extreme value distributions for offshore wind turbines international summer school on stochastic...

www.cesos.ntnu.no Author – Centre for Ships and Ocean Structureswww.cesos.ntnu.no CeSOS – Centre for Ships and Ocean Structures

2nd International Summer Schoolon Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers

Extreme value distributionsfor offshore wind turbines

by Arvid Naess a, b and Oleh Karpa b

a Department of Mathematical Sciences, NTNU.b CeSOS - Centre for Ships and Ocean Structures, NTNU.

www.cesos.ntnu.no Author – Centre for Ships and Ocean Structures

Outline

Introduction

Upcrossing Rate

Extreme values of Gaussian processes

Classical asymptotic Extreme Value Theory

The Average Conditional Exceedance Rate (ACER) method

Results and conclusions

The bivariate ACER method

References

A. Naess and O. Karpa, Extreme values by the ACER method. p. 2/72

Introduction

Storm hits the lighthouse, Seaham harbor, UK. February 2013.

Introduction

A skyscraper has severe glass damage after Hurricane Ike.Houston, USA, September 2008.

Introduction

A wind turbine caught fire in high winds. Scotland, December 2011.

Upcrossing Rate

Let us consider the response process X (t). The largest valueof X (t) during the time T is denoted by M(T ):M(T ) = max

{X (t); 0 ≤ t ≤ T

Let us define the event E ={

X (t) ≤ η for all t ∈ (0, T )}

This event can also be expressed asE =

{X (0) ≤ η and N+(η,T ) = 0

}, where N+(η,T ) is the

number of η-upcrossings of X (t) during time T .

Hence Pr{E}= Pr

{X (0) ≤ η and N+(η,T ) = 0

N+(η,T ) = 0}

when η →∞ because of the law ofmarginal probability.

The extreme values in most cases are much larger than thetypical values. Hence we introduce the approximationPr{E}= Pr

{N+(η,T ) = 0

}for η →∞.

Upcrossing Rate

{X (t); 0 ≤ t ≤ T

X (t) ≤ η for all t ∈ (0, T )}

{X (0) ≤ η and N+(η,T ) = 0

Hence Pr{E}= Pr

{X (0) ≤ η and N+(η,T ) = 0

N+(η,T ) = 0}

{N+(η,T ) = 0

}for η →∞.

Upcrossing Rate

{X (t); 0 ≤ t ≤ T

X (t) ≤ η for all t ∈ (0, T )}

{X (0) ≤ η and N+(η,T ) = 0

Hence Pr{E}= Pr

{X (0) ≤ η and N+(η,T ) = 0

N+(η,T ) = 0}

{N+(η,T ) = 0

}for η →∞.

Upcrossing Rate

{X (t); 0 ≤ t ≤ T

X (t) ≤ η for all t ∈ (0, T )}

{X (0) ≤ η and N+(η,T ) = 0

Hence Pr{E}= Pr

{X (0) ≤ η and N+(η,T ) = 0

N+(η,T ) = 0}

{N+(η,T ) = 0

}for η →∞.

Upcrossing Rate

{X (t); 0 ≤ t ≤ T

X (t) ≤ η for all t ∈ (0, T )}

{X (0) ≤ η and N+(η,T ) = 0

Hence Pr{E}= Pr

{X (0) ≤ η and N+(η,T ) = 0

N+(η,T ) = 0}

{N+(η,T ) = 0

}for η →∞.

Upcrossing Rate

Evidently, Pr{

M(T ) ≤ η}= Pr

{N+(η,T ) = 0

}. Both events

express the same: Level η was not exceeded during time T .

To determine Pr{

N+(η,T ) = 0}

, we shall introduce thefollowing simplifying assumption: Upcrossings of high levelsare statistically independent events.

This implies that the random number of upcrossings in anarbitrary time interval of length T is Poisson distributed withparameter E

[N+(η,T )

]= ν+X (η)T , assuming X (t) stationary.

This leads to the result

N+(η,T ) = 0}= exp

{− ν+X (η)T

}, η →∞

Finally,

FM(T )(η) = Pr{

M(T ) ≤ η}= exp

{− ν+X (η)T

}, η →∞

Upcrossing Rate

Evidently, Pr{

M(T ) ≤ η}= Pr

{N+(η,T ) = 0

}. Both events

To determine Pr{

N+(η,T ) = 0}

[N+(η,T )

N+(η,T ) = 0}= exp

{− ν+X (η)T

}, η →∞

Finally,

FM(T )(η) = Pr{

M(T ) ≤ η}= exp

{− ν+X (η)T

}, η →∞

Upcrossing Rate

Evidently, Pr{

M(T ) ≤ η}= Pr

{N+(η,T ) = 0

}. Both events

To determine Pr{

N+(η,T ) = 0}

[N+(η,T )

N+(η,T ) = 0}= exp

{− ν+X (η)T

}, η →∞

Finally,

FM(T )(η) = Pr{

M(T ) ≤ η}= exp

{− ν+X (η)T

}, η →∞

Upcrossing Rate

Evidently, Pr{

M(T ) ≤ η}= Pr

{N+(η,T ) = 0

}. Both events

To determine Pr{

N+(η,T ) = 0}

[N+(η,T )

