (886, 934) NaményTisza 906 (cm) (886, 933) Namény (Záhony)Tisza 903 (cm) (883, 931) CsengerSzamos2639 (m 3 /s) (2140, 3508) Záhony (Namény)Tisza 736 (cm) (716, 762) ZáhonyTisza 736 (cm) (716, 762) Záhony (Polgár)Tisza 736 (cm) (716, 762) BereckiBodrog1227 (m 3 /s) (1008, 1630) Polgár (Záhony)Tisza 759 (cm) (734, 791) PolgárTisza 778 (cm) (751, 818) Polgár (Szolnok)Tisza 766 (cm) (740, 798) Szolnok (Polgár)Tisza 942 (cm) (911, 983) SzolnokTisza 942 (cm) (911, 985) Szolnok (Szeged)Tisza 942 (cm) (909, 991) Szeged (Szolnok)Tisza 922 (cm) (886, 974) SzegedTisza 929 (cm) (890, 983)

Return level (100 years) Value Confidence interval (95%) GarbolcTúr314 (m 3 /s) (242, 431) Tiszabecs3745(m 3 /s) (3121, 4815) TivadarTisza 942 (cm) (888, 1017) Tivadar (Namény)Tisza 942 (cm) (888, 1018) Namény (Tivadar)Tisza 922 (cm) (901, 953) NaményTisza 922 (cm) (901, 953) Namény (Záhony)Tisza 919(cm) (898, 954) CsengerSzamos3167 (m 3 /s) (2423, 4407) Záhony (Namény)Tisza 752 (cm) (731, 780) ZáhonyTisza 752 (cm) (732, 781) Záhony (Polgár)Tisza 752 (cm) (731, 780) BereckiBodrog1351 (m 3 /s) (1078, 1956) Polgár (Záhony)Tisza 778 (cm) (751, 815) PolgárTisza 800 (cm) (769, 844) Polgár (Szolnok)Tisza 785 (cm) (757, 821) Szolnok (Polgár)Tisza 968 (cm) (931, 1017) SzolnokTisza 968 (cm) (931, 1020) Szolnok (Szeged)Tisza 967 (cm) (929, 1028) Szeged (Szolnok)Tisza 953 (cm) (910, 1014) SzegedTisza 961 (cm) (913, 1023)

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.