slides of my presentation at the ties 2015
TRANSCRIPT
Extreme Value Modelling with Application toEnvironmental Control
Laique Merlin Djeutchouang1 and Abdel Hameed El-Shaarawi2
1
2National Water Research Institute, Canada and The American University of Cairo, Egypt
November 23, 2015
The 25th Annual Conference of The International Environmetrics Society
Al Ain, 2015
Extreme Value Modelling with Application to Environmental Control
Overview of this talk
1 Motivations and Background
2 Statistical Modelling for ExtremesBlock Maximum ApproachPeak over Threshold ApproachNotion of Return Value and its estimation
3 Applications to the Sea DataThe dataResults
4 Current developments: Additional Remarks
5 Summary
2 / 39
Extreme Value Modelling with Application to Environmental Control
Overview of this talk
1 Motivations and Background
2 Statistical Modelling for ExtremesBlock Maximum ApproachPeak over Threshold ApproachNotion of Return Value and its estimation
3 Applications to the Sea DataThe dataResults
4 Current developments: Additional Remarks
5 Summary
2 / 39
Extreme Value Modelling with Application to Environmental Control
Overview of this talk
1 Motivations and Background
2 Statistical Modelling for ExtremesBlock Maximum ApproachPeak over Threshold ApproachNotion of Return Value and its estimation
3 Applications to the Sea DataThe dataResults
4 Current developments: Additional Remarks
5 Summary
2 / 39
Extreme Value Modelling with Application to Environmental Control
Overview of this talk
1 Motivations and Background
2 Statistical Modelling for ExtremesBlock Maximum ApproachPeak over Threshold ApproachNotion of Return Value and its estimation
3 Applications to the Sea DataThe dataResults
4 Current developments: Additional Remarks
5 Summary
2 / 39
Extreme Value Modelling with Application to Environmental Control
Overview of this talk
1 Motivations and Background
2 Statistical Modelling for ExtremesBlock Maximum ApproachPeak over Threshold ApproachNotion of Return Value and its estimation
3 Applications to the Sea DataThe dataResults
4 Current developments: Additional Remarks
5 Summary
2 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extreme frequency and magnitude of weather
I A combined effect of irrigation and global warming.
Figure : The lake Chad has lost about 80% of it surface area since the 1960s.
3 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extreme frequency and magnitude of weather
Figure : The lake Chad has lost about 80% of it surface area since the 1960s.
3 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extreme amount of precipitations
Figure : Flooding in Dar-es-Salaam, May 2015, Tanzania.
4 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal Natural Events
I Why do some Earthquakes cause Tsunamis but others don’t?
I Magnitude of the quake, which is a measure of the amplitude of thelargest seismic wave recorded for the earthquake, must exceed acertain threshold.
ExamplesI The unforgotten and unfortunate, Indian Ocean Earthquake and
Tsunami of the December 26, 2004.
I The Fukushima tsunami of March 2011.
I The Tropical Storm Katrina (Hurricane Katrina).
I etc.
5 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Indian Ocean Earthquake and Tsunami
Figure : December 26, 2004 Indian Ocean Tsunami.
I With its magnitude of 9.0 on the Rchter Scale, over 227 898 peoplehave been confirmed dead.
I Making this the fourth largest death toll from an earthquake inrecorded history.
6 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Indian Ocean Earthquake and Tsunami
I With its magnitude of 9.0 on the Rchter Scale, over 227 898 peoplehave been confirmed dead.
I Making this the fourth largest death toll from an earthquake inrecorded history.
6 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Indian Ocean Earthquake and Tsunami
I With its magnitude of 9.0 on the Rchter Scale, over 227 898 peoplehave been confirmed dead.
I Making this the fourth largest death toll from an earthquake inrecorded history.
6 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal events: Disasters
I Cancer rates at Fukushima worse than Chermobyl (November 112015, Fukushima Watch), Cancer rates spike in Fukushimaprefecture.
Figure : Fukushima-Fire-Explosion-Radiation.
7 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Why extreme value theory?
Extremal events: Disasters
Figure : Fukushima-Fire-Explosion-Radiation.
7 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
I Basically, we are interested in a threshold value XT , from which onecan predict an extreme event based on the observed data
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Objectives
Procedure for estimatingI Choose an appropriate parametric distribution function or model.
I Calibrate the model such that it describes reasonably available data.
Estimation of the Return valueI Find reliable estimates of XT for large T , i.e rare events.
I Even for T large than the period of observation.
I This implies an extrapolation from observed to unobserved values.
I Extreme value theory provides a class of models to handle suchextrapolation.
8 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Extremal types distribution
1 Gumbel distribution:
G(x) = exp
−exp
[−(
x−µ
σ
)], where x ∈R and σ > 0,µ ∈R.
2 Frechet distribution:
F(x) =
exp
−(
x−µ
σ
)−α, if x ≥ µ
0, if x < µ
, where α > 0,σ > 0,µ ∈R.
3 Weibull distribution:
W (x) =
exp
−(
µ− xσ
)α, if x < µ
1, if x ≥ µ
, where α > 0,σ > 0,µ ∈R.
9 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Extremal types distribution
1 Gumbel distribution:
G(x) = exp
−exp
[−(
x−µ
σ
)], where x ∈R and σ > 0,µ ∈R.
2 Frechet distribution:
F(x) =
exp
−(
x−µ
σ
)−α, if x ≥ µ
0, if x < µ
, where α > 0,σ > 0,µ ∈R.
3 Weibull distribution:
W (x) =
exp
−(
µ− xσ
)α, if x < µ
1, if x ≥ µ
, where α > 0,σ > 0,µ ∈R.
9 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Extremal types distribution
1 Gumbel distribution:
G(x) = exp
−exp
[−(
x−µ
σ
)], where x ∈R and σ > 0,µ ∈R.
2 Frechet distribution:
F(x) =
exp
−(
x−µ
σ
)−α, if x ≥ µ
0, if x < µ
, where α > 0,σ > 0,µ ∈R.
3 Weibull distribution:
W (x) =
exp
−(
µ− xσ
)α, if x < µ
1, if x ≥ µ
, where α > 0,σ > 0,µ ∈R.
9 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Generalized Extreme Value (GEV) distribution
I The cumulative distribution function is given as follows
GEV(x ,ξ ,µ,σ) =
exp
−(
1 + ξ
(x−µ
σ
))− 1ξ
, if ξ 6= 0,
exp
−exp
[−(
x−µ
σ
)], if ξ = 0.