N+(η,T ) = 0}= exp

{− ν+X (η)T

}, η →∞

Finally,

FM(T )(η) = Pr{

M(T ) ≤ η}= exp

{− ν+X (η)T

}, η →∞

Upcrossing Rate

Evidently, Pr{

M(T ) ≤ η}= Pr

{N+(η,T ) = 0

}. Both events

To determine Pr{

N+(η,T ) = 0}

[N+(η,T )

N+(η,T ) = 0}= exp

{− ν+X (η)T

}, η →∞

Finally,

FM(T )(η) = Pr{

M(T ) ≤ η}= exp

{− ν+X (η)T

}, η →∞

Extreme values of Gaussian processesFor a stationary Gaussian process with E

[X (t)

ν+X (η) = ν+X (0)exp(− η2

), where ν+X (0) =

σXσX

This gives

FM(T )(η) = exp{− ν+X (0)T exp

(− η2

)}, η →∞

A design value of interest is the level ηp = ηp(T ) not exceededduring the time T with probability p, that is FM(T )(ηp) = p.

ηp(T ) = σX

√2 ln

(− ν+X (0)T/ ln(p)

The expected value of the extreme value is obtained as:

E[M(T )] ≈ σX

{√2 ln

(ν+X (0)T

)+ 0.5772/

√2 ln

(ν+X (0)T

[X (t)

ν+X (η) = ν+X (0)exp(− η2

), where ν+X (0) =

σXσX

This gives

(− η2

)}, η →∞

ηp(T ) = σX

√2 ln

(− ν+X (0)T/ ln(p)

E[M(T )] ≈ σX

{√2 ln

(ν+X (0)T

)+ 0.5772/

√2 ln

(ν+X (0)T

[X (t)

ν+X (η) = ν+X (0)exp(− η2

), where ν+X (0) =

σXσX

This gives

(− η2

)}, η →∞

ηp(T ) = σX

√2 ln

(− ν+X (0)T/ ln(p)

E[M(T )] ≈ σX

{√2 ln

(ν+X (0)T

)+ 0.5772/

√2 ln

(ν+X (0)T

[X (t)

ν+X (η) = ν+X (0)exp(− η2

), where ν+X (0) =

σXσX

This gives

(− η2

)}, η →∞

ηp(T ) = σX

√2 ln

(− ν+X (0)T/ ln(p)

E[M(T )] ≈ σX

{√2 ln

(ν+X (0)T

)+ 0.5772/

√2 ln

(ν+X (0)T

[X (t)

ν+X (η) = ν+X (0)exp(− η2

), where ν+X (0) =

σXσX

This gives

(− η2

)}, η →∞

ηp(T ) = σX

√2 ln

(− ν+X (0)T/ ln(p)

E[M(T )] ≈ σX

{√2 ln

(ν+X (0)T

)+ 0.5772/

√2 ln

(ν+X (0)T

Let X1, . . . , Xn be a sequence of i.i.d. r.v.’s that follow Fdistribution. In applications, the Xi are values of the measuredprocess X(t).

The maximum of the process over n time units of observationsis Mn = max {X1, . . . ,Xn} .

The extreme value Mn follows the asymptotic GEV distributionas n→∞:

G(η) = exp{−(1 + γ η−µ

)−1/γ}

The exceedances above a high threshold u: {X − u |X > u}follow the GP distribution:

H(y) = Pr {X − u ≤ y |X > u} ≈ 1−(1 + γ y−µ

)−1/γ .

G(η) = exp{−(1 + γ η−µ

)−1/γ}

H(y) = Pr {X − u ≤ y |X > u} ≈ 1−(1 + γ y−µ

)−1/γ .

G(η) = exp{−(1 + γ η−µ

)−1/γ}

H(y) = Pr {X − u ≤ y |X > u} ≈ 1−(1 + γ y−µ

)−1/γ .

G(η) = exp{−(1 + γ η−µ

)−1/γ}

H(y) = Pr {X − u ≤ y |X > u} ≈ 1−(1 + γ y−µ

)−1/γ .

Classical asymptotic Extreme Value TheoryChallenges of the standard methods:

Extreme value statistics, even in applications, have very oftenbeen based on asymptotic results. The asymptotic EVT itselfcannot be used in practice to decide to what extent it isapplicable for the observed data.

Hence, the assumption that an asymptotic EVD is theappropriate distribution for the observed data is based more orless on faith or convenience

There is no general agreement on the method that wouldserve prediction purposes in a completely satisfactory manner.

None of the methods accounts for dependence effects in thedata time series. A number of approximate methods fordealing with dependence have been proposed over the years,but no good solution has been found so far.

The Average Conditional Exceedance Rate(ACER) method

The new ACER method:

The average conditional exceedance rate (ACER) methodavoids invoking the ultimate asymptotic distributions.

The method is able to provide an accurate estimate of the trueextreme value distribution given by the data.

Constructs a cascade of conditional distributions accountingfor dependence in the data with increasing accuracy.

When it has been ascertained that this cascade hasconverged, an estimate of the EVD has been obtained.

The ACER method: problem description

Let us consider values X1, . . . ,XN , which have been derivedfrom the observed stochastic process X (t) in a time interval(0,T ), and allocated to the discrete times0 ≤ t1 <, . . . , < tN ≤ T .

The extreme value is defined as MN = max{Xj ; j = 1, . . . ,N}.

The goal is to estimate the distribution functionP(η) = Pr(MN ≤ η) accurately for large values of η.

