1. Covariate-dependent modeling of
extreme events by nonstationary
Peaks Over Threshold analysis
A review and a case study in Spain
Santiago Beguería
santiago.begueria@csic.es
Estación Experimental de Aula Dei,
Consejo Superior de Investigaciones Cientícas (EEADCSIC), Zaragoza, España
Liberec, February 17th 2012
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 1 / 52

2. Motivation, I
We all love the extreme value theory (EVT) applied to the analysis of
climatic hazard: it's elegant, simple, and provides useful and
understandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation of
the EVT.
Relatively recent extensions of the EVT allow for non-stationary
analysis (Coles, 2001), and an increasing number of authors are
exploring their possibilities for the analysis of climatic hazard.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52

3. Motivation, I
We all love the extreme value theory (EVT) applied to the analysis of
climatic hazard: it's elegant, simple, and provides useful and
understandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation of
the EVT.
Relatively recent extensions of the EVT allow for non-stationary
analysis (Coles, 2001), and an increasing number of authors are
exploring their possibilities for the analysis of climatic hazard.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52

4. Motivation, I
We all love the extreme value theory (EVT) applied to the analysis of
climatic hazard: it's elegant, simple, and provides useful and
understandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation of
the EVT.
Relatively recent extensions of the EVT allow for non-stationary
analysis (Coles, 2001), and an increasing number of authors are
exploring their possibilities for the analysis of climatic hazard.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 2 / 52

5. Motivation, II
Most studies up to now focused on identifying temporal trends in the
occurrence of extreme events, i.e. making time a covariate.
In the last few years other covariates with an expected inuence on
the occurrence of extreme events are being used, too high
relevance for the statistical downscaling of reanalysis or model data,
which typically cannot be used for local impact studio.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 3 / 52

6. Motivation, II
Most studies up to now focused on identifying temporal trends in the
occurrence of extreme events, i.e. making time a covariate.
In the last few years other covariates with an expected inuence on
the occurrence of extreme events are being used, too high
relevance for the statistical downscaling of reanalysis or model data,
which typically cannot be used for local impact studio.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 3 / 52

7. Summary
In this talk I will present ongoing research on the relationship between
extreme precipitation and teleconnection indices in Spain, using
non-stationary EVT techniques. The talk is organized as follows:
A review of non-stationary Peaks Over Threshold analysis
Case study: relationship between teleconnection indices and extreme
rainfall events in Spain
Conclusions and future work
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52

8. Summary
In this talk I will present ongoing research on the relationship between
extreme precipitation and teleconnection indices in Spain, using
non-stationary EVT techniques. The talk is organized as follows:
A review of non-stationary Peaks Over Threshold analysis
Case study: relationship between teleconnection indices and extreme
rainfall events in Spain
Conclusions and future work
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52

9. Summary
In this talk I will present ongoing research on the relationship between
extreme precipitation and teleconnection indices in Spain, using
non-stationary EVT techniques. The talk is organized as follows:
A review of non-stationary Peaks Over Threshold analysis
Case study: relationship between teleconnection indices and extreme
rainfall events in Spain
Conclusions and future work
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52

10. Summary
In this talk I will present ongoing research on the relationship between
extreme precipitation and teleconnection indices in Spain, using
non-stationary EVT techniques. The talk is organized as follows:
A review of non-stationary Peaks Over Threshold analysis
Case study: relationship between teleconnection indices and extreme
rainfall events in Spain
Conclusions and future work
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 4 / 52

11. Short review of NSPOT analysis, I
200
150
Intensity (mm day-1)
100
50
0
1950 1960 1970 1980 1990 2000 2010
Time (n = 266)
Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 5 / 52

12. Short review of NSPOT analysis, II
200
150
Intensity (mm day-1)
100
50
0
1950 1960 1970 1980 1990 2000 2010
Time (n = 266)
Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 6 / 52

13. Short review of NSPOT analysis, III
Stationary POT: assuming independent inter-arrival times, the POT data
follows a Generalized-Pareto distribution.
Probability of exceedance:
−1/κ
x − x0
P (X x |X x0 ) = 1 − λ 1 + κ (1)
α
Quantile corresponding to a return period T:
κ
α 1
XT = x0 + 1− (2)
κ λT
(beware of alternative conventions: x0 = u , α = σ, κ = ξ )
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 7 / 52

