markov-chain monte carlo methods for flood data analysis
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
Markov-chain Monte Carlo
methods for flood data analysis
Anita Ivett Szabó, András Zempléni
ELTE TTK
2004.
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
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 0.662 Halfwidth test Mean Halfwidth Passed 174 0.987
Stationarity test Start iteration p-value Passed 1 0.943 Halfwidth test Mean Halfwidth Passed -0.493 0.00415
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).
The parameters of the MCMC algorithm:
Initial value:
Apriori distribution (Gaussian):
Distribution of the iterative step:
)0,200log,450(1
)1,0()2,200(log)200,450(~),log,( NNNN
)05.0,()1.0,(log)100,(~)|( NNNq iii
Heidelberger-WelchParameters
Stationarity test Start iteration p-value Passed 1 0.0954 Halfwidth test Mean Halfwidth Passed 437 1.88
Stationarity test Start iteration p-value Passed 1 0.927 Halfwidth test Mean Halfwidth Passed 204 1.42
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)
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:
)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
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%)
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)
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)