Note that P(η) = Pr(XN ≤ η, . . . ,X1 ≤ η). This joint distributionfunction cannot in general be estimated directly from the data.

Therefore, we introduce a cascade of approximations Pk (η) ofP(η), such that Pk (η)→ P(η) as k increases.

The ACER methodCascade of conditioning approximations

From the definition of P(η) and MN , we have:

P(η) = Pr(XN ≤ η, . . . ,X1 ≤ η)= Pr(XN ≤ η|XN−1 ≤ η, . . . ,X1 ≤ η) · Pr(XN−1 ≤ η, . . . ,X1 ≤ η)

Pr(Xj ≤ η |Xj−1 ≤ η, . . . ,X1 ≤ η) · Pr(X1 ≤ η)

Assuming Xj are independent and using (1− x) ≈ exp(−x), x → 0,the first approximation for η →∞ is:

P(η) ≈N∏

Pr(Xj ≤ η) =N∏

(1− α1j(η)

)≈ P1(η) = exp

N∑j=1

α1j(η))

, where α1j(η)def= Pr(Xj > η)

From the definition of P(η) and MN , we have:

P(η) = Pr(XN ≤ η, . . . ,X1 ≤ η)= Pr(XN ≤ η|XN−1 ≤ η, . . . ,X1 ≤ η) · Pr(XN−1 ≤ η, . . . ,X1 ≤ η)

Pr(Xj ≤ η |Xj−1 ≤ η, . . . ,X1 ≤ η) · Pr(X1 ≤ η)

Assuming Xj are independent and using (1− x) ≈ exp(−x), x → 0,the first approximation for η →∞ is:

P(η) ≈N∏

Pr(Xj ≤ η) =N∏

(1− α1j(η)

)≈ P1(η) = exp

N∑j=1

α1j(η))

, where α1j(η)def= Pr(Xj > η)

Let us consider a one-step memory conditioning:

Pr(Xj ≤ η|Xj−1 ≤ η, . . .,X1 ≤ η) ≈ Pr(Xj ≤ η|Xj−1 ≤ η)for j = 2, . . . ,N

This gives the second approximation:

P(η) ≈N∏

Pr(Xj ≤ η |Xj−1 ≤ η) · Pr(X1 ≤ η) =N∏

(1− α2j(η)

)·(1− α11(η)

)≈ P2(η) = exp

N∑j=2

α2j(η)− α11(η)), η →∞,

where α2j(η)def= Pr

(Xj > η |Xj−1 ≤ η

), 2 ≤ j ≤ N.

Let us consider a one-step memory conditioning:

Pr(Xj ≤ η|Xj−1 ≤ η, . . .,X1 ≤ η) ≈ Pr(Xj ≤ η|Xj−1 ≤ η)for j = 2, . . . ,N

This gives the second approximation:

P(η) ≈N∏

Pr(Xj ≤ η |Xj−1 ≤ η) · Pr(X1 ≤ η) =N∏

(1− α2j(η)

)·(1− α11(η)

)≈ P2(η) = exp

N∑j=2

α2j(η)− α11(η)), η →∞,

where α2j(η)def= Pr

(Xj > η |Xj−1 ≤ η

), 2 ≤ j ≤ N.

Further, we introduce notation :

αkj(η)def= Pr(Xj > η |Xj−1 ≤ η, . . . ,Xj−k+1 ≤ η), 1 ≤ k ≤ j ≤ N

So, αkj(η) denotes the probability of exceedance, conditioned on(k − 1) immediately preceding non-exceedances.

At the k ’th step (k ≥ 2) we shall obtain:

P(η) ≈N∏

Pr(Xj ≤ η |Xj−1 ≤ η, . . . ,Xj−k+1 ≤ η)

· Pr(Xk−1 ≤ η|Xk−2 ≤ η, . . . ,X1 ≤ η) · . . .· Pr(X3 ≤ η|X2 ≤ η,X1 ≤ η) · Pr(X2 ≤ η|X1 ≤ η) · Pr(X1 ≤ η)

≈ Pk (η) = exp(−

N∑j=k

αkj(η)−k−1∑j=1

αjj(η)), η →∞.

Further, we introduce notation :

αkj(η)def= Pr(Xj > η |Xj−1 ≤ η, . . . ,Xj−k+1 ≤ η), 1 ≤ k ≤ j ≤ N

So, αkj(η) denotes the probability of exceedance, conditioned on(k − 1) immediately preceding non-exceedances.

At the k ’th step (k ≥ 2) we shall obtain:

P(η) ≈N∏

Pr(Xj ≤ η |Xj−1 ≤ η, . . . ,Xj−k+1 ≤ η)

· Pr(Xk−1 ≤ η|Xk−2 ≤ η, . . . ,X1 ≤ η) · . . .· Pr(X3 ≤ η|X2 ≤ η,X1 ≤ η) · Pr(X2 ≤ η|X1 ≤ η) · Pr(X1 ≤ η)

≈ Pk (η) = exp(−

N∑j=k

αjj(η)), η →∞.