(1)
I The three families can be deduced as follows:ξ < 0⇒Weibull(ξ ,µ,σ), called GEV type I,
ξ = 0⇒ Gumbel(ξ ,µ,σ), called GEV type II,
ξ > 0⇒ Frechet(ξ ,µ,σ), called GEV type III.
10 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Generalized Extreme Value (GEV) distribution
I The cumulative distribution function is given as follows
GEV(x ,ξ ,µ,σ) =
exp
−(
1 + ξ
(x−µ
σ
))− 1ξ
, if ξ 6= 0,
exp
−exp
[−(
x−µ
σ
)], if ξ = 0.
(1)
I The three families can be deduced as follows:ξ < 0⇒Weibull(ξ ,µ,σ), called GEV type I,
ξ = 0⇒ Gumbel(ξ ,µ,σ), called GEV type II,
ξ > 0⇒ Frechet(ξ ,µ,σ), called GEV type III.
10 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Generalized Extreme Value (GEV) distribution
−4 0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
GEV: pdf
xAll with: mu=−1, sigma=1
Pro
babili
ty d
ensty
ξ < 0ξ = 0ξ > 0
−4 0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
GEV: cdf
xAll with: mu=−1, sigma=1
Pro
babili
ty d
istr
ibution
ξ < 0ξ = 0ξ > 0
Figure : PDF and the corresponding CDF, ξ varying with 0,−0.5,0.5. 11 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Height Endpoint of a distribution: Limit distribution
I Let X1, . . . ,Xk be iid with same distribution as X with the underlyingCDF F .
Height Endpoint of F
x∗ = supx ∈ R : F(x) < 1 ≤+∞. (2)
I AndZk = max
1≤i≤k(Xi)
L−→ x∗, as k −→+∞ (3)
I Then
FZk (z) −→k→∞G(z) =
0, if z < x∗
1, if z ≥ x∗, (4)
12 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Height Endpoint of a distribution: Limit distribution
I Let X1, . . . ,Xk be iid with same distribution as X with the underlyingCDF F .
Height Endpoint of F
x∗ = supx ∈ R : F(x) < 1 ≤+∞. (2)
I AndZk = max
1≤i≤k(Xi)
L−→ x∗, as k −→+∞ (3)
I Then
FZk (z) −→k→∞G(z) =
0, if z < x∗
1, if z ≥ x∗, (4)
12 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Height Endpoint of a distribution: Limit distribution
I Let X1, . . . ,Xk be iid with same distribution as X with the underlyingCDF F .
Height Endpoint of F
x∗ = supx ∈ R : F(x) < 1 ≤+∞. (2)
I AndZk = max
1≤i≤k(Xi)
L−→ x∗, as k −→+∞ (3)
I Then
FZk (z) −→k→∞G(z) =
0, if z < x∗
1, if z ≥ x∗, (4)
12 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
Height Endpoint of a distribution: Limit distribution
I Let X1, . . . ,Xk be iid with same distribution as X with the underlyingCDF F .
Height Endpoint of F
x∗ = supx ∈ R : F(x) < 1 ≤+∞. (2)
I AndZk = max
1≤i≤k(Xi)
L−→ x∗, as k −→+∞ (3)
I Then
FZk (z) −→k→∞G(z) =
0, if z < x∗
1, if z ≥ x∗, (4)
12 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
General Theorem in EVTFisher-Tippett-Gendenko theorem
1 Let X1, . . . ,Xk be a sequence of independent and identicallydistributed (iid) real random variables and
Zk = max1≤i≤k
(Xi) .
2 If there exists (ak ,bk )k∈N such that each ak > 0, k ∈ N and
limk−→+∞
P
Zk −bk
ak≤ x
= F(x) (5)
where F is a non degenerate cumulative function,3 Then the limit distribution F belongs to either Gumbel, Frechet or
Weibull family, that is
F(x)≡ GEV(x ,ξ ,µ,σ).
13 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
General Theorem in EVTFisher-Tippett-Gendenko theorem
1 Let X1, . . . ,Xk be a sequence of independent and identicallydistributed (iid) real random variables and
Zk = max1≤i≤k
(Xi) .
2 If there exists (ak ,bk )k∈N such that each ak > 0, k ∈ N and
limk−→+∞
P
Zk −bk
ak≤ x
= F(x) (5)
where F is a non degenerate cumulative function,
3 Then the limit distribution F belongs to either Gumbel, Frechet orWeibull family, that is
F(x)≡ GEV(x ,ξ ,µ,σ).
13 / 39
Extreme Value Modelling with Application to Environmental Control
Motivations and Background
Theoretical Motivations
General Theorem in EVTFisher-Tippett-Gendenko theorem
1 Let X1, . . . ,Xk be a sequence of independent and identicallydistributed (iid) real random variables and
Zk = max1≤i≤k
(Xi) .
2 If there exists (ak ,bk )k∈N such that each ak > 0, k ∈ N and
limk−→+∞
P
Zk −bk
ak≤ x
= F(x) (5)
where F is a non degenerate cumulative function,3 Then the limit distribution F belongs to either Gumbel, Frechet or
Weibull family, that is
F(x)≡ GEV(x ,ξ ,µ,σ).
13 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Extreme value modelling
I Let us consider an iid observations of X1,X2, . . . ,Xk , with k = m×n.
Block maxima method (BM)I The daily maximum temperature or precipitation,
I The monthly or annual maximum temperature or precipitation.
peak above the threshold method (POT)
Others modelling approaches of extremesI The Poison point process method, etc.
14 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Extreme value modelling
I Let us consider an iid observations of X1,X2, . . . ,Xk , with k = m×n.
Block maxima method (BM)I The daily maximum temperature or precipitation,
I The monthly or annual maximum temperature or precipitation.
peak above the threshold method (POT)
Others modelling approaches of extremesI The Poison point process method, etc.
14 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Extreme value modelling
I Let us consider an iid observations of X1,X2, . . . ,Xk , with k = m×n.
Block maxima method (BM)I The daily maximum temperature or precipitation,
I The monthly or annual maximum temperature or precipitation.
peak above the threshold method (POT)
Others modelling approaches of extremesI The Poison point process method, etc.
