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Covariate-dependent modeling of
extreme events by non�stationary
Peaks Over Threshold analysis
A review and a case study in Spain
Santiago Beguerí[email protected]
Estación Experimental de Aula Dei,Consejo Superior de Investigaciones Cientí�cas (EEAD�CSIC), Zaragoza, España
Liberec, February 17th 2012
([email protected]) NSPOT with covariates Liberec, 17/02/2012 1 / 52
Motivation, I
We all love the extreme value theory (EVT) applied to the analysis ofclimatic hazard: it's elegant, simple, and provides useful andunderstandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation ofthe EVT.
Relatively recent extensions of the EVT allow for non-stationaryanalysis (Coles, 2001), and an increasing number of authors areexploring their possibilities for the analysis of climatic hazard.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 2 / 52
Motivation, I
We all love the extreme value theory (EVT) applied to the analysis ofclimatic hazard: it's elegant, simple, and provides useful andunderstandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation ofthe EVT.
Relatively recent extensions of the EVT allow for non-stationaryanalysis (Coles, 2001), and an increasing number of authors areexploring their possibilities for the analysis of climatic hazard.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 2 / 52
Motivation, I
We all love the extreme value theory (EVT) applied to the analysis ofclimatic hazard: it's elegant, simple, and provides useful andunderstandable results in terms of magnitude / frequency curves.
The stationarity assumption, though, was an important limitation ofthe EVT.
Relatively recent extensions of the EVT allow for non-stationaryanalysis (Coles, 2001), and an increasing number of authors areexploring their possibilities for the analysis of climatic hazard.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 2 / 52
Motivation, II
Most studies up to now focused on identifying temporal trends in theoccurrence of extreme events, i.e. making time a covariate.
In the last few years other covariates with an expected in�uence onthe occurrence of extreme events are being used, too �> highrelevance for the statistical downscaling of reanalysis or model data,which typically cannot be used for local impact studio.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 3 / 52
Motivation, II
Most studies up to now focused on identifying temporal trends in theoccurrence of extreme events, i.e. making time a covariate.
In the last few years other covariates with an expected in�uence onthe occurrence of extreme events are being used, too �> highrelevance for the statistical downscaling of reanalysis or model data,which typically cannot be used for local impact studio.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 3 / 52
Summary
In this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-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 extremerainfall events in Spain
Conclusions and future work
([email protected]) NSPOT with covariates Liberec, 17/02/2012 4 / 52
Summary
In this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-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 extremerainfall events in Spain
Conclusions and future work
([email protected]) NSPOT with covariates Liberec, 17/02/2012 4 / 52
Summary
In this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-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 extremerainfall events in Spain
Conclusions and future work
([email protected]) NSPOT with covariates Liberec, 17/02/2012 4 / 52
Summary
In this talk I will present ongoing research on the relationship betweenextreme precipitation and teleconnection indices in Spain, usingnon-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 extremerainfall events in Spain
Conclusions and future work
([email protected]) NSPOT with covariates Liberec, 17/02/2012 4 / 52
Short review of NSPOT analysis, I
1950 1960 1970 1980 1990 2000 2010
050
100
150
200
Time (n = 266)
Inte
nsity
(mm
day
-1)
Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0
([email protected]) NSPOT with covariates Liberec, 17/02/2012 5 / 52
Short review of NSPOT analysis, II
1950 1960 1970 1980 1990 2000 2010
050
100
150
200
Time (n = 266)
Inte
nsity
(mm
day
-1)
Peaks-over-threshold (POT) sampling: take only exccedances over a threshold, x − x0
([email protected]) NSPOT with covariates Liberec, 17/02/2012 6 / 52
Short review of NSPOT analysis, III
Stationary POT: assuming independent inter-arrival times, the POT datafollows a Generalized-Pareto distribution.
Probability of exceedance:
P (X > x |X > x0) = 1− λ(1+ κ
x − x0α
)−1/κ
(1)
Quantile corresponding to a return period T :
XT = x0 +α
κ
[1−
(1
λT
)κ](2)
(beware of alternative conventions: x0 = u, α = σ, κ = ξ)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 7 / 52
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
([email protected]) NSPOT with covariates Liberec, 17/02/2012 8 / 52
Short review of NSPOT analysis, V
1 2 5 10 20 50 100
050
010
0015
0020
0025
00
Return period curves, GPar distribution
return period (years)
daily
EI
NAO−
NAO+
Split-sample approach: independent models for positive and negative phases of NAO (Angulo et al., 2011).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 9 / 52
Short review of NSPOT analysis, VI2110 S. BEGUERIA et al.