The ACER method

In most practical cases N � k , therefore for η →∞ holds:

P1(η) = exp(−

N∑j=1

α1j(η));

P2(η) = exp(−

N∑j=2

α2j(η)− α11(η))≈ exp

N∑j=2

α2j(η));

P3(η) = exp(−

N∑j=3

α3j(η)− α22(η)− α11(η))≈ exp

N∑j=3

α3j(η));

· · ·

Pk (η) = exp(−

N∑j=k

αjj(η))≈ exp

N∑j=k

αkj(η));

↓ (k increases)

P(η) = Pr(XN ≤ η, . . . ,X1 ≤ η)A. Naess and O. Karpa, Extreme values by the ACER method. p. 14/72

The ACER methodEmpirical estimation of the ACER

Estimation of the EV distribution reduces to estimation of theset of

{αkj(η)

}functions (Naess and Gaidai, 2009).

Let us introduce the Average Conditional Exceedance Rates(ACER):

εk (η)def=

1N − k + 1

N∑j=k

αkj(η) , k = 1, 2, . . .

Hence, for high η levels we may write:

P(η) ≈ exp{− (N − k + 1) εk (η)

}; η →∞.

{αkj(η)

εk (η)def=

1N − k + 1

N∑j=k

αkj(η) , k = 1, 2, . . .

P(η) ≈ exp{− (N − k + 1) εk (η)

}; η →∞.

{αkj(η)

εk (η)def=

1N − k + 1

N∑j=k

αkj(η) , k = 1, 2, . . .

P(η) ≈ exp{− (N − k + 1) εk (η)

}; η →∞.

The numerical estimation of the ACER functions starts by definingindicator functions:

Akj(η) = 1{Xj > η,Xj−1 ≤ η, ..., Xj−k+1 ≤ η}

Bkj(η) = 1{Xj−1 ≤ η, ..., Xj−k+1 ≤ η},

Then evidently:

αjk (η) =E[Akj(η)]

E[Bkj(η)], k ≥ 2, j = k , . . . ,N ,

Assuming X (t) is ergodic process, then obviouslyεk (η) = αkk (η) = . . . = αkN(η) and it may be assumed that for thetime series,

εk (η) = limN→∞

∑Nj=k Akj(η)∑Nj=k Bkj(η)

Akj(η) = 1{Xj > η,Xj−1 ≤ η, ..., Xj−k+1 ≤ η}

Bkj(η) = 1{Xj−1 ≤ η, ..., Xj−k+1 ≤ η},

Then evidently:

E[Bkj(η)], k ≥ 2, j = k , . . . ,N ,

Akj(η) = 1{Xj > η,Xj−1 ≤ η, ..., Xj−k+1 ≤ η}

Bkj(η) = 1{Xj−1 ≤ η, ..., Xj−k+1 ≤ η},

Then evidently:

E[Bkj(η)], k ≥ 2, j = k , . . . ,N ,

The ACER method

500 700 900 1100 1300

Time [s]

For R realizations of X (t), the sample estimate of εk (η):

εk (η) =1R

R∑r=1

ε(r)k (η).

The sample standard deviation sk (η):

sk (η) =1

R − 1

R∑r=1

(ε(r)k (η)− εk (η)

where index (r) refers to sample no. r .

The 95% Confidence Interval for independent realizations:

CI±(η) = εk (η) ± τ ·sk (η)√

where τ – corresponding quantile of the Student’st-distribution.

εk (η) =1R

R∑r=1

ε(r)k (η).

sk (η) =1

R − 1

R∑r=1

(ε(r)k (η)− εk (η)

CI±(η) = εk (η) ± τ ·sk (η)√

εk (η) =1R

R∑r=1

ε(r)k (η).

sk (η) =1

R − 1

R∑r=1

(ε(r)k (η)− εk (η)

CI±(η) = εk (η) ± τ ·sk (η)√

The ACER methodThe T -year return level by the ACER

The T -year return level ηT is represented by the distributionfunction of the annual extreme value F 1yr :

1− F 1yr (ηT ) = 1/T .

The expression (N − k + 1) · εk (η) is the expected effectivenumber of exceedances, subjected to k − 1 immediatelypreceding non-exceedances during time T .

Let us assume that the duration of the process (0,T ) takes nyyears (or corresponding periods), then

F 1yr (η) = exp {− (N − k + 1) εk (η)/ny} .

Thus, the level ηT is obtained as solution of the equation:

εk (ηT ) = − ln

(1− 1

N − k + 1.

1− F 1yr (ηT ) = 1/T .

F 1yr (η) = exp {− (N − k + 1) εk (η)/ny} .

εk (ηT ) = − ln

(1− 1

N − k + 1.

1− F 1yr (ηT ) = 1/T .

F 1yr (η) = exp {− (N − k + 1) εk (η)/ny} .

εk (ηT ) = − ln

(1− 1

N − k + 1.

1− F 1yr (ηT ) = 1/T .

F 1yr (η) = exp {− (N − k + 1) εk (η)/ny} .

εk (ηT ) = − ln

(1− 1

N − k + 1.

The ACER methodOptimal curve fitting to empirically estimated ACER functions

Based on the sampled time series only, possiblesub-asymptotic functional forms of εk (η) cannot easily bedecided.

However, the fact that the asymptotic extreme valuedistribution is of Gumbel type can be used as a guide.

Thus, it is argued (Naess and Gaidai, 2009) that the ACERfunctions can be represented in the tail as follows,

εk (η) ≈ qk exp{−ak (η − bk )ck}, η ≥ η1 ≥ b

with suitable constants ak , bk , ck , and qk .Note: ck = qk = 1 corresponds to the Gumbel distribution.