14 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Extreme value modelling
I Let us consider an iid observations of X1,X2, . . . ,Xk , with k = m×n.
Block maxima method (BM)I The daily maximum temperature or precipitation,
I The monthly or annual maximum temperature or precipitation.
peak above the threshold method (POT)
Others modelling approaches of extremesI The Poison point process method, etc.
14 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Extreme value modelling
I Let us consider an iid observations of X1,X2, . . . ,Xk , with k = m×n.
Block maxima method (BM)I The daily maximum temperature or precipitation,
I The monthly or annual maximum temperature or precipitation.
peak above the threshold method (POT)
Others modelling approaches of extremesI The Poison point process method, etc.
14 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Extreme value modellingBlock Maxima Approach (BM)
I So let us set
Zt,n = max(t−1)n+1≤j≤tn
(Xj) , t = 1,2, . . . ,m, (6)
I Assume the new observations of Z1,n, . . . ,Zm,n are iid with thedistribution GEV(ξ ,µ,σ).
I Therefore the log-likelihood function for BM approach gives us
L1(θ) =−m logσ −m
∑t=1
B− 1
ξ
t −(
1ξ
+ 1
) m
∑t=1
logBt , (7)
I Where Bt = 1 + ξ
(zt −µ
σ
).
15 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Extreme value modellingBlock Maxima Approach (BM)
I So let us set
Zt,n = max(t−1)n+1≤j≤tn
(Xj) , t = 1,2, . . . ,m, (6)
I Assume the new observations of Z1,n, . . . ,Zm,n are iid with thedistribution GEV(ξ ,µ,σ).
I Therefore the log-likelihood function for BM approach gives us
L1(θ) =−m logσ −m
∑t=1
B− 1
ξ
t −(
1ξ
+ 1
) m
∑t=1
logBt , (7)
I Where Bt = 1 + ξ
(zt −µ
σ
).
15 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Extreme value modellingBlock Maxima Approach (BM)
I So let us set
Zt,n = max(t−1)n+1≤j≤tn
(Xj) , t = 1,2, . . . ,m, (6)
I Assume the new observations of Z1,n, . . . ,Zm,n are iid with thedistribution GEV(ξ ,µ,σ).
I Therefore the log-likelihood function for BM approach gives us
L1(θ) =−m logσ −m
∑t=1
B− 1
ξ
t −(
1ξ
+ 1
) m
∑t=1
logBt , (7)
I Where Bt = 1 + ξ
(zt −µ
σ
).
15 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Extreme value modellingBlock Maxima Approach (BM)
I So let us set
Zt,n = max(t−1)n+1≤j≤tn
(Xj) , t = 1,2, . . . ,m, (6)
I Assume the new observations of Z1,n, . . . ,Zm,n are iid with thedistribution GEV(ξ ,µ,σ).
I Therefore the log-likelihood function for BM approach gives us
L1(θ) =−m logσ −m
∑t=1
B− 1
ξ
t −(
1ξ
+ 1
) m
∑t=1
logBt , (7)
I Where Bt = 1 + ξ
(zt −µ
σ
).
15 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Extreme value modellingBlock Maxima Approach (BM)
I So let us set
Zt,n = max(t−1)n+1≤j≤tn
(Xj) , t = 1,2, . . . ,m, (6)
I Assume the new observations of Z1,n, . . . ,Zm,n are iid with thedistribution GEV(ξ ,µ,σ).
I Therefore the log-likelihood function for BM approach gives us
L1(θ) =−m logσ −m
∑t=1
B− 1
ξ
t −(
1ξ
+ 1
) m
∑t=1
logBt , (7)
I Where Bt = 1 + ξ
(zt −µ
σ
).
15 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Block Maxima Approach (BM)Estimating of model parameters by ML
I The ML estimates θ =(
ξ , µ, σ)
are then found by solving theoptimization problem
arg maxθ=(ξ ,µ,σ)
L1(θ) (8)
I For inference and confidence interval estimation, by applying thecentral limit theorem in multi-dimension,
√m(
Θ−θ
)∼N (O,Ω) (9)
I Where Ω is the variance-covariance matrix of Θ:
Ω = [E(I1(θ))]−1 .
16 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Block Maxima Approach (BM)Estimating of model parameters by ML
I The ML estimates θ =(
ξ , µ, σ)
are then found by solving theoptimization problem
arg maxθ=(ξ ,µ,σ)
L1(θ) (8)
I For inference and confidence interval estimation, by applying thecentral limit theorem in multi-dimension,
√m(
Θ−θ
)∼N (O,Ω) (9)
I Where Ω is the variance-covariance matrix of Θ:
Ω = [E(I1(θ))]−1 .
16 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
Block Maxima Approach (BM)Estimating of model parameters by ML
I The ML estimates θ =(
ξ , µ, σ)
are then found by solving theoptimization problem
arg maxθ=(ξ ,µ,σ)
L1(θ) (8)
I For inference and confidence interval estimation, by applying thecentral limit theorem in multi-dimension,
√m(
Θ−θ
)∼N (O,Ω) (9)
I Where Ω is the variance-covariance matrix of Θ:
Ω = [E(I1(θ))]−1 .
16 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
ML estimates of θ = (ξ ,µ,σ)How accurate are parameter estimates
I From the approximate normality of ML estimators, a 100(1−α)%confidence interval for GEV parameter θ1 = ξ ,θ2 = µ,θ3 = σ isgiven as follows
θi ±Φ−1(
1− α
2
)√ωii
m, with i = 1,2,3 , (10)
I Where Φ is the cumulative distribution function of the standardnormal distribution, and ωii denotes the i-th diagonal coefficient of Ω
17 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Block Maximum Approach
ML estimates of θ = (ξ ,µ,σ)How accurate are parameter estimates
I From the approximate normality of ML estimators, a 100(1−α)%confidence interval for GEV parameter θ1 = ξ ,θ2 = µ,θ3 = σ isgiven as follows
θi ±Φ−1(
1− α
2
)√ωii
m, with i = 1,2,3 , (10)
I Where Φ is the cumulative distribution function of the standardnormal distribution, and ωii denotes the i-th diagonal coefficient of Ω
17 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Peak over Threshold Approach
POT framework
I Let us consider a high threshold u. We want to use all the excessesXi −u for those Xi > u as well as the indicator of Xi < u for thoseobservations below u to estimate the parameters.
