peaks-over-threshold models szabolcs erdélyi research assistant vituki plc
TRANSCRIPT
Peaks-over-threshold modelsPeaks-over-threshold models
Szabolcs ErdélyiSzabolcs Erdélyi
research assistantresearch assistant
VITUKI Plc. VITUKI Plc.
AbstractAbstract
– Used dataUsed data
– POT modelPOT model
– Choosing thresholds Choosing thresholds
– ResultsResults
– SummarySummary
Used dataUsed dataSTATION DATATYPE FROM TO
Tiszabecs H 1924 2000
Tivadar H 1901 2000
Tivadar Q 1951 2000
Vásárosnamény H 1901 2000
Vásárosnamény Q 1901 2000
Záhony H 1901 1999
Záhony Q 1931 1999
Polgár H 1901 2000
Polgár Q 1931 2000
Szolnok H 1901 1999
Szolnok Q 1920 1999
Szeged H 1901 2000
Szeged Q 1921 2000
POT modelPOT model
XX1 1 ,, X X2 2 , … , … independence, identically distributed independence, identically distributed
random variablesrandom variables
uu high enough thresholdhigh enough threshold
HH((zz)) distribution function of GPDdistribution function of GPD
when when yy > 0, and > 0, and
1
~11
yyHuXuXP
0~/1 y
POT modelPOT model
– Choosing thresholdChoosing threshold
– Selecting data over threshold from daily maximum Selecting data over threshold from daily maximum valuesvalues
– DeclusteringDeclustering
– Time of declustering (It’s necessary because of Time of declustering (It’s necessary because of independence): 30-60 daysindependence): 30-60 days
– Calculate model parameters with maximum Calculate model parameters with maximum likelihood functionlikelihood function
– Representing results: return levels and confidence Representing results: return levels and confidence intervals with profile likelihoodintervals with profile likelihood
Choosing thresholdChoosing threshold
Expected value of GPD, when threshold is Expected value of GPD, when threshold is uu00::
when when < 1< 1 ( (else infinity). Every else infinity). Every uu > > uu00::
Expected value is linear, the shape parameter isExpected value is linear, the shape parameter is
constant function in constant function in uu..
1
0
00
uuXuXE
1100
uuuXuXE uu
Average exceed curveAverage exceed curve
Szeged(H)Szeged(H)
y = -0.2844x + 289.2
50
100
150
200
250
100 200 300 400 500 600 700 800 900
Küszöbérték (cm)
Átla
gos
meg
hala
dás
(cm
)
Average exceed curveAverage exceed curve
Szeged(Q)Szeged(Q)
y = -0.1741x + 970.2
400
500
600
700
800
900
1000
0 500 1000 1500 2000 2500 3000
Küszöbérték (m3/s)
Átla
gos
meg
hala
dás
(m 3/s
)
Average exceed curveAverage exceed curve
Polgár(H)Polgár(H)
y = -0.247x + 219.4
0
50
100
150
200
250
300
0 100 200 300 400 500 600 700 800
Küszöbérték (cm)
Átla
gos
meg
hala
dás
(cm
)
Average exceed curveAverage exceed curve
Polgár(Q)Polgár(Q)
y = -0.2677x + 1237.4
300
400
500
600
700
800
0 500 1000 1500 2000 2500 3000 3500
Küszöbérték (m 3/s)
Átla
go
s m
eg
ha
lad
ás
(m
3/s
)
Shape parameterShape parameter
Shape parameterShape parameter
Záhony(H)Záhony(H)
Záhony(H)Záhony(H)
Záhony(Q)Záhony(Q)
Záhony(Q)Záhony(Q)
Polgár(H)Polgár(H)
Polgár(Q)Polgár(Q)
Results, VásárosnaményResults, Vásárosnamény
Datatype ThresholdScale
parameter
Shapeparamete
r
Return level in 100 years
Confidenceinterval (95%)
H 300 cm 345.4 -0.5422 908 cm [893, 944]
H 400 cm 289 -0.5372 908 cm [893, 948]
H 500 cm 238.8 -0.5474 908 cm [892, 946]
H 600 cm 174.3 -0.5108 908 cm [889, 956]
Q 800 m3/s 836.4 -0.1904 3735 m3/s [3426, 4307]
Q 1100 m3/s 781 -0.1936 3727 m3/s [3427, 4395]
Q 1300 m3/s 797 -0.2346 3682 m3/s [3434, 4258]
Q 1500 m3/s 772 -0.2493 3677 m3/s [3441, 4253]
Other resultsOther results
Station Datatype ThresholdReturn level in 100 years
Confidenceinterval (95%)
Tiszabecs H 300 cm 679 cm [616, 864]
Tivadar H 500 cm 912 cm [875, 994]
Tivadar Q 800 m3/s 3188 m3/s [2692, 4680]
Záhony H 450 cm 744 cm [718, 810]
Záhony Q 1500 m3/s 3683 m3/s [3351, 4627]
Polgár H 470 cm 789 cm [759, 871]
Szolnok H 600 cm 949 cm [921, 1031]
Szeged H 550 cm 937 cm [908, 1014]
Szeged Q 1500 m3/s 4150 m3/s [3746, 5522]
SummarySummary
– On the majotity of data series the fitting is On the majotity of data series the fitting is appropriate, the results are resonableappropriate, the results are resonable
– The final result is slighty affected by the selection The final result is slighty affected by the selection of thresholdsof thresholds
– In the cause of the data of Polgár(Q) and In the cause of the data of Polgár(Q) and Szolnok(Q) the model does not fit properlySzolnok(Q) the model does not fit properly
– The reason for that can be found in the incidental The reason for that can be found in the incidental errors of the calculation of dataerrors of the calculation of data