The optimal values of the parameters q,b,a, c are obtained byminimizing a mean square error function (Naess and Gaidai, 2009);

The objective function is written as:

F (a,b, c,q) =n∑

w ′i (log ε(ηi)− log q + a(ηi − b)c)2 −→ min,

where ηi , i = 1, . . . ,n, are the levels at which the ACER functionshave been empirically estimated, and

w ′i = wi/∑n

j=1 wj , with wi =(

log CI+(ηi)− log CI−(ηi))−2

denoting weight factors.

F (a,b, c,q) =n∑

w ′i = wi/∑n

j=1 wj , with wi =(

F (a,b, c,q) =n∑

w ′i = wi/∑n

j=1 wj , with wi =(

The Levenberg-Marquardt Least Squares optimization methodcombined with the objective function seems to work quite well(Naess and Gaidai, 2009);

The Nonlinear Constrained Optimization is also well suited for thetask:

F (a,b, c,q)→ min ;

log q − a(ηi − b)c ≤ 0 , i = 1, . . . ,n ;

{a,b, c,q} ∈ S(4) ,

where the constraints domain S(4) is defined as:

S(4) ={{a,b, c,q} ∈ R4 ∣∣ a, c ∈ (0,+∞);b ∈ (bmin, η1];q ∈ [ε(η1),+∞]

The Levenberg-Marquardt Least Squares optimization methodcombined with the objective function seems to work quite well(Naess and Gaidai, 2009);

The Nonlinear Constrained Optimization is also well suited for thetask:

F (a,b, c,q)→ min ;

log q − a(ηi − b)c ≤ 0 , i = 1, . . . ,n ;

{a,b, c,q} ∈ S(4) ,

where the constraints domain S(4) is defined as:

S(4) ={{a,b, c,q} ∈ R4 ∣∣ a, c ∈ (0,+∞);b ∈ (bmin, η1];q ∈ [ε(η1),+∞]

To estimate the confidence interval for the predicted EV:

the empirical confidence band is reanchored to the optimalcurve;

next, the optimal curve fitting procedure is applied to thereanchored confidence band;

finally, both fitted curves extrapolated to the level of interest willdetermine an optimized confidence interval of the predictedreturn value.

Results and conclusions: synthetic data

Let us consider the underlying normalized stationary Gaussianprocess X (t) with E

[X (t)

Let us assume that on average, there are 103 zeroupcrossings per year, that is ν+(0)T = 103, where T = 1 year.

Therefore, the distribution F 1yr (η) of the yearly extreme valueof X (t) is:

F 1yr (η) = exp{−ν+(η)T

}= exp

{−ν+(0)T exp

(−η

)}= exp

{−103 exp

(−η

The 100-year return period value η100yr is the solution ofF 1yr (η100yr ) = 1− 1/100. Thus, the 100-year value isη100yr = 4.80.

[X (t)

}= exp

{−ν+(0)T exp

(−η

)}= exp

{−103 exp

(−η

[X (t)

}= exp

{−ν+(0)T exp

(−η

)}= exp

{−103 exp

(−η

[X (t)

}= exp

{−ν+(0)T exp

(−η

)}= exp

{−103 exp

(−η

The peak events extracted from measurements of the windspeed process should be separated by 3-4 days to obtainapproximately independent data.

The synthetic peak event data separated by 3.65 days weregenerated from the F 3d(η) extreme value distribution:

F 3d(η) = exp{−q exp

(−η

where q = ν+(0)T = 10, which corresponds to T = 3.65 days,so that F 1yr (η) =

(F 3d(η)

Therefore, we generated 20 years of 100 peak event datapoints from one year, so in total 2000 data points.

(−η

(F 3d(η)

(−η

(F 3d(η)

In order to get an idea about the performance of the ACER, POTand Gumbel methods, 200 independent 20 years Monte Carlosimulations were done. We keep in mind the exact η100yr = 4.80.

XXXXXXXXXXXMethodAv. 200

η100yr CI BCI

ACER, η1 = 2.3 4.82 (4.40, 5.09) (4.47, 5.21)Gumbel, MM 4.86 – (4.39, 5.44)POT, threshold u0 = 3 4.72 – (4.27, 5.28)

ACER CI – estimated by the ACER extrapolation approach;ACER BCI – estimated by the non param. bootstrap (103 replicates);Gumbel BCI – estimated by the param. bootstrap (104 replicates);POT BCI – estimated by the param. bootstrap (103 replicates).

In order to get an idea about the performance of the ACER, POTand Gumbel methods, 200 independent 20 years Monte Carlosimulations were done. We keep in mind the exact η100yr = 4.80.

XXXXXXXXXXXMethodAv. 200

η100yr CI BCI

ACER, η1 = 2.3 4.82 (4.40, 5.09) (4.47, 5.21)Gumbel, MM 4.86 – (4.39, 5.44)POT, threshold u0 = 3 4.72 – (4.27, 5.28)

ACER CI – estimated by the ACER extrapolation approach;ACER BCI – estimated by the non param. bootstrap (103 replicates);Gumbel BCI – estimated by the param. bootstrap (104 replicates);POT BCI – estimated by the param. bootstrap (103 replicates).

Results and conclusions: real wind dataHourly maximum of the three seconds wind gust (10 m above theground). (Norwegian Meteorological Institute, eKlima.no).A – Sula (12 years) and B – Nordøyan Fyr (13 years).

Results and conclusions: real wind data

12 years records from

the Sula station. Data

series are divided into 12

annual records.

13 years records from

Nordøyan Fyr station. Data

series are divided into

13 annual records.

Results and conclusions: real wind dataPlot for Sula of εk (η) against η on a logarithmic scale, fork = 1,2. . ., 96. k = 1 uppermost, then k = 2, etc.; σ = 5.49.

3 4 5 6 7

10−4

10−3

10−2

10−1

k=1k=2k=4k=24k=48k=72k=96

Results and conclusions: real wind dataPlot for Nordøyan Fyr of εk (η) against η on a logarithmic scale, fork = 1,2. . ., 96. k = 1 uppermost, then k = 2, etc.; σ = 6.01.

3 4 5 6 7

10−4

10−3

10−2

10−1

k=1k=2k=4k=24k=48k=72k=96

Results and conclusions: real wind dataPlot for Sula of ε1(η) against η on a logarithmic scale of theoptimized fit to the data with the 95% conf. band. with η1 = 12m/s;σ = 5.49.

3 4 5 6 7 8 9

10−6

10−5

10−4

10−3

10−2

10−1

Results and conclusions: real wind dataPlot for Nordøyan of ε1(η) against η on a logarithmic scale of theoptimized fit to the data with the 95% conf. band. with η1 = 10m/s;σ = 6.01.

3 4 5 6 7 8 9

10−6

10−5

10−4

10−3

10−2

10−1

The 100-year return period levels for Sula station by theACER-method for different degrees of conditioning, Annual maximaand POT methods, respectively:

Method Spec η100yr , m/s 95% CI (η100yr ), m/s

ACER, various k

1 46.33 (43.41, 47.77)2 46.81 (44.08, 49.04)4 47.99 (44.80, 50.57)24 46.65 (44.10, 48.07)48 46.83 (44.28, 48.03)72 45.80 (43.01, 46.96)96 45.69 (42.32, 47.01)

Annual maximaMM 48.66 (41.58, 57.58)GL 52.90 (44.29, 63.39)

POT – 43.42 (39.07, 47.80)

The 100-year return period levels for Nordøyan station by theACER-method for different degrees of conditioning, Annual maximaand POT methods, respectively:

Method Spec η100yr , m/s 95% CI (η100yr ), m/s

ACER, various k

1 51.85 (48.4, 53.1)2 51.48 (46.1, 54.1)4 52.56 (46.7, 55.7)24 52.90 (47.0, 56.2)48 54.62 (47.7, 57.6)72 53.81 (46.9, 58.3)96 54.97 (47.5, 60.5)

Annual maximaMM 51.5 (45.2, 59.3)GL 55.5 (48.0, 64.9)

POT – 47.8 (44.8, 52.7)

Results and conclusions

The converged empirical ACER functions provide an accurateestimate of the extreme value distribution inherent in the data;

the ACER method appears to be robust with respect to thechoice of parameter values. Even if the parameter values maydeviate somewhat from the optimal values, the obtainedsolutions are stable;

it has been observed that the ACER method seems to giveconsistent and accurate results compared with e.g. the POTmethod.

The bivariate ACER methodIntroduction

Challenges in bivariate extreme value modelling:

the general result on possible asymptotic bivariate extremevalue distributions is in a sense too general to be of muchpractical value;

focus on bivariate copula models;

the main problem with the current copula approach is that it isto some extent ad hoc;

there are no precise estimation tools that allow you to decideon the joint extreme value distribution (or copula) from a givenset of bivariate data.

The bivariate ACER methodProblem description

Consider a bivariate stochastic process Z (t) =(X (t),Y (t)

0 ≤ t ≤ T ; the sampled values(X1,Y1

), . . . ,

(XN ,YN

allocated to the times 0 ≤ t1 < . . . < tN ≤ T .

Objective is to determine the distribution function of theextreme value vector ZN =

(XN , YN

), where

XN = max1≤i≤N Xi , YN = max1≤i≤N Yi .

Specifically, we want to estimate P(ξ, η) = Pr(XN ≤ ξ, YN ≤ η

)accurately for large values of ξ and η (and N).

), . . . ,

(XN ,YN

(XN , YN

), where

), . . . ,

(XN ,YN

(XN , YN

), where

The bivariate ACER methodCascade of conditioning approximations

To ease the notation we introduce the non-exceedance event

Ckj ={

Xj−1 ≤ ξ,Yj−1 ≤ η, . . . ,Xj−k+1 ≤ ξ,Yj−k+1 ≤ η}

for 1 ≤ k ≤ j ≤ N + 1.

Then from the definition of P(ξ, η) follows:

P(ξ, η) = Pr(XN ≤ ξ,YN ≤ η . . . ,X1 ≤ ξ,Y1 ≤ η

(CN+1,N+1

(XN ≤ ξ,YN ≤ η | CNN

)· Pr

N∏j=2

Pr(Xj ≤ ξ,Yj ≤ η | Cjj

)· Pr

(C22).

To ease the notation we introduce the non-exceedance event

Ckj ={

Xj−1 ≤ ξ,Yj−1 ≤ η, . . . ,Xj−k+1 ≤ ξ,Yj−k+1 ≤ η}

for 1 ≤ k ≤ j ≤ N + 1.