I We have shown that
Fu (y) = PXi −u ≤ y | Xi > u= 1−(
1 +ξ yσ
)− 1ξ
, (11)
I Which is the CDF a generalized Pareto distribution (GPD) (Pickands,1975) with parameters: location 0, scale σ = σ + ξ (u−µ) andshape ξ .
I On the other hand, Fu (y) can be interpreted like the probability thata value, temperature or precipitation amount, exceeds the thresholdu by no more than an amount value y , given that the threshold u isexceeded.
18 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Peak over Threshold Approach
POT framework
I We have shown that
Fu (y) = PXi −u ≤ y | Xi > u= 1−(
1 +ξ yσ
)− 1ξ
, (11)
I Which is the CDF a generalized Pareto distribution (GPD) (Pickands,1975) with parameters: location 0, scale σ = σ + ξ (u−µ) andshape ξ .
I On the other hand, Fu (y) can be interpreted like the probability thata value, temperature or precipitation amount, exceeds the thresholdu by no more than an amount value y , given that the threshold u isexceeded.
18 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Peak over Threshold Approach
POT framework
I We have shown that
Fu (y) = PXi −u ≤ y | Xi > u= 1−(
1 +ξ yσ
)− 1ξ
, (11)
I Which is the CDF a generalized Pareto distribution (GPD) (Pickands,1975) with parameters: location 0, scale σ = σ + ξ (u−µ) andshape ξ .
I On the other hand, Fu (y) can be interpreted like the probability thata value, temperature or precipitation amount, exceeds the thresholdu by no more than an amount value y , given that the threshold u isexceeded.
18 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Peak over Threshold Approach
POT framework
I We have shown that
Fu (y) = PXi −u ≤ y | Xi > u= 1−(
1 +ξ yσ
)− 1ξ
, (11)
I Which is the CDF a generalized Pareto distribution (GPD) (Pickands,1975) with parameters: location 0, scale σ = σ + ξ (u−µ) andshape ξ .
I On the other hand, Fu (y) can be interpreted like the probability thata value, temperature or precipitation amount, exceeds the thresholdu by no more than an amount value y , given that the threshold u isexceeded.
18 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Model adequacy or goodness of fit
Model adequacy or goodness of fit
I As with all statistical models, there are various goodness-of-fitproperties that should be considered to check the overall adequacyof the fitted model:
1 The probability plots: it is the plot of the points(F(x(i)),
in + 1
): i = 1,2, . . . ,n
, (12)
2 quantile-quantile (Q-Q plots): it is the plot of the points(F−1
(i
n + 1
),x(i)
): i = 1,2, . . . ,n
, (13)
3 and simply plotting a histogram of the data against the fitted density.
19 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Model adequacy or goodness of fit
Model adequacy or goodness of fit
I As with all statistical models, there are various goodness-of-fitproperties that should be considered to check the overall adequacyof the fitted model:
1 The probability plots: it is the plot of the points
(F(x(i)),
in + 1
): i = 1,2, . . . ,n
, (12)
2 quantile-quantile (Q-Q plots): it is the plot of the points(F−1
(i
n + 1
),x(i)
): i = 1,2, . . . ,n
, (13)
3 and simply plotting a histogram of the data against the fitted density.
19 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Model adequacy or goodness of fit
Model adequacy or goodness of fit
I As with all statistical models, there are various goodness-of-fitproperties that should be considered to check the overall adequacyof the fitted model:
1 The probability plots: it is the plot of the points(F(x(i)),
in + 1
): i = 1,2, . . . ,n
, (12)
2 quantile-quantile (Q-Q plots): it is the plot of the points
(F−1
(i
n + 1
),x(i)
): i = 1,2, . . . ,n
, (13)
3 and simply plotting a histogram of the data against the fitted density.
19 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Model adequacy or goodness of fit
Model adequacy or goodness of fit
I As with all statistical models, there are various goodness-of-fitproperties that should be considered to check the overall adequacyof the fitted model:
1 The probability plots: it is the plot of the points(F(x(i)),
in + 1
): i = 1,2, . . . ,n
, (12)
2 quantile-quantile (Q-Q plots): it is the plot of the points(F−1
(i
n + 1
),x(i)
): i = 1,2, . . . ,n
, (13)
3 and simply plotting a histogram of the data against the fitted density.
19 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Return value with the corresponding return period
I Let p be the probability that Zt,n exceeds a level Xp, that is
p = PZt,n > Xp
I p is called the upper tail probability and we deduce that
Xp =
µ +σ
ξ
([− log(1−p)]−ξ −1
), for ξ 6= 0,
µ−σ log(− log(1−p)) , for ξ = 0.(14)
I XT ≡ Xp is referred to return value with return period of T = 1p which
is exceeded on average every T period of time.I Equivalently, the average time for a very rare event to exceed XT is
every T period of time.
20 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Return value with the corresponding return period
I Let p be the probability that Zt,n exceeds a level Xp, that is
p = PZt,n > Xp
I p is called the upper tail probability and we deduce that
Xp =
µ +σ
ξ
([− log(1−p)]−ξ −1
), for ξ 6= 0,
µ−σ log(− log(1−p)) , for ξ = 0.(14)
I XT ≡ Xp is referred to return value with return period of T = 1p which
is exceeded on average every T period of time.I Equivalently, the average time for a very rare event to exceed XT is
every T period of time.
20 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Return value with the corresponding return period
I Let p be the probability that Zt,n exceeds a level Xp, that is
p = PZt,n > Xp
I p is called the upper tail probability and we deduce that
Xp =
µ +σ
ξ
([− log(1−p)]−ξ −1
), for ξ 6= 0,
µ−σ log(− log(1−p)) , for ξ = 0.(14)
I XT ≡ Xp is referred to return value with return period of T = 1p which
is exceeded on average every T period of time.
I Equivalently, the average time for a very rare event to exceed XT isevery T period of time.
20 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Return value with the corresponding return period
I Let p be the probability that Zt,n exceeds a level Xp, that is
p = PZt,n > Xp
I p is called the upper tail probability and we deduce that
Xp =
µ +σ
ξ
([− log(1−p)]−ξ −1
), for ξ 6= 0,
µ−σ log(− log(1−p)) , for ξ = 0.(14)
I XT ≡ Xp is referred to return value with return period of T = 1p which
is exceeded on average every T period of time.I Equivalently, the average time for a very rare event to exceed XT is
every T period of time.20 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Estimation of Return LevelHow accurate is the return value estimates?