0.00
0.02
0.04
0.06
0.08
0.10
1930 1950 1970 1990 2010
X287
0.00
0.05
0.10
0.15
0.20
0.25
1930 1950 1970 1990 2010
X164
0.0
0.1
0.2
0.3
0.4
0.5
1930 1950 1970 1990 2010
X9198
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1930 1950 1970 1990 2010
X2234
Figure 6. Model uncertainty: time variation of the 10-year return period event intensity standard error (mm d−1) by a nonparametric NSPOTmodel (dots), a linear NSPOT model (solid line), a high order polynomial NSPOT model (short-dashed line) and a stationary POT model
(long-dashed line) in four stations. Location of the stations is shown in Figure 1.
by more than 50% (e.g. series X164). The time evolutionof q10 depended on the evolution of both α (scale) andκ (shape), although the effect of the latter parameter wasmuch stronger. The amplitude of the confidence bandwas also well correlated to the value of κ , with a negativerelationship. It is noteworthy that, although a linear modelwas used for both parameters, their combined effect canlead to a curvilinear relationship between q10 and time,although the highest/lowest values of q10 always occurredat the beginning/end of the time period.
Comparison of the nonparametric and the parametricapproaches to NSPOT modelling (dots and lines inFigures 4 and 5) shows the main advantages of the latterapproach: (1) to provide estimates of extreme events atthe beginning and the end of the time series, whichare lost in the nonparametric model due to the movingwindow necessary to obtain sub-samples and (2) to offerrobustness against the occurrence of very extreme events,which have a major impact on the nonparametric modeldue to the shorter time span of the sub-samples. Thisis clearly shown by the magnitude of the standard errorof q10, which was much higher for the nonparametricmodel (Figure 6). The standard error of the parametricNSPOT model was, in average, equivalent to that of the(parametric) stationary POT model, illustrating one of theprincipal advantages of the parametric NSPOT approach.
High order polynomial NSPOT models were also fittedto the data (Figure 7). Models of increasing complexitywere tested for significance against the stationary model,
and the most parsimonious (i.e. with the lowest polyno-mial order) was chosen. Following this approach, statis-tical significance was found in 45 (70.3%) models forintensity and 42 (65.5%) for magnitude. The models hadorders between 5 and 9 for both α and κ , which reflectsquite a high complexity. It is evident that high order poly-nomials are more flexible for fitting complex data, butthat comes at the expense of reduced reliability. In fact,the models have the same drawbacks as the nonpara-metric approach: (1) excessive impact of outlier observa-tions resulting in over-fitting and (2) non-reliability at thebeginning and the end of the study period, as shown bythe large increase in the uncertainty bands in many cases(e.g. series X9198 in Figure 7 and short-dashed lines inFigure 6).
With respect to the time series of q10, differenceswere occasionally found between adjacent observatories,reflecting the influence of single, very localized extremeevents. Maps of q10 were produced to visualize the spatialdistribution of the q10 difference at the beginning and theend of the period of study, and to check the regionalcoherence of the stations for which a linear NSPOTmodel was significant (Figures 8 and 9). Differences inq10 were not very high for both the event intensity andmagnitude at the annual level (ranging between −10 and+30% in most cases), and the spatial distribution of thefew significant series was random, suggesting that notrends exist in the study area.