Then from the definition of P(ξ, η) follows:

(CN+1,N+1

(XN ≤ ξ,YN ≤ η | CNN

)· Pr

N∏j=2

Pr(Xj ≤ ξ,Yj ≤ η | Cjj

)· Pr

(C22).

Conditioning on not more than k − 1 previous data points gives:

P(ξ, η) ≈N∏

Pr(Xj ≤ ξ,Yj ≤ η | Ckj

)· Pr

{1− Pr(Xj > ξ | Ckj)−Pr(Yj > η | Ckj)

+Pr(Xj > ξ,Yj > η | Ckj)}· Pr

(Ckk). k = 2, · · · ,N,

where k = 2 means conditioning only on the previous observation.

For j = 1, . . . ,N we introduce the notation

Pr(Xj > ξ | Ckj

):= αk j(ξ; η);

Pr(Yj > η | Ckj

):= βk j(η; ξ);

Pr(Xj > ξ,Yj > η | Ckj

):= γk j(ξ, η).

Conditioning on not more than k − 1 previous data points gives:

P(ξ, η) ≈N∏

Pr(Xj ≤ ξ,Yj ≤ η | Ckj

)· Pr

{1− Pr(Xj > ξ | Ckj)−Pr(Yj > η | Ckj)

+Pr(Xj > ξ,Yj > η | Ckj)}· Pr

(Ckk). k = 2, · · · ,N,

where k = 2 means conditioning only on the previous observation.

For j = 1, . . . ,N we introduce the notation

Pr(Xj > ξ | Ckj

):= αk j(ξ; η);

Pr(Yj > η | Ckj

):= βk j(η; ξ);

Pr(Xj > ξ,Yj > η | Ckj

):= γk j(ξ, η).

Using the approximation (1− x) ≈ exp(−x) when x → 0, for thehigh values of ξ and η we shall obtain:

P(ξ, η) ≈ exp{−

N∑j=k

(αkj(ξ; η)+βkj(η; ξ)− γkj(ξ, η)

·exp{−

k−1∑j=1

(αjj(ξ; η) + βjj(η; ξ)− γjj(ξ, η)

N� k≈ Pk (ξ, η) = exp{−

N∑j=k

(αkj(ξ; η)+βkj(η; ξ)− γkj(ξ, η)

The converging sequence:

P1(ξ, η) ≈ exp{−

N∑j=1

(α1j(ξ) + β1j(η)− γ1j(ξ, η)

P2(ξ, η) ≈ exp{−

N∑j=2

(α2j(ξ; η) + β2j(η; ξ)− γ2j(ξ, η)

Pk (ξ, η) ≈ exp{−

N∑j=k

(αkj(ξ; η) + βkj(η; ξ)− γkj(ξ, η)

↓ (k increases)

)A. Naess and O. Karpa, Extreme values by the ACER method. p. 41/72

The bivariate ACER methodEmpirical estimation of the ACER

Estimation of the bivariate EV distribution reduces to estimation ofthe set of functions

{(αkj(ξ; η) + βkj(η; ξ)− γkj(ξ, η)

Let us introduce the Average Conditional Exceedance Rates(ACER) surface:

Ek (ξ, η)def=

1N − k + 1

N∑j=k

(αkj(ξ; η) + βkj(η; ξ)−γkj(ξ, η)

), k = 1, 2, . . .

Since N � k , hence we may write:

P(ξ, η) ≈ exp{− (N − k + 1) Ek (ξ, η)

}; ξ, η →∞.

Ek (ξ, η)def=

1N − k + 1

N∑j=k

), k = 1, 2, . . .

P(ξ, η) ≈ exp{− (N − k + 1) Ek (ξ, η)

}; ξ, η →∞.

Ek (ξ, η)def=

1N − k + 1

N∑j=k

), k = 1, 2, . . .

P(ξ, η) ≈ exp{− (N − k + 1) Ek (ξ, η)

}; ξ, η →∞.

The numerical estimation of the ACER functions starts byintroducing a set of indicator functions:

Remember thatCkj(ξ, η) = {Xj−1 ≤ ξ,Yj−1 ≤ η, . . . ,Xj−k+1 ≤ ξ,Yj−k+1 ≤ η}

Akj(ξ; η) = 1{Xj > ξ ∩ Ckj(ξ, η)};

Bkj(η; ξ) = 1{Yj > η ∩ Ckj(ξ, η)};

Gkj(ξ, η) = 1{Xj > ξ,Yj > η ∩ Ckj(ξ, η)};

Ckj(ξ, η) = 1{Ckj(ξ, η)},

for k ≥ 2, j = k , . . . ,N, where 1{A} denotes the indicator functionof some event A.