I The return level XT is often interpreted as the expected waiting valueuntil another exceedance event.
I So for ξ 6= 0, using the estimates θ =(
ξ , µ, σ)
above, uT can be
estimated as follows
xT ≡ xp = µ +σ
ξ
([− log(1−p)]−ξ −1
)(15)
I We have Xp = g(θ) = µ +σ
ξ
([− log(1−p)]−ξ −1
), (for ξ 6= 0),
I So the delta method gives us
Var(
Xp
)≈ (∇g(θ))T Ω∇g(θ),
21 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Estimation of Return LevelHow accurate is the return value estimates?
I The return level XT is often interpreted as the expected waiting valueuntil another exceedance event.
I So for ξ 6= 0, using the estimates θ =(
ξ , µ, σ)
above, uT can be
estimated as follows
xT ≡ xp = µ +σ
ξ
([− log(1−p)]−ξ −1
)(15)
I We have Xp = g(θ) = µ +σ
ξ
([− log(1−p)]−ξ −1
), (for ξ 6= 0),
I So the delta method gives us
Var(
Xp
)≈ (∇g(θ))T Ω∇g(θ),
21 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Estimation of Return LevelHow accurate is the return value estimates?
I The return level XT is often interpreted as the expected waiting valueuntil another exceedance event.
I So for ξ 6= 0, using the estimates θ =(
ξ , µ, σ)
above, uT can be
estimated as follows
xT ≡ xp = µ +σ
ξ
([− log(1−p)]−ξ −1
)(15)
I We have Xp = g(θ) = µ +σ
ξ
([− log(1−p)]−ξ −1
), (for ξ 6= 0),
I So the delta method gives us
Var(
Xp
)≈ (∇g(θ))T Ω∇g(θ),
21 / 39
Extreme Value Modelling with Application to Environmental Control
Statistical Modelling for Extremes
Notion of Return Value and its estimation
Estimation of Return LevelHow accurate is the return value estimates?
I The return level XT is often interpreted as the expected waiting valueuntil another exceedance event.
I So for ξ 6= 0, using the estimates θ =(
ξ , µ, σ)
above, uT can be
estimated as follows
xT ≡ xp = µ +σ
ξ
([− log(1−p)]−ξ −1
)(15)
I We have Xp = g(θ) = µ +σ
ξ
([− log(1−p)]−ξ −1
), (for ξ 6= 0),
I So the delta method gives us
Var(
Xp
)≈ (∇g(θ))T Ω∇g(θ),
21 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
The data
Origin of the data and Description
I Data (1912 - 2003): Daily flow rate (m3/s) of Fraser River (Station ofHope) which drains a 220 000Km2 area.
I Variables: Daily maximum flow rate.
Figure : The map of Fraser River.
22 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
The data
Origin of the data and Description
Figure : The map of Fraser River.22 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Description of the data and visualization
Description of the data and visualization
0 200 400 600 800 1000
05000
10000
15000
Time (days)
Flo
w R
ate
(m
3/s
)
Figure : Scatter plot of the data.
Months
Flo
w R
ate
(m
3/s
)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
05
00
01
00
00
15
00
0
Figure : Monthly time series plots of the data.
23 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Description of the data and visualization
Description of the data and visualization
Months
Flo
w R
ate
(m
3/s
)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
05
00
01
00
00
15
00
0
Figure : Monthly time series plots of the data.
23 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Description of the data and visualization
BM Approach: Yearly maxima
1920 1940 1960 1980 2000
60
00
10
00
01
40
00
Year
Da
ily F
low
Ra
te (
m3
/s)
Yearly Maxima
Figure : Yearly maxima plot of the daily flow rate of Fraser River in BritishColumbia, 1912-2003.
24 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
BM Approach: estimate parameters
Parameters
Methods shape(
ξ
)location(µ) scale(σ)
MLEstimates -0.076 7963.50 1412.97
95%CI (−0.192,0.041) (7644.52,8282.48) (1195.35,1630.58)
L-MomentsEstimates -0.068 7963.50 1412.96
95%CI (−0.220,0.070) (7647.26,8261.46) (1190.40,1661.41)
Table : Yearly maxima: estimated parameters θ = (ξ , µ, σ) and their 95%confidence interval of the GEV distribution, using both ML and L-Momentsmethods.
25 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
Adequacy of fit for GEV model with the yearly maxima
6000 8000 10000 12000
6000
10000
14000
Model Quantiles
Em
pir
ical Q
uantile
s
6000 8000 12000
6000
10000
14000
DailyFlowRate Empirical Quantiles
Quantile
s fro
m M
odel S
imula
ted D
ata
1−1 line
regression line
95% confidence bands
4000 8000 12000 160000.0
0000
0.0
0015
N = 92 Bandwidth = 588.6
Density
Empirical
Modeled
2 5 10 50 200 1000
8000
12000
16000
Return Period (years)
Retu
rn L
eve
l
fevd(x = DailyFlowRate, data = BM.FR.Hope1, method = "MLE")
Figure : Diagnostic plots of the fit of the GEV distribution.26 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
Return level estimation: Friser River
I We are now able to provide an estimate of the daily flow rate ofFraser River to protect the cities around the river against the floodswe would expect to see,
I once in T = 10 years;I once in a T = 100 years, etc.
T (in year) Estimate 95%CI2
T -year return level (in m3/s)
8 474.247 (8 130.125, 8 818.369)5 9 966.811 (9 508.747, 10 424.876)10 10 886.754 (10 305.199, 11 468.308)20 11 721.342 (10 964.387, 12 478.297)50 12 736.158 (11 664.145, 13 808.172)
100 13 450.993 (12 083.758, 14 818.227)
Table : Some estimate Return levels or values with their 95% confidenceinterval.
27 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
Return level estimation: Friser River
T (in year) Estimate 95%CI2
T -year return level (in m3/s)
8 474.247 (8 130.125, 8 818.369)5 9 966.811 (9 508.747, 10 424.876)10 10 886.754 (10 305.199, 11 468.308)20 11 721.342 (10 964.387, 12 478.297)50 12 736.158 (11 664.145, 13 808.172)
100 13 450.993 (12 083.758, 14 818.227)
Table : Some estimate Return levels or values with their 95% confidenceinterval.
