Az előadások a következő témára: "Markov-chain Monte Carlo methods for flood data analysis Anita Ivett Szabó, András Zempléni ELTE TTK 2004."— Előadás másolata:
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 Tivadar1901-2000 Vásárosnamény1990-2000 Záhony1901-1998 Polgár1991-2000 Szolnok1991-1999 Szeged1991-2000 Runoff maximum Csenger1920-2002 Garbolc1950-2002 Felsőberecki1939-2001 Tiszabecs1938-2002
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:
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: