Bevacqua Emanuele
A model based on Pair-Copula Construction to analyze and represent Compound Flooding E. Bevacqua1 , D. Maraun1 , I.H. Haff2 , M. Widmann3 , M. Vrac4 , and C. Manning3 1 Wegener
Center for Climate and Global Change University of Graz
2 Norwegian
Computing Center, Oslo
3 University
of Birmingham
4 LSCE,
Paris
The 13th Internation Meeting on Statistical Climatology, 2016
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
1 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
1 / 12
Compound Events
Category of extreme events which has received little attention so far [IPCC SREX, 2012]:
Definition A compound event is an extreme impact that depends on multiple statistically dependent variables or events. [Leonard et al., 2013]
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
2 / 12
Compound Events
Category of extreme events which has received little attention so far [IPCC SREX, 2012]:
Definition A compound event is an extreme impact that depends on multiple statistically dependent variables or events. [Leonard et al., 2013]
−→ Compound Floods [Seth Westra, talk yesterday]
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
2 / 12
Case Study: Ravenna (Italy) Compound Flood, 6-2-2015
Yi = variables
Xi = predictors
Data Set: Daily, Winter (Nov-March), Period: 2009-2015 Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
3 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
3 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
Emanuele Bevacqua
Statistical Model for Compound Events
(1)
13th IMSC, 2016
4 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
2
(1)
The Y variables follow a joint probability distribution function (pdf): fY (y1 , y2 , y3 )
Emanuele Bevacqua
fY|X (y1 , y2 , y3 |x1 , x2 )
Statistical Model for Compound Events
13th IMSC, 2016
4 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
2
The Y variables follow a joint probability distribution function (pdf): fY (y1 , y2 , y3 )
3
(1)
fY|X (y1 , y2 , y3 |x1 , x2 )
Through simulating the Y we get simulated values of the impact: hsim. := h(Y1sim. , Y2sim. , Y3sim. )
4
(2)
Statistical analysis: study the statistical characteristics of the events and asses their associated risk. Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
4 / 12
Copula theory Consider the random variables Y = (Y1 , ..., Yn ), with continuous marginal cumulative distribution functions F1 (y1 ), ..., Fn (yn ). Using the Sklar theorem [Sklar, 1959], we get: f (y1 , ..., yn ) = f1 (y1 ) · ... · fn (yn ) · c(F1 (y1 ), ..., Fn (yn ))
c describes the dependence, independently from the marginal structures. Decomposition → flexibility.
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
5 / 12
Pair-Copula Construction Many available bivariate copulas but limited choice for higher dimensions (n > 2). Copulas for n > 2 generally assume same kind of dependence for all the pairs (e.g. either all tail dependent or not).
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Pair-Copula Construction Many available bivariate copulas but limited choice for higher dimensions (n > 2). Copulas for n > 2 generally assume same kind of dependence for all the pairs (e.g. either all tail dependent or not). Pair-Copula Construction theory. 3-Dim example: f123 (y1 , y2 , y3 ) = f3 (y3 ) · f1 (y1 ) · f2 (y2 ) · c31 (u3 , u1 ) · c12 (u1 , u2 ) · c32|1 (u3|1 , u2|1 ) Further decomposition → more flexibility!
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Impact function
2 h = a1 Y1Sea + a21 Y2River + a22 Y2River + a3 Y3River
(3)
1.0
1.5
Predicted (m) 2.0 2.5
3.0
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1.0
Emanuele Bevacqua
● ●
1.5
2.0 2.5 Observed (m)
3.0
Statistical Model for Compound Events
13th IMSC, 2016
7 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level YSea (t) = a SLPRavenna (t)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Meteorological Predictor of River Levels (one predictor) wT = Ptotal (T ) − E (T ) + Smelt (T ) − Sfall (T ) 0 YRiver (t) = a
t X T =t−1
Emanuele Bevacqua
wT + b
t X
wT
T =t−10
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Meteorological predictor of River Levels (one predictor) wT = Ptotal (T ) − E (T ) + Smelt (T ) − Sfall (T ) 0 YRiver (t) = a
t X T =t−1
Emanuele Bevacqua
wT + b
t X
wT
T =t−10
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Water level (m) 2.0 3.0 4.0
Non-stationary model (5-dim): Time series
1.0
Observed Mean Predicted 95% Prediction Interval
0
100
200 Time (days)
300
400
the non-stationary model captures the autocorrelation the observed signal is usually inside the prediction interval
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
9 / 12
Return Level (m) 2.5 3.5
Return levels
1.5
Observed 5−Dim Model 5−Dim Semi−Independent Model 3−Dim Model 3−Dim Semi−Independent Model
5
Emanuele Bevacqua
10 15 Return Period (Years)
Statistical Model for Compound Events
20
25
13th IMSC, 2016
10 / 12
Return Level (m) 2.5 3.5
Return levels
1.5
Observed 5−Dim Model 5−Dim Semi−Independent Model 3−Dim Model 3−Dim Semi−Independent Model
10 15 Return Period (Years)
20
25
5
10 15 Return Period (Years)
20
25
∆ Return Level (%) 0 5 10 15
5
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
10 / 12
Extension of the analysis to 1979-2015 (Winter)
4.0 2.0
3.0
Mean Predicted 95% Prediction Interval
1.0
Water level (m)
The highest Compound flood ever registered in 2009-2015 it is also the highest ever simulated from our model in the period 1979-2015.
1979
1985
1991
1997
2003
2009
2015
Time (years)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
11 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
11 / 12
Summary
New conceptual model - implemented via PCC - to study CEs. Extending the analysis to 1979-2015, when observed data are not available, we get a more robust estimation of the risk. The highest compound flood in the period 1979-2015 was simulated in 2015. Without considering the dependence between Sea and River Levels the risk is underestimated of ∼ 10% (∼ 0.3m).
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
2.0
0.0 0.2 0.4 0.6 ●
Y1 Sea
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2.0 1.5
Y2 River
1.0 0.5
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0.44
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0.6
1.5
0.2
1.0
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−0.2
0.5
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● ● ●● ● ●
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● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ●● ● ●●● ●● ●●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●
●
3.0
Y3 River
● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ●●● ● ●● ● ● ● ● ● ● ●● ●●● ●●●●● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●
2.0
0.79
0.45
● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●●● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ●● ● ● ●● ● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ●●● ● ● ●● ● ●●●● ●● ● ● ●●●●● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ●●● ● ● ●● ●● ● ● ● ●● ● ● ● ●●● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●●●● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ●● ●●●●● ● ● ●● ●
4.0
● ● ●
●●
0.81
0.22
0.72
0.59
X1 Sea
0.24
●
●
0.58
X23 Rivers
0.5
0.61
1.0
1.5
0.0 0.2 0.4 0.6
● ● ●● ●● ●● ● ● ●● ●● ● ● ●● ●● ●●● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●
−0.2
0.2
0.6
2.0
3.0
4.0
0.5
1.0
1.5
Representing the variables directly by predictors we would wrong represent the dependence among the variables, which is crucial for modelling compound events. Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
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0.5
1.0
0.6 ●
● ●
● ●● ●●
1.5
●
●
Y1Sea
●
0.2
● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ●● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●●●● ● ●● ● ●● ●● ● ● ●● ●● ●● ●● ●● ● ● ● ●●● ●● ●●● ●● ● ●● ● ● ●● ● ●● ●●●● ● ●●● ●● ● ● ● ●● ● ● ●●●●● ● ●● ● ●●●● ● ● ● ● ● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ● ●● ●● ●●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●●●●● ● ● ●● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ●●● ●● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●
●
−0.2
0.2 −0.2
Y1Sea
0.6
Stationary Model (3-dim)
2.0
2.5
● ● ● ● ● ● ● ●● ●● ● ● ●●●● ● ● ●●● ● ●● ●● ●●●● ● ●●● ● ●●● ● ● ●●●● ●● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ●●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●● ● ●● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ●●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ●●● ● ● ● ●● ●●● ● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ●● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●
2.0
2.5
3.0
3.5
●
●
4.0
4.5
Y2River
Y3River
0.5 1.0 1.5 2.0 2.5
Y2River
● ●
●
● ●
●
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Y3River
Autocorrelation is not reproduced by construction Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
Extension of the analysis to 1979-2015 (Winter)
4.0 2.0
3.0
Mean Predicted 95% Prediction Interval
1.0
Water level (m)
The highest Compound flood ever registered (February 2015) it is also the highest ever simulated from our model in the period 1979-2015.