27 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
Data: Monthly maximaAdequacy of fit for GEV model with the monthly maxima
Date JulianDay True.Date Year Month.num Month.cha Day DailyFlowRate21938 72-03-23 83.00 812.00 1972.00 3.00 Mar 23.00 3170.008096 34-04-30 120.00 -13030.00 1934.00 4.00 Apr 30.00 8240.00
13241 48-05-31 152.00 -7885.00 1948.00 5.00 May 31.00 15200.0013242 48-06-01 153.00 -7884.00 1948.00 6.00 Jun 1.00 14800.0015828 55-07-01 182.00 -5298.00 1955.00 7.00 Jul 1.00 11100.003077 20-08-02 215.00 -18049.00 1920.00 8.00 Aug 2.00 7650.00
25764 82-09-13 256.00 4638.00 1982.00 9.00 Sep 13.00 6260.003148 20-10-12 286.00 -17978.00 1920.00 10.00 Oct 12.00 5320.00
17416 59-11-05 309.00 -3710.00 1959.00 11.00 Nov 5.00 4130.0025139 80-12-27 362.00 4013.00 1980.00 12.00 Dec 27.00 4210.0025144 81-01-01 1.00 4018.00 1981.00 1.00 Jan 1.00 2760.0018242 62-02-08 39.00 -2884.00 1962.00 2.00 Feb 8.00 2940.00
Table : Monthly Maxima of the daily flow rate of Fraser, British Columbia; datarecorded from March 1912 to December 2003.
5000 15000 25000
4000
10000
Model Quantiles
Em
pir
ical Q
uantile
s
4000 8000 12000
5000
20000
DailyFlowRate Empirical Quantiles
Quantile
s fro
m M
odel S
imula
ted D
ata
1−1 line
regression line
95% confidence bands
0 5000 150000.0
0000
0.0
0010
N = 12 Bandwidth = 2070
Density
Empirical
Modeled
2 5 10 50 200 1000
−1.0
e+
07
1.0
e+
07
Return Period (years)
Retu
rn L
eve
l
fevd(x = DailyFlowRate, data = BM.FR.Hope2, method = "MLE")
Figure : Diagnostic plots of the fit of the GEV model to the monthly maxima ofthe daily flow rate of Fraser River in British Columbia, 1912-2003.
28 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
Data: Monthly maximaAdequacy of fit for GEV model with the monthly maxima
5000 15000 25000
4000
10000
Model Quantiles
Em
pir
ical Q
uantile
s
4000 8000 12000
5000
20000
DailyFlowRate Empirical Quantiles
Quantile
s fro
m M
odel S
imula
ted D
ata
1−1 line
regression line
95% confidence bands
0 5000 150000.0
0000
0.0
0010
N = 12 Bandwidth = 2070
Density
Empirical
Modeled
2 5 10 50 200 1000
−1.0
e+
07
1.0
e+
07
Return Period (years)
Retu
rn L
eve
l
fevd(x = DailyFlowRate, data = BM.FR.Hope2, method = "MLE")
Figure : Diagnostic plots of the fit of the GEV model to the monthly maxima ofthe daily flow rate of Fraser River in British Columbia, 1912-2003.
28 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Threshold selection
I Only 14 observation exceed 12 500m3/s, and 99 values exceed 10500m3/s .
I Therefore, in order to ensure we have enough data and to moreeasily interpret the mean residual life plot, we will restrict it to therange of 350 (minimum value) to 10 500m3/s.
0 2000 4000 6000 8000 10000
−1
00
00
10
00
300
0
threshrange.plot(x = DailyFlowRate.FR, r = c(350, 10500), type = "GP", set.panels = F)
Threshold
repa
ram
ete
rize
d s
cale
0 2000 4000 6000 8000 10000
−0.2
−0
.10
.00
.10.2
Threshold
sha
pe
0 2000 4000 6000 8000 10000
02
00
06
00
0
u
Me
an
Exce
ss
Figure : Threshold selection diagnostic plots.
29 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Threshold selection
0 2000 4000 6000 8000 10000
−100
00
100
03
000
threshrange.plot(x = DailyFlowRate.FR, r = c(350, 10500), type = "GP", set.panels = F)
Threshold
rep
ara
mete
rize
d s
cale
0 2000 4000 6000 8000 10000
−0.2
−0.1
0.0
0.1
0.2
Threshold
sh
ape
0 2000 4000 6000 8000 10000
020
00
600
0
u
Me
an
Exce
ss
Figure : Threshold selection diagnostic plots.
29 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Estimate parameters
I The selected threshold POT model estimation is u =7 550m3/s.
Parameters
shape(
ξ
)scale
(σu)
Estimates -0.0734 1282.0895%CI (−0.120,−0.027) (1192.55,1371.61)
Table : Estimated parameters ξ and σu and confidence intervals, with 95% levelof confidence, of the GPD fitted to the daily flow rate of Fraser River, havingexceeded the threshold u = 7 550m3/s.
30 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Estimate parameters
I The selected threshold POT model estimation is u =7 550m3/s.
Parameters
shape(
ξ
)scale
(σu)
Estimates -0.0734 1282.0895%CI (−0.120,−0.027) (1192.55,1371.61)
Table : Estimated parameters ξ and σu and confidence intervals, with 95% levelof confidence, of the GPD fitted to the daily flow rate of Fraser River, havingexceeded the threshold u = 7 550m3/s.
30 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Adequacy of the model
I The Model.1, that incorporates the seasonality, models the scaleparameter σu as follows
σu(t) = exp(φ0 + φ1 cos(2π× t/365.25) + φ2 sin(2π× t/365.25)) ,(16)
where t = 1,2, . . . ,365.
Figure : Diagnostic plots for the fitting of GPD model.
31 / 39
Extreme Value Modelling with Application to Environmental Control
Applications to the Sea Data
Results
POT Approach: Adequacy of the model
Figure : Diagnostic plots for the fitting of GPD model.
31 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of MLE for BM method: Additional Remarks
I The idea is to use the likelihood based to estimate the nonstationarity effects (time dependence).