14. Short review of NSPOT analysis, IV
Approaches for assessing non-stationarity in POT modeling:
Split-sample approach (Li et al., 2005)
Moving kernel (Hall and Tajvidi, 2000)
Non-stationary POT (NSPOT) modeling
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 8 / 52

15. Short review of NSPOT analysis, GPar distribution
Return period curves,
V
2500
NAO−
2000
1500
daily EI
NAO+
1000
500
0
1 2 5 10 20 50 100
return period (years)
Split-sample approach: independent models for positive and negative phases of NAO (Angulo et al., 2011).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 9 / 52

16. Short review of NSPOT analysis, VI
2110 ´
S. BEGUERIA et al.
X287 X164
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.05 0.10 0.15 0.20 0.25
1930 1950 1970 1990 2010 1930 1950 1970 1990 2010
X9198 X2234
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
0.4
0.3
0.2
0.1
0.0
1930 1950 1970 1990 2010 1930 1950 1970 1990 2010
Figure 6. Model uncertainty: time variation of the 10-year return period event intensity standard error (mm by a nonparametric NSPOT d−1 )
model (dots), a linear NSPOT model (solid line), a high order polynomial NSPOT model (short-dashed line) and a stationary POT model
Moving kernel approach: time variability in the stations. Location of the stations a moving Figure 1. of the previous 20 years of
(long-dashed line) in four P10 quantile, based on is shown in window
data (Beguería et al., 2011).
by more than 50% (e.g. series X164). The time evolution and the most parsimonious (i.e. with the lowest polyno-
of q10 depended on the evolution
(santiago.begueria@csic.es) of both α (scale) and mial order) was chosen. Following this approach, statis- 10 / 52
NSPOT with covariates Liberec, 17/02/2012

17. Short review of NSPOT analysis, VII
Stationary POT:
−1/κ
x − x0
P (X x |X x0 ) = 1 − λ 1 + κ
α
Non-stationary POT:
−1/κ(c )
x − x0 (c )
P (X x |X x0 , C ) = 1 − λ(c ) 1 + κ(c ) (3)
α(c )
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 11 / 52

18. Short review of NSPOT analysis, VIII
Some examples of NSPOT analysis of climatic variables:
Time dependence of T and P (Smith, 1999)
Nogaj et al. (2006) time trends of T extremes over the NA region
Laurent and Parey (2007), Parey et al. (2007), T extremes in France
Méndez et al. (2006), trends and seasonality of POT wave height
Yiou et al. (2006) trends of POT discharge in the Czech Republic
Abaurrea et al. (2007) trends of POT T in the IP
Acero et al. (2011), Beguería et al. (2011), trends in POT P, IP
Friederichs (2010), Kallache et al. (2011), downscaling of POT P based on
reanalysis / GCM data
Tramblay et al. (2012), covariation between POT P extremes and
atmospheric covariates, SE France
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 12 / 52

19. Teleconnections aecting precipitation in IP, I
The North Atlantic Oscillation (NAO).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 13 / 52

20. Teleconnections aecting precipitation in IP, II
The North Atlantic Oscillation (NAO).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 14 / 52

21. Teleconnections aecting precipitation in IP, III
The Mediterranean Oscillation (MO, Palutikof 2003).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 15 / 52

22. Teleconnections aecting precipitation in IP, IV
The Mediterranean Oscillation (MO, Palutikof 2003).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 16 / 52

23. Teleconnections aecting precipitation in IP, V
The Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 17 / 52

24. Teleconnections aecting precipitation in IP, VI
The Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 18 / 52

25. Teleconnections aecting precipitation in IP, VII
1.5
1.5
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q
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q q q qq q
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q
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q q
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q
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q q q q q q
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q qq q qqqq qq qq q q q qq q q
q q qqq q q q
q q q q q qqq q qqqq q q qq
qq q qqq q q
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q q
q q
q
q q q q q q q q qq q q
q q qq q
q q qq q q
q q q
q q
q q qqq
q q q
q q q
q q
q q
qq q q
−1.5
r2=0.047
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
MOi
Correlations between teleconnection indices.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 19 / 52