Copyright 2010 Royal Meteorological Society Int. J. Climatol. 31: 2102–2114 (2011)
Moving kernel approach: time variability in the P10 quantile, based on a moving window of the previous 20 years ofdata (Beguería et al., 2011).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 10 / 52
Short review of NSPOT analysis, VII
Stationary POT:
P (X > x |X > x0) = 1− λ(1+ κ
x − x0α
)−1/κ
Non-stationary POT:
P (X > x |X > x0,C ) = 1− λ(c)(1+ κ(c)
x − x0(c)α(c)
)−1/κ(c)
(3)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 11 / 52
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 onreanalysis / GCM data
Tramblay et al. (2012), covariation between POT P extremes and
atmospheric covariates, SE France
([email protected]) NSPOT with covariates Liberec, 17/02/2012 12 / 52
Teleconnections a�ecting precipitation in IP, I
The North Atlantic Oscillation (NAO).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 13 / 52
Teleconnections a�ecting precipitation in IP, II
The North Atlantic Oscillation (NAO).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 14 / 52
Teleconnections a�ecting precipitation in IP, III
The Mediterranean Oscillation (MO, Palutikof 2003).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 15 / 52
Teleconnections a�ecting precipitation in IP, IV
The Mediterranean Oscillation (MO, Palutikof 2003).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 16 / 52
Teleconnections a�ecting precipitation in IP, V
The Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 17 / 52
Teleconnections a�ecting precipitation in IP, VI
The Western Mediterranean Oscillation (WEMO, Martín-Vide and López-Bustins 2006).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 18 / 52
Teleconnections a�ecting precipitation in IP, VII
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−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
MOi
NA
Oi
r2=0.21
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−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
WEMOi
NA
Oi
r2=0.005
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−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
−1.
5−
0.5
0.0
0.5
1.0
1.5
MOi
WE
MO
i
r2=0.047
Correlations between teleconnection indices.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 19 / 52
Dataset, I
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
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4400
000
4600
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4800
000
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Stations network
106 stations, 58 daily precipitation series reconstructed for the period 1950-2009 (source: AEMET).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 20 / 52
Dataset, II
Teleconnection indices (Reykjavik, Padova, Lod and Gibraltar). Sources: http://www.cru.uea.ac.uk,http://www.ub.es.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 21 / 52
Dataset, III
0 20 40 60 80
−2
−1
01
NA
Oi
0 20 40 60 80
−2
−1
01
2
WE
MO
i
0 20 40 60 80
05
1015
Time (days since 01/11/2001)
P (
mm
/day
)
Declustering: intensity and magnitude series and associated teleconnection indices.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 22 / 52
Dataset, IV
0 20 40 60 80
−2
−1
01
NA
Oi
0 20 40 60 80
−2
−1
01
2
WE
MO
i
0 20 40 60 80
05
1020
Time (days since 01/11/2001)
P (
mm
/day
)
Declustering: intensity and magnitude series and associated teleconnection indices.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 23 / 52
Dataset, V
0 20 40 60 80
−2
−1
01
NA
Oi
0 20 40 60 80
05
1020
Time (days since 01/11/2001)
P (
mm
/day
)
0 20 40 60 80
−2
−1
01
2
WE
MO
i
0 20 40 60 80
05
1020
Time (days since 01/11/2001)
P (
mm
/day
)
Declustering: intensity and magnitude series and associated teleconnection indices.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 24 / 52
Dataset, VI
0 20 40 60 80
−2
−1
01
NA
Oi
0 20 40 60 80
05
1015
Time (days since 01/11/2001)
P (
mm
/day
)
0 20 40 60 80
−2
−1
01
2
WE
MO
i
0 20 40 60 80
05
1015
Time (days since 01/11/2001)
P (
mm
/day
)
Declustering: intensity and magnitude series and associated teleconnection indices.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 25 / 52
Analysis, I
M0: P (X > x |X > x0) = 1− λ(1+ κ x−x0
α
)−1/κ
M1: P (X > x |X > x0,C ) = 1− λ(1+ κ x−x0(c)
α(c)
)−1/κ
x0 = β0 + βic (4)
α = γ0γci (5)
κ = δ (6)
λ = ε (7)
Likelihood ratio test:
D = − 2 (`1(M1)− `0(M0)) (8)
distributed according to χ2k(with d.f. k = 4).