I For the location parameter µ we propose on the one hand (trendeffect):
µ(t) = φ0 + φ1t, for linear trend
or
µ(t) = φ0 + φ1t + φ2t2, for quadratic trend
(16)
I And on the other hand using (seasonality effect):
µ(t) = φ0 + φ1 cos(2π t) + φ2 sin(2π t) (17)
I So, θ = (φ0,φ1,φ2,σ0,ξ0) with φ0,φ1,φ2 the parameters related tothe location µ(t) via the above regression model.
32 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of MLE for BM method: Additional Remarks
I The idea is to use the likelihood based to estimate the nonstationarity effects (time dependence).
I For the location parameter µ we propose on the one hand (trendeffect):
µ(t) = φ0 + φ1t, for linear trend
or
µ(t) = φ0 + φ1t + φ2t2, for quadratic trend
(16)
I And on the other hand using (seasonality effect):
µ(t) = φ0 + φ1 cos(2π t) + φ2 sin(2π t) (17)
I So, θ = (φ0,φ1,φ2,σ0,ξ0) with φ0,φ1,φ2 the parameters related tothe location µ(t) via the above regression model.
32 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of MLE for BM method: Additional Remarks
I The idea is to use the likelihood based to estimate the nonstationarity effects (time dependence).
I For the location parameter µ we propose on the one hand (trendeffect):
µ(t) = φ0 + φ1t, for linear trend
or
µ(t) = φ0 + φ1t + φ2t2, for quadratic trend
(16)
I And on the other hand using (seasonality effect):
µ(t) = φ0 + φ1 cos(2π t) + φ2 sin(2π t) (17)
I So, θ = (φ0,φ1,φ2,σ0,ξ0) with φ0,φ1,φ2 the parameters related tothe location µ(t) via the above regression model.
32 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of MLE for BM method: Additional Remarks
I The idea is to use the likelihood based to estimate the nonstationarity effects (time dependence).
I For the location parameter µ we propose on the one hand (trendeffect):
µ(t) = φ0 + φ1t, for linear trend
or
µ(t) = φ0 + φ1t + φ2t2, for quadratic trend
(16)
I And on the other hand using (seasonality effect):
µ(t) = φ0 + φ1 cos(2π t) + φ2 sin(2π t) (17)
I So, θ = (φ0,φ1,φ2,σ0,ξ0) with φ0,φ1,φ2 the parameters related tothe location µ(t) via the above regression model.
32 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea
Recall: James Pickands, 1975
Xi −u > y | Xi > u, i = 1,2 . . .n follows a generalized Pareto distributionwith parameters: location 0, scale σu = σ + ξ (u−µ) and the shape ξ .
I To estimate our model parameters θ = (ξ ,µ,σ), we want to use allthe extremes Yi = Xi −u for those Xi > u as well as those Xi ≤ u tocompute the likelihood function L2(θ).
I Let us consider the indicator variable:
∆i = 1Xi>u =
1, if Xi exceeds u,
0, otherwise.(18)
33 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea
Recall: James Pickands, 1975
Xi −u > y | Xi > u, i = 1,2 . . .n follows a generalized Pareto distributionwith parameters: location 0, scale σu = σ + ξ (u−µ) and the shape ξ .
I To estimate our model parameters θ = (ξ ,µ,σ), we want to use allthe extremes Yi = Xi −u for those Xi > u as well as those Xi ≤ u tocompute the likelihood function L2(θ).
I Let us consider the indicator variable:
∆i = 1Xi>u =
1, if Xi exceeds u,
0, otherwise.(18)
33 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea
Recall: James Pickands, 1975
Xi −u > y | Xi > u, i = 1,2 . . .n follows a generalized Pareto distributionwith parameters: location 0, scale σu = σ + ξ (u−µ) and the shape ξ .
I To estimate our model parameters θ = (ξ ,µ,σ), we want to use allthe extremes Yi = Xi −u for those Xi > u as well as those Xi ≤ u tocompute the likelihood function L2(θ).
I Let us consider the indicator variable:
∆i = 1Xi>u =
1, if Xi exceeds u,
0, otherwise.(18)
33 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
I So we have P∆i = 1= PXi > u= pn, from Eqn. (??), andthen P∆i = 0= 1−pn.
I Actually our observed data with which we are going to work are theiid observations (x1−u,δ1), . . . ,(xn−u,δn) of respectively(Y1,∆1), . . . ,(Yn,∆n).
I Since we need to work with our new data, we are going to define thecorresponding function f (y ,δ ) joint probability density function of(Y ,∆) = (X −u,∆).
I In order to derive this density, we assume that for a given thresholdu:
Py ≤ Y < y + h |∆ = 0= Py ≤ Y < y + h | X ≤ u ≈ h (19)
for any h positively close to 0.
34 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
I So we have P∆i = 1= PXi > u= pn, from Eqn. (??), andthen P∆i = 0= 1−pn.
I Actually our observed data with which we are going to work are theiid observations (x1−u,δ1), . . . ,(xn−u,δn) of respectively(Y1,∆1), . . . ,(Yn,∆n).
I Since we need to work with our new data, we are going to define thecorresponding function f (y ,δ ) joint probability density function of(Y ,∆) = (X −u,∆).
I In order to derive this density, we assume that for a given thresholdu:
Py ≤ Y < y + h |∆ = 0= Py ≤ Y < y + h | X ≤ u ≈ h (19)
for any h positively close to 0.
34 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
I So we have P∆i = 1= PXi > u= pn, from Eqn. (??), andthen P∆i = 0= 1−pn.
I Actually our observed data with which we are going to work are theiid observations (x1−u,δ1), . . . ,(xn−u,δn) of respectively(Y1,∆1), . . . ,(Yn,∆n).
I Since we need to work with our new data, we are going to define thecorresponding function f (y ,δ ) joint probability density function of(Y ,∆) = (X −u,∆).
I In order to derive this density, we assume that for a given thresholdu:
Py ≤ Y < y + h |∆ = 0= Py ≤ Y < y + h | X ≤ u ≈ h (19)
for any h positively close to 0.
34 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
I So we have P∆i = 1= PXi > u= pn, from Eqn. (??), andthen P∆i = 0= 1−pn.
I Actually our observed data with which we are going to work are theiid observations (x1−u,δ1), . . . ,(xn−u,δn) of respectively(Y1,∆1), . . . ,(Yn,∆n).
I Since we need to work with our new data, we are going to define thecorresponding function f (y ,δ ) joint probability density function of(Y ,∆) = (X −u,∆).
