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Trends in the Occurrence of Trends in the Occurrence of Extreme Events:Extreme Events:
An Example From the North SeaAn Example From the North Sea
Manfred Mudelsee
Department of Earth Sciences
Boston University, USA
2
ResultsResults
• Computer program XTREND estimates trends in occurrence rate (risk)
• Can be applied to occurrence of extreme climate events (floods, storms, etc.)
• Example: major windstorms in North Sea region over past 500 years
• Preliminary result, occurrence rate: (1) low at 1800, (2) recent upward trend
3
Background—Background—StatisticalStatistical
• Risk = adverse probability
• Occurrence rate = probability per year
• Occurrence rate may be time-dependent
• Statistical model: inhomogeneous Poisson process
4
Background—ClimatologicalBackground—Climatological
• Climate system is complex (atmosphere, ocean, surface; nonlinear interactions)
• Intergovernmental Panel on Climate Change (IPCC) (Houghton et al. 2001):- changed atmosphere (greenhouse gases)- radiative effects- concern: increased risk of extreme climate
5
Relevance to (re)insurers (1)Relevance to (re)insurers (1)
• Losses in Europe caused by extreme climate events:
Event Deaths Damages ($)
Oder flood 1997 114 4.4 billion
Elbe flood 2002 36 13.2 billion
Windstorms 1990-2001
>430 30 billion
6
Relevance to (re)insurers (2)Relevance to (re)insurers (2)
• Trends in the occurrence rate of extreme climate events should be estimated and tested before an extreme value analysis.
nonstationarity
• Extrapolation of trends: risk prediction !?
7
The Rest of This TalkThe Rest of This Talk
• Method: occurrence rate estimation
• Method: testing for trend
• Example: winter floods in Elbe
• Example: windstorms in North Sea (RPI)
• Demonstration (XTREND):estimating/testing occurrences of major windstorms in North Sea
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Occurrence Rate Estimation (1)Occurrence Rate Estimation (1)
• Dates of extreme events:T1, T2,…,TN
• Observation interval [TS; TE]
• Inhomogeneous Poisson process:- independent events- no simultaneous events
- Prob(event in [t; t+]0 [TS; TE]) = · (t)
- occurrence rate or intensity (t) (unit:1/yr)
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1500 2000
Elbe, winter floods
Occurrence Rate Estimation (2)Occurrence Rate Estimation (2)
10
1500 2000
Elbe, winter floods
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Elbe, winter floods1500 2000
0
5
10
15
4 12 2 6 3
12
Elbe, winter floods1500 2000
0
5
10
15
4 12 2 6 3
Steps toward a better method
13
Elbe, winter floods1500 2000
0
5
10
15
4 12 2 6 3
Steps toward a better method Advantage
1. continuous shifting more estimation points (kernel estimation) no ambiguity
14
Elbe, winter floods1500 2000
0
5
10
15
4 12 2 6 3
Steps toward a better method Advantage
1. continuous shifting more estimation points (kernel estimation) no ambiguity
2. Gaussian (not uniform) smooth estimatekernel
15
Elbe, winter floods1500 2000
0
5
10
15
4 12 2 6 3
Steps toward a better method Advantage
1. continuous shifting more estimation points (kernel estimation) no ambiguity
2. Gaussian (not uniform) smooth estimatekernel
3. cross-validated minimal estimation bandwidth error
16
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
17
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
OK, how significant is that trend ??
18
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
19
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
1500 2000
bootstrap resample (with replacement, same size)
20
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
1500 2000
bootstrap resample (with replacement, same size)
21
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
1500 2000
bootstrap resample (with replacement, same size)
1500 2000
2nd bootstrap resample
22
Elbe, winter floods1500 2000
00.10.20.30.4
occu
rren
ce r
ate
(yr-1
)
1500 2000
bootstrap resample (with replacement, same size)
1500 2000
2nd bootstrap resample
take 2000 bootstrap resamples
23
Elbe, winter floods1500 2000
00.10 .20 .30 .4
occu
rre
nce
rate
(yr
-1)
90% percentile confidence band
24
Elbe, winter floods1500 2000
00.10 .20 .30 .4
occu
rre
nce
rate
(yr
-1)
90% percentile confidence band
Method:
Cowling et al. (1996) Journal of the American Statistical Association 91: 1516–1524.
Mudelsee M (2002) Sci. Rep. Inst. Meteorol. Univ. Leipzig 26: 149–195. [available online]
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Testing for TrendTesting for Trend
• Null hypothesis H0: “(t) is constant”
• Test statistic:
u = [∑i Ti /N−(TS+TE)/2] / [(TS−TE)/(12 N)1/2]
• Under H0: u ~ N(0; 1)• Cox & Lewis (1966) The Statistical Analysis of
Series of Events. Methuen, London.
26
Winter Floods in ElbeWinter Floods in Elbe
1000 1200 1400 1600 1800 2000Year
0 .00 .10 .20 .30 .4
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
Mudelsee et al. (2003) Nature 425: 166–169.
test
27
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
• Acknowledgments:- RPI- Jens Neubauer, Institute of Meteorology,
University of Leipzig, Germany- Frank Rohrbeck, Institute of Meteorology,
Free University Berlin, Germany
28
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
29
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
• Long-term perspective (last 500 yr)
• Information: historical documents- Lamb H (1991) Historic Storms of the North Sea.
Cambridge University Press, Cambridge.- Weikinn C (1958–2002) Quellentexte zur
Witterungsgeschichte Europas von der Zeitwende bis zum Jahre 1850: Hydrographie. Vols. 1–4, Akademie-Verlag, Berlin, Vols. 5–6, Gebrüder Borntraeger, Berlin.
30
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
10–12 December 1792Area: Whole North Sea [...]Maximum wind strength: The strongest gusts of the surface wind probably exceeded 100 knots over both these regions [southern North Sea near Dutch and German coast].Minimal pressure estimate: 945 mbar.
[From Lamb 1991]
31
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
1792 & 10. Dez. & Gegend von Hamburg & Sturmflut & & 1 & I, 5: 539 (4260)10. Dez. Der Sturm trieb das Wasser zu Hamburg 20 F 6 Z über die ordin. Ebbe, eine Höhe, wie sie daselbst, soweit die Nachrichten reichen, noch nie gehabt, zu Cuxhafen 20 F 3 Z. Sie richtete in [...] (Fr. Arends 1833 “Physische Geschichte d. Nordsee-Küste etc.” II. S. 305.)
[From Weikinn 1958–2002]
32
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
33
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
34
WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
1500 1600 1700 1800 1900 2000Year
35
Demonstration (XTREND):Demonstration (XTREND):WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
1500 1600 1700 1800 1900 2000Year
36
Demonstration (XTREND): Demonstration (XTREND): WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
• All regions, 1500–1990, both magnitudes
1500 1600 1700 1800 1900 2000Year
0
0.2
0.4
0.6
0.8
1 Occurrencerate (1/yr)
90%
37
Next Steps: Next Steps: WWindstorms in North Sea (RPI)indstorms in North Sea (RPI)
• Inter-check (Lamb vs. Weikinn)• Homogeneity problem: document loss• Extension 1990–2003 using measurements• Differentiation: region, magnitude