([email protected]) NSPOT with covariates Liberec, 17/02/2012 26 / 52
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)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 27 / 52
Analysis, III
Histogram of tel$naoi
tel$naoi
Frequency
-2 0 2 4 6
01000
2000
3000
4000
Histogram of pnorm(tel$naoi)
pnorm(tel$naoi)
Frequency
0.0 0.2 0.4 0.6 0.8 1.0
0200
400
600
800
1000
1200
Covariates: NAOi and pnorm(NAOi)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 28 / 52
Example: Valencia, I
0e+00 5e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
●
Location of station: VALENCIA, X8416
Spatial location
([email protected]) NSPOT with covariates Liberec, 17/02/2012 29 / 52
Example: Valencia, II
10 20 30 40 50 60 70 80
2025
3035
4045
50
u
Mea
n E
xces
s
Threshold (u = q0.9 = 23.5, n = 229)
50 100 150 200 250
Intensity (mm day−1)Intensity (mm)Intensity (days)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Stationary model (AIC=1875)
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([email protected]) NSPOT with covariates Liberec, 17/02/2012 30 / 52
Example: Valencia, III
10 20 30 40 50 60 70 80
2025
3035
4045
50
u
Mea
n E
xces
s
Threshold (u = q0.9 = 23.5, n = 229)
50 100 150 200 250
Intensity (mm day−1)Intensity (mm)Intensity (days)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Stationary model (AIC=1875)
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([email protected]) NSPOT with covariates Liberec, 17/02/2012 31 / 52
Example: Valencia, IV
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
WEMOi (n = 200)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1015
2025
3035
WEMOi (AIC=1569*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
WEMOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30 40
50 60
70 80
90 100
110 120 130
140 150 160 170
180 190 200 210
220 Non-stationary model: threshold model
([email protected]) NSPOT with covariates Liberec, 17/02/2012 32 / 52
Example: Valencia, V
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
WEMOi (n = 200)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1015
2025
3035
WEMOi (AIC=1569*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
WEMOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30 40
50 60
70 80
90 100
110 120 130
140 150 160 170
180 190 200 210
220 Non-stationary model: scale parameter model
([email protected]) NSPOT with covariates Liberec, 17/02/2012 33 / 52
Example: Valencia, VI
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
WEMOi (n = 200)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1015
2025
3035
WEMOi (AIC=1569*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
WEMOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30 40
50 60
70 80
90 100
110 120 130
140 150 160 170
180 190 200 210
220
Non-stationary model: quantile plot
([email protected]) NSPOT with covariates Liberec, 17/02/2012 34 / 52
Example: Valencia, VII
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
WEMOi (n = 200)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1015
2025
3035
WEMOi (AIC=1569*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
WEMOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30 40
50 60
70 80
90 100
110 120 130
140 150 160 170
180 190 200 210
220
Non-stationary model: quantile plot
([email protected]) NSPOT with covariates Liberec, 17/02/2012 35 / 52
Example: Valencia, VIII
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
NAOi (n = 221)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1416
1820
NAOi (AIC=1817*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2 −1 −0.5 0 0.5 1 2
−2 −1 0 1 2
NAOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30
40
50
60
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100
110
120 130
140
150 160
170 180 190
200
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
MOi (n = 223)
Inte
nsity
(m
m d
ay−
1)
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−2 −1 0 1 2
1416
1820
MOi (AIC=1831*)
Sca
le p
aram
eter
50 100 150 200 250 300 350
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
MOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30
40
50
60
70
80 90
100
110
120
130
140 150 160
170 180 190
200 210
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0.0 0.2 0.4 0.6 0.8 1.0
050
100
150
200
WEMOi (n = 200)
Inte
nsity
(m
m d
ay−
1)
●
●
●
●
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●
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●
−2 −1 0 1 2
1015
2025
3035
WEMOi (AIC=1569*)
Sca
le p
aram
eter
50 100 150 200 250 300
Intensity (mm day−1)
Ret
urn
perio
d (y
ears
)
12
510
2550
100 −2−1−0.500.512
−2 −1 0 1 2
WEMOi
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30 40
50 60
70 80
90 100
110 120 130
140 150 160 170
180 190 200 210
220
Non-stationary model: NAO (left), MO (center), WEMO (right)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 36 / 52