I In order to derive this density, we assume that for a given thresholdu:
Py ≤ Y < y + h |∆ = 0= Py ≤ Y < y + h | X ≤ u ≈ h (19)
for any h positively close to 0.
34 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
Recall
Let us recall that the density function of (Y ,∆) is defined as:
f (y ,δ ) = limh→0
Py ≤ Y < y + h,∆ = δh
, with δ ∈ 0,1 . (20)
I Case 1: δ = 1, that is y > 0, we have: f (y ,1) = fu(y)pn.I Case 2: δ = 0, that is y ≤ 0, we have: f (y ,0) = 1−pn.
I Therefore, the probability density function of (Y ,∆) is given by:
f (y ,δ ) = (fu(y)pn)δ (1−pn)
1−δ . (21)
I Recall that pn and fu(y) are respectively ...
35 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
Recall
Let us recall that the density function of (Y ,∆) is defined as:
f (y ,δ ) = limh→0
Py ≤ Y < y + h,∆ = δh
, with δ ∈ 0,1 . (20)
I Case 1: δ = 1, that is y > 0, we have: f (y ,1) = fu(y)pn.
I Case 2: δ = 0, that is y ≤ 0, we have: f (y ,0) = 1−pn.
I Therefore, the probability density function of (Y ,∆) is given by:
f (y ,δ ) = (fu(y)pn)δ (1−pn)
1−δ . (21)
I Recall that pn and fu(y) are respectively ...
35 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
Recall
Let us recall that the density function of (Y ,∆) is defined as:
f (y ,δ ) = limh→0
Py ≤ Y < y + h,∆ = δh
, with δ ∈ 0,1 . (20)
I Case 1: δ = 1, that is y > 0, we have: f (y ,1) = fu(y)pn.I Case 2: δ = 0, that is y ≤ 0, we have: f (y ,0) = 1−pn.
I Therefore, the probability density function of (Y ,∆) is given by:
f (y ,δ ) = (fu(y)pn)δ (1−pn)
1−δ . (21)
I Recall that pn and fu(y) are respectively ...
35 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
Recall
Let us recall that the density function of (Y ,∆) is defined as:
f (y ,δ ) = limh→0
Py ≤ Y < y + h,∆ = δh
, with δ ∈ 0,1 . (20)
I Case 1: δ = 1, that is y > 0, we have: f (y ,1) = fu(y)pn.I Case 2: δ = 0, that is y ≤ 0, we have: f (y ,0) = 1−pn.
I Therefore, the probability density function of (Y ,∆) is given by:
f (y ,δ ) = (fu(y)pn)δ (1−pn)
1−δ . (21)
I Recall that pn and fu(y) are respectively ...
35 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: Formulation of the idea...
Recall
Let us recall that the density function of (Y ,∆) is defined as:
f (y ,δ ) = limh→0
Py ≤ Y < y + h,∆ = δh
, with δ ∈ 0,1 . (20)
I Case 1: δ = 1, that is y > 0, we have: f (y ,1) = fu(y)pn.I Case 2: δ = 0, that is y ≤ 0, we have: f (y ,0) = 1−pn.
I Therefore, the probability density function of (Y ,∆) is given by:
f (y ,δ ) = (fu(y)pn)δ (1−pn)
1−δ . (21)
I Recall that pn and fu(y) are respectively ...35 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: ML Estimation
I So, given the new iid observations (x1−u,δ1), . . . ,(xn−u,δn),L2(θ) is defined by (with yi = xi −u,):
L2(θ) =n
∏i=1
f (yi ,δi) =n
∏i=1
(fu (xi −u)pn)δi (1−pn)1−δi ,
I Therefore the log-likelihood is:
L2(θ) = (n−Nu) log(1−pn)−Nu log(nσ)−(
1ξ
+ 1
) n
∑i=1δi=1
logDi (22)
I where Di = 1 + ξ
(xi −µ
σ
), and Nu =
n
∑i=1
δi is the number of
observations that have exceeded the threshold u
36 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: ML Estimation
I So, given the new iid observations (x1−u,δ1), . . . ,(xn−u,δn),L2(θ) is defined by (with yi = xi −u,):
L2(θ) =n
∏i=1
f (yi ,δi) =n
∏i=1
(fu (xi −u)pn)δi (1−pn)1−δi ,
I Therefore the log-likelihood is:
L2(θ) = (n−Nu) log(1−pn)−Nu log(nσ)−(
1ξ
+ 1
) n
∑i=1δi=1
logDi (22)
I where Di = 1 + ξ
(xi −µ
σ
), and Nu =
n
∑i=1
δi is the number of
observations that have exceeded the threshold u
36 / 39
Extreme Value Modelling with Application to Environmental Control
Current developments: Additional Remarks
Review of POT approach: ML Estimation
I So, given the new iid observations (x1−u,δ1), . . . ,(xn−u,δn),L2(θ) is defined by (with yi = xi −u,):
L2(θ) =n
∏i=1
f (yi ,δi) =n
∏i=1
(fu (xi −u)pn)δi (1−pn)1−δi ,
I Therefore the log-likelihood is:
L2(θ) = (n−Nu) log(1−pn)−Nu log(nσ)−(
1ξ
+ 1
) n
∑i=1δi=1
logDi (22)
I where Di = 1 + ξ
(xi −µ
σ
), and Nu =
n
∑i=1
δi is the number of
observations that have exceeded the threshold u36 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Modelling Challenges
Covariable effects and Cluster dependenceI Location, direction, seasonality,...
I Multicovariables in practice
I e.g: Storms independent observed many times at many locations
Others challengesI Threshold estimation
I Parameters estimation
37 / 39
Extreme Value Modelling with Application to Environmental Control
Summary
Key References
J. Pickands.Statistical inference using extremes order statistics.The Annals of Statistics, 3(1): 119 â131, 1975.
D. Depuis.Exceedances over high threshold: A guide to threshold selection.DalTech, Dalhousie University, 1:111â121, 1998.
A. J. McNeil and T. Saladin.The peaks over thresholds method for estimating high quantiles ofloss distributions.In Proceedings of 27-th International ASTIN Colloquium, pages23â43, CiteSeer 5M, 1997.
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Extreme Value Modelling with Application to Environmental Control
Summary
THANK YOU FOR YOUR KINDATTENTION
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