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 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 X1,…,Xn be independent identically

distributed random variables (the annual river level maxima).

The joint distribution of the sample is

),,(

)(g

n

iixfxf

1

)|()|(

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.

dxfg

xfgxf

)|()(

)|()()|( (1)

)|( zf

dxfzfxzf )|()|()|( (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 stationarydistribution is the aposteriori distribution.

i

1

1 ii 1i )|( iq

)|( xq

},)|()()|(

)|()()|(,1min{

iii

ii qgxf

qgxf

ii

iii

1y probabilit with

y probabilit with 11

MCMC method

is the distribution of the future maxima given the sample and the

apriori information.

dxfzMPxxzMP nnn )|()|(),...,|( 1

s

iin zMP

s 1

)|(1 (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.

DataSettlement time period

level maximumTivadar 1901-2000Vásárosnamény 1990-2000Záhony 1901-1998Polgár 1991-2000Szolnok 1991-1999Szeged 1991-2000

Runoff maximumCsenger 1920-2002Garbolc 1950-2002Felsőberecki 1939-2001Tiszabecs 1938-2002

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:

)0,200log,500(1

)1,0()2,200(log)200,500(~),log,( NNNN

)07.0,()08.0,(log)24,(~)|( NNNq iii

Geweke-diagnostic

Geweke-diagnostic

Geweke-diagnostic

Heidelberger-Welch

Parameters

Stationarity test Start iteration p-value Passed 1 0.735 Halfwidth test Mean Halfwidth Passed 603 1.17


Stationarity test Start iteration p-value Passed 1 0.943 Halfwidth test Mean Halfwidth Passed -0.493 0.00415

Empirical density functions

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).

)ˆ,ˆ,ˆ(

)ˆ,ˆ,ˆ(

)ˆ,ˆ,ˆ(

Confidence intervals

95% empirical confidence interval for the parameters

(563.932; 638.251)

(149.99; 205.628)

(-0.604; -0.378)

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).


Initial value:



)0,200log,450(1

)1,0()2,200(log)200,450(~),log,( NNNN

)05.0,()1.0,(log)100,(~)|( NNNq iii

Geweke diagnostic

Geweke diagnostic

Geweke diagnostic

Heidelberger-WelchParameters



Stationarity test Start iteration p-value Passed 1 0.899 Halfwidth test Mean Halfwidth Passed -0.0194 0.00995


Parameter estimation and the confidence intervals

95% confidence interval for the parameters

(389.8745, 487.212)

(170.4324, 242.1865)

(-0.1987, 0.2075)

)ˆ,ˆ,ˆ( =(436.7372; 204.4207; -0.0194)

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)


Initial value:


~N(500,200)*N(log 200, 2)*N(0,1)*N(500,200)*N(log 200, 2)*N(0,1)


)0,200log,500,0,200log,500(1

~),,,,,( 222111

where),,(),(),(~),( iiii NNNq

Tiii ),( 21 T

iii )log,(loglog 21 and ,),( 21 Tiii

1515

1520

04.007.0

07.0045.0

05.003.0

03.006.0

Geweke diagnostic

Geweke diagnostic

Geweke diagnostic

Heidelberger-Welch

Vásárosnamény:

Stationarity test Start iteration p-value Passed 1 0.184 (0.735) Halfwidth test Mean Halfwidth Passed 602 (603) 2.19 (1.17)

Stationarity test Start iteration p-value Passed 1 0.821 (0.662) Halfwidth test Mean Halfwidth Passed 173 (174) 1.46 (0.987)

Stationarity test Start iteration p-value Passed 1 0.922 (0.943) Halfwidth test Mean Halfwidth Passed -0.491 (-0.493) 0.00597 (0.00415)


Parameter estimation and the confidence intervals

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)

1

1

1

)ˆ,ˆ,ˆ( 111 Vásárosnamény =(601.744; 173.37; -0.491)

Tivadar =(502.041; 172.91; -0.293) )ˆ,ˆ,ˆ( 222

22

2

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) TivadarReturn 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)River Value Confidence interval (95%)

Garbolc Túr 242 (m3/s) (202, 306)Tiszabecs 3225 (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) Tisza 721 (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)

Return level (50 years)River Value Confidence interval (95%)


Return level (100 years)Value Confidence interval (95%)


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.

Markov-chain Monte Carlo methods for flood data analysis

Documents

stationary distribution

joint distribution

apriori distribution

known apriori distribution

standard normal distribution

halfwidth test mean

convergence test

iteration pvalue