Results: event's magnitude, I
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
E�ect of NAO on the 100-years return period event:
([email protected]) NSPOT with covariates Liberec, 17/02/2012 37 / 52
Results: event's magnitude, II
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
E�ect of MO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 38 / 52
Results: event's magnitude, III
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of WEMO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 39 / 52
Results: event's intensity
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of NAO, MO and WEMO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 40 / 52
Results: event's magnitude, winter
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of NAO, MO and WEMO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 41 / 52
Results: event's magnitude, spring
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of NAO, MO and WEMO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 42 / 52
Results: event's magnitude, autumn
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of NAO, MO and WEMO on the 100-years return period event
([email protected]) NSPOT with covariates Liberec, 17/02/2012 43 / 52
Results: threshold independence, I
10 20 30 40 50 60 70 80
2030
4050
60
u
Mea
n E
xces
s
Threshold (u = q0.85 = 16.945, n = 345)
50 100 150 200 250 300
Intensity (mm day−1)Intensity (mm)Intensity (days)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Stationary model (AIC=2766)
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2025
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u
Mea
n E
xces
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Threshold (u = q0.9 = 23.5, n = 229)
50 100 150 200 250
Intensity (mm day−1)Intensity (mm)Intensity (days)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Stationary model (AIC=1875)
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Mea
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xces
s
Threshold (u = q0.95 = 36.1, n = 113)
50 100 150 200 250
Intensity (mm day−1)Intensity (mm)Intensity (days)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Stationary model (AIC=994)
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Quantile plots for rainfall intensity in Valencia, thresholds at u=q85, u=q90 and u=q95
([email protected]) NSPOT with covariates Liberec, 17/02/2012 44 / 52
Results: threshold independence, II
−2 −1 0 1 2
NAOi (AIC=2695*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
20
30
40
50 60
70 80
90 100
110 120
130
140 150
160 170
180
−2 −1 0 1 2
WEMOi (AIC=2476*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Intensity, non−stationary model
20
30
40
50
60
70
80
90 100 110
120 130
140 150
160 170 180 190
200 210
−2 −1 0 1 2
MOi (AIC=2609*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
20
30
40
50 60
70 80
90 100
110 120
130
140
150
160
170 180
−2 −1 0 1 2
NAOi (AIC=1817*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30
40
50
60 70
80
90
100
110
120
130
140
150 160
170 180 190
200
−2 −1 0 1 2
WEMOi (AIC=1569*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Intensity, non−stationary model
30 40
50
60 70
80 90 100
110 120
130
140 150
160 170
180 190
200 210
−2 −1 0 1 2
MOi (AIC=1831*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
30
40
50 60
70
80 90
100
110 120
130 140
150 160 170 180
190 200
−2 −1 0 1 2
NAOi (AIC=1033)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
40
50
60 70
80
90
100
110
120
130
140
150
160
170 180
190 200
210 220
230 240 250
−2 −1 0 1 2
WEMOi (AIC=953*)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
Intensity, non−stationary model
40
60
80
100
120
140
160 180
200 220 240
260 280
300
−2 −1 0 1 2
MOi (AIC=1046)
Ret
urn
perio
d (y
ears
)
12
510
2550
100
40
50
60 70
80
90
100
110 120
130
140 150
160
170 180
190 200 210
220 230 240
Quantile plots for rainfall intensity in Valencia, thresholds at u=q85, u=q90 and u=q95
([email protected]) NSPOT with covariates Liberec, 17/02/2012 45 / 52
Results: threshold independence, III
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
NAO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
MO effect on q100
x times
1.25234
0e+00 2e+05 4e+05 6e+05 8e+05 1e+06
4000
000
4200
000
4400
000
4600
000
4800
000
WEMO effect on q100
x times
1.25234
E�ect of WEMO in rainfall magnitude, thresholds at u=q85, u=q90 and u=q95
([email protected]) NSPOT with covariates Liberec, 17/02/2012 46 / 52
Projected evolution of NAOi, MOi and WEMOi
NA
Oi
1950 1960 1970 1980 1990 2000
−0.
20.
00.
10.
2
NA
Oi
2000 2020 2040 2060 2080 2100−
0.4
0.0
0.2
0.4
MO
i1950 1960 1970 1980 1990 2000
−0.
3−
0.1
0.1
0.2
0.3
MO
i
2000 2020 2040 2060 2080 2100
−0.
20.
00.
20.
4
WE
MO
i
1950 1960 1970 1980 1990 2000
−0.
3−
0.1
0.1
0.2
WE
MO
i
2000 2020 2040 2060 2080 2100
−0.