26. Dataset, I
Stations network
4800000
q q q
q q
q q q qq
q
q q
q q q
q qq q q
q
q q
q q
q q
q q
q q
4600000
q q q q
q
q
q
q
q q
q q
q q
qq
q
q
q q
4400000
q q
q
q
q
qq
qq
q q
qq q q q
q
q
4200000
q q
q
q q q
qq
q
q qq
q q
q
q
q
q
4000000
q
q
q
q
106 stations, 58 daily precipitation 2e+05 reconstructed for6e+05 period 8e+05
0e+00 series 4e+05 the 1950-2009 1e+06
(source: AEMET).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 20 / 52

27. Dataset, II
Teleconnection indices (Reykjavik, Padova, Lod and Gibraltar). Sources: http://www.cru.uea.ac.uk,
http://www.ub.es.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 21 / 52

28. Dataset, III
1
0
NAOi
−1
−2
0 20 40 60 80
2
1
WEMOi
0
−1
−2
0 20 40 60 80
15
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
Declustering: intensity and magnitude series and associated teleconnection indices.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 22 / 52

29. Dataset, IV
1
0
NAOi
−1
−2
0 20 40 60 80
2
1
WEMOi
0
−1
−2
0 20 40 60 80
20
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
Declustering: intensity and magnitude series and associated teleconnection indices.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 23 / 52

30. Dataset, V
1
0
NAOi
−1
−2
0 20 40 60 80
20
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
2
1
WEMOi
0
−1
−2
0 20 40 60 80
20
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
Declustering: intensity and magnitude series and associated teleconnection indices.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 24 / 52

31. Dataset, VI
1
0
NAOi
−1
−2
0 20 40 60 80
15
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
2
1
WEMOi
0
−1
−2
0 20 40 60 80
15
P (mm/day)
10
5
0
0 20 40 60 80
Time (days since 01/11/2001)
Declustering: intensity and magnitude series and associated teleconnection indices.
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 25 / 52

32. Analysis, I
x |X x0 ) = 1 − λ 1 + κ x −x0
−1/κ
M0: P (X α
−1/κ
M1: P (X x |X x0 , C ) = 1 − λ 1 + κ x −x0 (c )
α(c )
x0 = β0 + βi c (4)
α = γ0 γic (5)
κ = δ (6)
λ = ε (7)
Likelihood ratio test:
D = − 2 ( 1 (M1 ) − 0( M0 )) (8)
distributed according to χ2 (with d.f.
k k = 4).
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 26 / 52

33. Analysis, II
R, package ismev (Stuart Coles, ported to R by Alec Stephenson).
...
m0 gpd.t(xdat=dat, threshold=u, npy=rate)
...
uu predict(lm(v∼cdat))
m1 gpd.t(xdat=dat, threshold=uu, npy=rate, ydat=cdat,
sigl=1, siglink=exp)
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 27 / 52

34. Analysis, III
Histogram of tel$naoi Histogram of pnorm(tel$naoi)
4000
1000 1200
3000
800
Frequency
Frequency
2000
600
400
1000
200
0
0
-2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0
tel$naoi pnorm(tel$naoi)
Covariates: NAOi and pnorm(NAOi)
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 28 / 52

35. Example: Valencia, I
Location of station: VALENCIA, X8416
4800000
4600000
4400000
q
4200000
4000000
Spatial location 0e+00 5e+05 1e+06
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 29 / 52

36. Example: Valencia, II
Threshold (u = q0.9 = 23.5, n = 229)
50
100
45
50
Return period (years)
40
25
Mean Excess
35
10
30
5
25
2
20
1
q
q
q
q
qqq
qq
q
q
qq
q
q
q
q
q
10 20 30 40 50 60 70 80 q
q
q
q
q
50
q
q
q
q
q
q
q
qq
q
q
u q
q
q
q
Stationary model: xed threshold
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 30 / 52

37. Example: Valencia, III
29) Stationary model (AIC=1875)
100
q
50
Return period (years)
q
25
q
q
q
10
q
q
q
q
q
q
q q
5
q q
q
qq
q q
q
q
q
q
qqq
q
2
q
q
qq
q
q
qq
q
qq
q
qq
qq
q
qq
1
q
q
q
q
qqq
qq
q
q
qq
qq
q
q
q
70 80 q
q
q
q
q
50 100 150 200 250
qq
q
q
q
q
q
q
q
q
q
q
q
q
q Intensity (mm day−1)
Intensity (mm)
Intensity (days)
Stationary model: quantile plot
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 31 / 52