1979
1985
1991
1997
2003
2009
2015
Time (years)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
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0
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1979
Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
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Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
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Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
0.0
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Emanuele Bevacqua
0.0
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Statistical Model for Compound Events
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(u1,u2)
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(u5,u3)
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13th IMSC, 2016
0.6
0.8
1.0
12 / 12
Tree 1: V 4, V 5: V 5, V 3: V 3, V 1: V 1, V 2:
Frank with par=4.11 Gumbel with par=1.79 Survival Clayton-Gumbel with par=0.49, par2=1.15 Joe-Frank with par=4.01, par2=0.6
Tree 2: V 4, V 3|V 5: Rotated Joe-Frank 270 deg with par=-1.37, par2=-0.95 V 5, V 1|V 3: Survival Joe-Frank with par=6, par2=0.57 V 3, V 2|V 1: Gumbel with par=2.01 Tree 3: V 4, V 1|V 3, V 5: Gaussian with par=0.73 V 5, V 2|V 1, V 3: Survival Clayton with par=0.2 Tree 4: V 4, V 2|V 1, V 3, V 5: Rotated Clayton 90 deg with par=-0.2
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
Center for Climate and Global Change University of Graz
2 Norwegian
Computing Center, Oslo
3 University
of Birmingham
4 LSCE,
Paris
The 13th Internation Meeting on Statistical Climatology, 2016
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
1 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
1 / 12
Compound Events
Category of extreme events which has received little attention so far [IPCC SREX, 2012]:
Definition A compound event is an extreme impact that depends on multiple statistically dependent variables or events. [Leonard et al., 2013]
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
2 / 12
Compound Events
Category of extreme events which has received little attention so far [IPCC SREX, 2012]:
Definition A compound event is an extreme impact that depends on multiple statistically dependent variables or events. [Leonard et al., 2013]
−→ Compound Floods [Seth Westra, talk yesterday]
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
2 / 12
Case Study: Ravenna (Italy) Compound Flood, 6-2-2015
Yi = variables
Xi = predictors
Data Set: Daily, Winter (Nov-March), Period: 2009-2015 Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
3 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
3 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
Emanuele Bevacqua
Statistical Model for Compound Events
(1)
13th IMSC, 2016
4 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
2
(1)
The Y variables follow a joint probability distribution function (pdf): fY (y1 , y2 , y3 )
Emanuele Bevacqua
fY|X (y1 , y2 , y3 |x1 , x2 )
Statistical Model for Compound Events
13th IMSC, 2016
4 / 12
Conceptual model for studying Compound Events 1
h-function. The impact of the compound events (here h) is a function of the contributing variables Y = (Y1 , Y2 , Y3 ): h = h(Y1 , Y2 , Y3 )
2
The Y variables follow a joint probability distribution function (pdf): fY (y1 , y2 , y3 )
3
(1)
fY|X (y1 , y2 , y3 |x1 , x2 )
Through simulating the Y we get simulated values of the impact: hsim. := h(Y1sim. , Y2sim. , Y3sim. )
4
(2)
Statistical analysis: study the statistical characteristics of the events and asses their associated risk. Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
4 / 12
Copula theory Consider the random variables Y = (Y1 , ..., Yn ), with continuous marginal cumulative distribution functions F1 (y1 ), ..., Fn (yn ). Using the Sklar theorem [Sklar, 1959], we get: f (y1 , ..., yn ) = f1 (y1 ) · ... · fn (yn ) · c(F1 (y1 ), ..., Fn (yn ))
c describes the dependence, independently from the marginal structures. Decomposition → flexibility.
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
5 / 12
Pair-Copula Construction Many available bivariate copulas but limited choice for higher dimensions (n > 2). Copulas for n > 2 generally assume same kind of dependence for all the pairs (e.g. either all tail dependent or not).
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Pair-Copula Construction Many available bivariate copulas but limited choice for higher dimensions (n > 2). Copulas for n > 2 generally assume same kind of dependence for all the pairs (e.g. either all tail dependent or not). Pair-Copula Construction theory. 3-Dim example: f123 (y1 , y2 , y3 ) = f3 (y3 ) · f1 (y1 ) · f2 (y2 ) · c31 (u3 , u1 ) · c12 (u1 , u2 ) · c32|1 (u3|1 , u2|1 ) Further decomposition → more flexibility!