4−
0.2
0.0
0.2
0.4
Time variation of NAOi, MOi and WEMOi in the 21th Century, INMCM3.0 model output, 48-months convolution
(envelope of SRES A1b, A2 and B1 scenarios)
([email protected]) NSPOT with covariates Liberec, 17/02/2012 47 / 52
Conclusions and future work I
NSPOT analysis is good at capturing the relationship betweenextreme precipitation processes and atmospheric circulation indices.
The results are promising for a variety of applications, includingshort-term warning systems and the statistical downscaling ofGCM/RCM outputs.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 48 / 52
Conclusions and future work II
Clustering methods based on the series of covariates (and not on P).
Other covariates: synoptic scale air�ow parameters (direction,strength, vorticity), speci�c humidity, etc.
Multi-covariate analysis.
Spatial model: take advantage of spatial dependence to reduceuncertainty.
([email protected]) NSPOT with covariates Liberec, 17/02/2012 49 / 52
References I
Smith RL. Extreme value analysis of environmental time series: an application to trend detection inground-level ozone. Statistical Science 4: 367�393 (1989).
Katz RW. Extreme value theory for precipitation: sensitivity analysis for climate change. Adv. Water Resour.23: 133�139 (1999).
Smith RL. Trends in rainfall extremes. Unpublished paper, available at http://www.stat.unc.edu/ (1999).
Hall P, Tajvidi N. Non-parametric analysis of temporal trend when �tting parametric models toextreme-value data. Stat. Science 15: 153�167 (2000).
Coles S. An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag: London (2001).
Palutikof J.P. Analysis of Mediterranean climate data: measured and modelled. In: Bolle H.J. (ed):Mediterranean climate: Variability and trends. Springer-Verlag, Berlin (2003)
Li Y, Cai W, Campbell EP. Statistical modeling of extreme rainfall in southwest Western Australia. J. Clim.18: 852�863 (2005).
Martín-Vide J., López-Bustins J.-A. The Western Mediterranean Oscillation and rainfall in the IberianPeninsula. Int. J. Climatol. 26: 1455�1475 (2006).
Nogaj M, Yioui P, Parey S, Malek F, Naveau P. Amplitude and frequency of temperature extremes over theNorth Atlantic region. Geophys. Res. Lett. 33: L10801 (2006).
Laurent C, Parey S. Estimation of 100-year return-period temperatures in France in a non-stationary climate:results from observations and IPCC scenarios. Global and Planetary Change 57: 177�188 (2007).
Parey S, Malek F, Laurent C, Dacunha-Castelle D. Trends and climate evolution: statistical approach forvery high temperatures in France. Climatic Change 81: 331�352 (2007).
Méndez FJ, Menéndez M, Luceño A, Losada IJ. Estimation of the long-term variability of extreme signi�cantwave height using a time-dependent Peak Over Threshold (POT) model. Journal of Geophysical Research C:Oceans 111: C07024 (2006).
Yiou P, Ribereau P, Naveau P, Nogaj M, Brazdil R. Statistical analysis of �oods in Bohemia (CzechRepublic) since 1825. Hydrological Sciences Journal 51: 930�945 (2006).
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References II
Abaurrea J, Asín J, Cebrián AC, Centelles A. Modeling and forecasting extreme hot events in the centralEbro valley, a continental-Mediterranean area. Global and Planetary Change 57: 43�58 (2007).
Angulo et al. (2011)
Beguería S, Angulo-Martínez M, Vicente-Serrano S, López-Moreno JI, El-Kenawy A. Assessing trends inextreme precipitation events intensity and magnitude using non-stationary peaks-over-threshold analysis: acase study in northeast Spain from 1930 to 2006. International Journal of Climatology 31: 2102�2114(2011).
Acero FJ, García JA, Gallego MC. Peaks-over-Threshold Study of Trends in Extreme Rainfall over theIberian Peninsula. Journal of Climate 24: 1089�1105 (2011).
Friederichs P. Statistical downscaling of extreme precipitation events using extreme value theory. Extremes13: 109�132 (2010).
Kallache M, Vrac M, Naveau P, Michelangeli P-A. Non-stationary probabilistic downscaling of extremeprecipitation. Journal of Geophysical Research 116, D05113 (2011).
Tramblay Y, Neppel L, Carreau J, Sánchez-Gómez E. Extreme value modelling of daily areal rainfall overMediterranean catchments in a changing climate. Hydrological Processes, DOI: 10.1002/hyp.8417 (2012).
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