38. Example: Valencia, IV
200
q
35
q
30
150 q q
Intensity (mm day−1)
Scale parameter
q
q
25
q q q q
q q q
100
qq
qq qq
q
20
q
qq q q q
qq q q q q
q q
q
qq q
q q q q q q
qq q qq qq q q q
qq q q q
50
qq q q
15
q q q q q q q q q
q q qq
q q q
qq qqqq
q q
qq q
q q q
q q qq qq q q q q q q q q q q q
q qqq qq q
q
q q q q q qq q q q q q q q
q q qqq q
q q q q q q qq q q q q q
qqqq qq q q q q q q q q qqq q q qq q q
qqqq qq q
q q q
qqq q q q qq q q q
q qq q q q q qq q q q
q qq
q qq q
q q q
qq q q
q qq q qqq q q q q q q qq qqqq qqq q q q qqq q qq qq q qqq qq q q q qqq
q qqq qq q q q q qq qq q qq q q qq q
q q q qq q qq q
qq q q q
q q q q
q q
q qq qq q q q qqq q q q q qq q
10
q qq qq qqqqq qq q qqq q q q q q qq qq qq qqq qqq q qqq q qq qqq qqq
q q q q qq qq q qq qq
q
q q
q q q q qq q q qqq qq q qqq q
q q q q qq q q qq q q q q q
q qq q q q qqq q q q q qqq q q
q qqqqqqqqqqqqq qqqqqqqqqq qqqqqq qqqqqqqqqq q qqqqqqqqqqq qqqq q qq
qqq q qq qq q qq q q qqq qqqq q qqq q qqqq q q qqqq q q
qqqqqqq qqqq qqqqq qqqqq qqqqqq qqq q qqq q q q qqqq qqqq qqqqq qq
q q q q q q q q qqq q q
q
q qqqqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqqqqqq q qqqqqqqqqqqqq
qqq
q q q q q q q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q qq qqq q qq q qqqqqq q q q qq qq qq q qq q q q qqqqq q q qqqqqqqq qq
qqq q
qq qqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq q qq qqqqqqqq qqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqqqqqqqq
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q q
q q q q q q q q q qq q q q q q qq q
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q
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q q qq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqq
qq q qq q q qq qq q
q q q q
q qq q qqqqq
qq q qq q
q qqqqqqqqq qqqq qqqq q qqqq qqqqqqqq qq qqq qqqqqqq q q qqqqqqq qq qq
q q qq q qq q qqqq qqq q qqq qq q qqq qqqqq qq q qqq q q qq qq q
0
0.0 0.2 0.4 0.6 0.8 1.0
WEMOi (n = 200)
100
Non-stationary model: threshold model
100
2 1 0.5 0 −0.5 −1 −2
50
50
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 32 / 52
s)
s)

39. Example: Valencia, V
35
30
Scale parameter
25
20
15
q
q
q q
q q
q qq q q
q q qqq
q
10
q qqq qqq
qq
q qq q q q
qq q q
q qq qq qqq
qqqqqqq qq
qq q qq qqq
q q qq qq
q q qq q
q
qqqqqqqqqq
qqqqqqqqqq
q
qqqqqqqq
qqqq qqqqq
q q q
qqqqqqqqqq
q qq q qqq
qqqqqqqqq
q q q qqq
qqqqqqqqqq
q qq
1.0 −2 −1 0 1 2
WEMOi (AIC=1569*)
100
Non-stationary model: scale parameter model
−2 220
210
200
50
190
180
(santiago.begueria@csic.es) 0
17
160 NSPOT with covariates Liberec, 17/02/2012 33 / 52
)

40. 0.0 0.2 0.4 0.6 0.8 1.0
Example: Valencia, VI
WEMOi (n = 200)
100
100
2 1 0.5 0 −0.5 −1 −2
50
50
Return period (years)
Return period (years)
25
25
10
10
5
5
2
2
1
1
50 100 150 200 250 300
Intensity (mm day−1)
Non-stationary model: quantile plot
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 34 / 52

41. 1.0 −2 −1 0 1 2
Example: Valencia, VII
WEMOi (AIC=1569*)
100
−2 220
210
200
50
190
180
170
160
Return period (years)
25
150 140
130
120
10
110
100
90
5
80
70
60
50
2
40
30
1
300 −2 −1 0 1 2
WEMOi
Non-stationary model: quantile plot
(santiago.begueria@csic.es) NSPOT with covariates Liberec, 17/02/2012 35 / 52