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
6 / 12
Impact function
2 h = a1 Y1Sea + a21 Y2River + a22 Y2River + a3 Y3River
(3)
1.0
1.5
Predicted (m) 2.0 2.5
3.0
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1.0
Emanuele Bevacqua
● ●
1.5
2.0 2.5 Observed (m)
3.0
Statistical Model for Compound Events
13th IMSC, 2016
7 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level YSea (t) = a SLPRavenna (t)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Meteorological Predictor of River Levels (one predictor) wT = Ptotal (T ) − E (T ) + Smelt (T ) − Sfall (T ) 0 YRiver (t) = a
t X T =t−1
Emanuele Bevacqua
wT + b
t X
wT
T =t−10
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Meteorological Predictors (Data: ERA-Interim Reanalysis) Meteorological Predictor of Sea Level
48
YSea (t) = a SLPRavenna (t) + b SLP(t) · RMAP + c sin(ω1Year t + φ) 1500
46
0
1000
−500
44
● 10
00
Latitude 42
500
40
0
500
0 00 −1
36
38
−500
−1000
5
10
15
20
Longitude
Meteorological predictor of River Levels (one predictor) wT = Ptotal (T ) − E (T ) + Smelt (T ) − Sfall (T ) 0 YRiver (t) = a
t X T =t−1
Emanuele Bevacqua
wT + b
t X
wT
T =t−10
Statistical Model for Compound Events
13th IMSC, 2016
8 / 12
Water level (m) 2.0 3.0 4.0
Non-stationary model (5-dim): Time series
1.0
Observed Mean Predicted 95% Prediction Interval
0
100
200 Time (days)
300
400
the non-stationary model captures the autocorrelation the observed signal is usually inside the prediction interval
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
9 / 12
Return Level (m) 2.5 3.5
Return levels
1.5
Observed 5−Dim Model 5−Dim Semi−Independent Model 3−Dim Model 3−Dim Semi−Independent Model
5
Emanuele Bevacqua
10 15 Return Period (Years)
Statistical Model for Compound Events
20
25
13th IMSC, 2016
10 / 12
Return Level (m) 2.5 3.5
Return levels
1.5
Observed 5−Dim Model 5−Dim Semi−Independent Model 3−Dim Model 3−Dim Semi−Independent Model
10 15 Return Period (Years)
20
25
5
10 15 Return Period (Years)
20
25
∆ Return Level (%) 0 5 10 15
5
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
10 / 12
Extension of the analysis to 1979-2015 (Winter)
4.0 2.0
3.0
Mean Predicted 95% Prediction Interval
1.0
Water level (m)
The highest Compound flood ever registered in 2009-2015 it is also the highest ever simulated from our model in the period 1979-2015.
1979
1985
1991
1997
2003
2009
2015
Time (years)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
11 / 12
Outline
1
Introduction
2
Conceptual model for Compound Events via PCC
3
Results
4
Summary
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
11 / 12
Summary
New conceptual model - implemented via PCC - to study CEs. Extending the analysis to 1979-2015, when observed data are not available, we get a more robust estimation of the risk. The highest compound flood in the period 1979-2015 was simulated in 2015. Without considering the dependence between Sea and River Levels the risk is underestimated of ∼ 10% (∼ 0.3m).
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
2.0
0.0 0.2 0.4 0.6 ●
Y1 Sea
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2.0 1.5
Y2 River
1.0 0.5
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0.44
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0.6
1.5
0.2
1.0
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−0.2
0.5
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3.0
Y3 River
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2.0
0.79
0.45
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4.0
● ● ●
●●
0.81
0.22
0.72
0.59
X1 Sea
0.24
●
●
0.58
X23 Rivers
0.5
0.61
1.0
1.5
0.0 0.2 0.4 0.6
● ● ●● ●● ●● ● ● ●● ●● ● ● ●● ●● ●●● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●
−0.2
0.2
0.6
2.0
3.0
4.0
0.5
1.0
1.5
Representing the variables directly by predictors we would wrong represent the dependence among the variables, which is crucial for modelling compound events. Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
●
0.5
1.0
0.6 ●
● ●
● ●● ●●
1.5
●
●
Y1Sea
●
0.2
● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ●● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●●●● ● ●● ● ●● ●● ● ● ●● ●● ●● ●● ●● ● ● ● ●●● ●● ●●● ●● ● ●● ● ● ●● ● ●● ●●●● ● ●●● ●● ● ● ● ●● ● ● ●●●●● ● ●● ● ●●●● ● ● ● ● ● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ● ●● ●● ●●●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●●●●● ● ● ●● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ●●● ●● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●
●
−0.2
0.2 −0.2
Y1Sea
0.6
Stationary Model (3-dim)
2.0
2.5
● ● ● ● ● ● ● ●● ●● ● ● ●●●● ● ● ●●● ● ●● ●● ●●●● ● ●●● ● ●●● ● ● ●●●● ●● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● ●●● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ●●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ●● ● ●● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ●●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ● ● ● ●●● ● ● ● ●● ●●● ● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ●● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●
2.0
2.5
3.0
3.5
●
●
4.0
4.5
Y2River
Y3River
0.5 1.0 1.5 2.0 2.5
Y2River
● ●
●
● ●
●
●
●
● ● ● ● ● ● ●●● ●● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●●●●●●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ●
2.0
2.5
3.0
●
● ● ●
●
●●
3.5
● ●
●
●
●
4.0
4.5
Y3River
Autocorrelation is not reproduced by construction Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
Extension of the analysis to 1979-2015 (Winter)
4.0 2.0
3.0
Mean Predicted 95% Prediction Interval
1.0
Water level (m)
The highest Compound flood ever registered (February 2015) it is also the highest ever simulated from our model in the period 1979-2015.
1979
1985
1991
1997
2003
2009
2015
Time (years)
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
●
●
● ●
●
0
●
1979
Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
●
●
● ●● ●
● ●
●
●
0
●●
1979
Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
Return Period (years) 20 40 60 80 120
Return period for the maximum observed water level
●
●
● ●● ●
● ●
●
●
0
●●
1979
Emanuele Bevacqua
1985
1991
1997 2003 Time (Years)
Statistical Model for Compound Events
2009
2015
13th IMSC, 2016
12 / 12
0.0
0.2
0.4
0.6
0.8
1.0
Emanuele Bevacqua
0.0
0.2
0.4
0.6
0.8
0.8
1.0
0.0
0.2
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0 0.8 0.6 0.4 0.2
1.0
0.0
0.2
● ●
●
● ● ●
Statistical Model for Compound Events
0.8
1.0
1.0
● ●
● ●
●
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ●● ● ●
●
●
0.2
0.6
(u4|513,u2|513)
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●● ● ●● ● ●● ●● ●● ●● ●● ●● ● ●● ● ● ●
●
0.0
0.4
0.8
1.0 0.8
1.0
0.4
(u5|31,u2|31)
● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ● ●
0.0
0.0
1.0 0.8 0.6 0.4 0.2
1.0
0.0
●
0.6
(u4|53,u1|53)
● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ●●● ●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●
0.8
1.0 0.8 0.6 0.4 0.2 ●
0.4
● ●● ● ●
0.6
0.2
●
●
0.4
0.0
0.0
1.0 0.8 0.6 0.4
1.0
● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ●● ●● ●● ●
●
0.2
0.8
●
0.6
0.6
(u3|1,u2|1)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ● ● ●● ●● ●● ● ●● ● ● ● ●
0.0
0.0
0.2
0.4
0.6
0.8
1.0
(u5|3,u1|3)
0.4
●● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ● ●● ● ●● ● ●● ● ●● ●●
0.4
0.2
●
0.2
0.0
0.2
1.0 0.8 0.6 0.4
1.0
● ● ● ●
(u4|5,u3|5)
(u1,u2)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ●● ●● ● ●● ●● ● ● ● ●
0.0
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
●
0.0
●
(u3,u1)
● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●●● ●● ●● ● ●● ●● ●● ● ● ● ●● ● ●● ●● ●● ● ● ● ●
0.2
0.8 0.6 0.4 0.2 0.0
(u5,u3)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ●●● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ●
0.0
1.0
(u4,u5)
0.4
0.6
0.8
1.0
● ●
0.0
0.2
0.4
13th IMSC, 2016
0.6
0.8
1.0
12 / 12
Tree 1: V 4, V 5: V 5, V 3: V 3, V 1: V 1, V 2:
Frank with par=4.11 Gumbel with par=1.79 Survival Clayton-Gumbel with par=0.49, par2=1.15 Joe-Frank with par=4.01, par2=0.6
Tree 2: V 4, V 3|V 5: Rotated Joe-Frank 270 deg with par=-1.37, par2=-0.95 V 5, V 1|V 3: Survival Joe-Frank with par=6, par2=0.57 V 3, V 2|V 1: Gumbel with par=2.01 Tree 3: V 4, V 1|V 3, V 5: Gaussian with par=0.73 V 5, V 2|V 1, V 3: Survival Clayton with par=0.2 Tree 4: V 4, V 2|V 1, V 3, V 5: Rotated Clayton 90 deg with par=-0.2
Emanuele Bevacqua
Statistical Model for Compound Events
13th IMSC, 2016
12 / 12