The numerical estimation of the ACER functions starts byintroducing a set of indicator functions:Remember thatCkj(ξ, η) = {Xj−1 ≤ ξ,Yj−1 ≤ η, . . . ,Xj−k+1 ≤ ξ,Yj−k+1 ≤ η}

Akj(ξ; η) = 1{Xj > ξ ∩ Ckj(ξ, η)};

Bkj(η; ξ) = 1{Yj > η ∩ Ckj(ξ, η)};

The numerical estimation of the ACER functions starts byintroducing a set of indicator functions:Remember thatCkj(ξ, η) = {Xj−1 ≤ ξ,Yj−1 ≤ η, . . . ,Xj−k+1 ≤ ξ,Yj−k+1 ≤ η}

Akj(ξ; η) = 1{Xj > ξ ∩ Ckj(ξ, η)};

Bkj(η; ξ) = 1{Yj > η ∩ Ckj(ξ, η)};

Hence, it follows that,

αkj(ξ; η) =E[Akj(ξ; η)]

E[Ckj(ξ, η)], βkj(η; ξ) =

E[Bkj(η; ξ)]

E[Ckj(ξ, η)], γkj(ξ, η) =

E[Gkj(ξ, η)]

E[Ckj(ξ, η)],

k ≥ 2, j = k , . . . ,N.

Assuming ergodicity of the bivariate process it may be assumedthat,

Ek (ξ, η) ≈

N∑j=k

(Akj(ξ; η) + Bkj(η; ξ)−Gkj(ξ, η)

j=kCkj(ξ, η)

, N � k .

Hence, it follows that,

αkj(ξ; η) =E[Akj(ξ; η)]

E[Ckj(ξ, η)], βkj(η; ξ) =

E[Bkj(η; ξ)]

E[Ckj(ξ, η)], γkj(ξ, η) =

E[Gkj(ξ, η)]

E[Ckj(ξ, η)],

k ≥ 2, j = k , . . . ,N.

Assuming ergodicity of the bivariate process it may be assumedthat,

Ek (ξ, η) ≈

N∑j=k

(Akj(ξ; η) + Bkj(η; ξ)−Gkj(ξ, η)

j=kCkj(ξ, η)

, N � k .

The bivariate ACER methodSurface matching

The T -year return level (ξT , ηT ) is represented by the joint DF ofthe annual maxima:

1− F 1yr (ξT , ηT ) = 1/T ,

If duration of the bivariate process Z (t) is ny years, then

F 1yr (ξ, η) = exp{− N − k + 1

nyEk (ξ, η)

Thus, the joint level (ξT , ηT ) is obtained as solution of the equation:

Ek (ξT , ηT ) = − log

(1− 1

N − k + 1.

The sub-asymptotic functional form of the ACER surface Ek (ξ, η)can possibly be obtained approximately by the copularepresentation of a bivariate EVD and the Sklar’s theorem.

The bivariate EVD with marginals Fx(ξ) and Gy (η) can bepresented through the associated extreme-value copula:

H(ξ, η) = exp

{log(Fx(ξ)Gy (η)

log(Fx(ξ)

)log(Fx(ξ)Gy (η)

))} ,where the dependence function D(x) : [0,1] 7−→ [max(x , 1− x), 1]

Type B distribution – Gumbel logistic (GL) model – setsD(x) = [xm + (1− x)m]1/m for m ≥ 1;

The asymmetric logistic (AL) model setsD(x) = [φmxm + θm(1− x)m]1/m + (θ − φ)x + 1− θ for0 ≤ θ ≤ 1, 0 ≤ φ ≤ 1, m ≥ 1.

H(ξ, η) = exp

{log(Fx(ξ)Gy (η)

log(Fx(ξ)

)log(Fx(ξ)Gy (η)

H(ξ, η) = exp

{log(Fx(ξ)Gy (η)

log(Fx(ξ)

)log(Fx(ξ)Gy (η)

H(ξ, η) = exp

{log(Fx(ξ)Gy (η)

log(Fx(ξ)

)log(Fx(ξ)Gy (η)

We assume that the asymptotically consistent marginal EVD arepresented through the univariate ACER functions:

Fx(ξ) ≈ exp{− (N − k + 1)εx

k (ξ)}, ξ ≥ ξ1 ;

Gy (η) ≈ exp{− (N − k + 1)εy

k (η)}, η ≥ η1 ,

where εk (θ) = qk exp{−ak (θ − bk )ck}.

As we have discovered, the bivariate EVD can be expressed as:

H(ξ, η) = exp {− (N − k + 1) Ek (ξ, η)}.

This implies that the following representation of the bivariate ACERsurface can be derived:

Eck (ξ, η) =

k (ξ) + εyk (η)

εxk (ξ)

εxk (ξ) + εy

k (η)

); ξ ≥ ξ1, η ≥ η1.

Fx(ξ) ≈ exp{− (N − k + 1)εx

k (ξ)}, ξ ≥ ξ1 ;

Gy (η) ≈ exp{− (N − k + 1)εy

k (η)}, η ≥ η1 ,

H(ξ, η) = exp {− (N − k + 1) Ek (ξ, η)}.

Eck (ξ, η) =

k (ξ) + εyk (η)

εxk (ξ)

εxk (ξ) + εy

k (η)

); ξ ≥ ξ1, η ≥ η1.

Fx(ξ) ≈ exp{− (N − k + 1)εx

k (ξ)}, ξ ≥ ξ1 ;

Gy (η) ≈ exp{− (N − k + 1)εy

k (η)}, η ≥ η1 ,

H(ξ, η) = exp {− (N − k + 1) Ek (ξ, η)}.

Eck (ξ, η) =

k (ξ) + εyk (η)

εxk (ξ)

εxk (ξ) + εy

k (η)

); ξ ≥ ξ1, η ≥ η1.

The functional form of the bivariate ACER in case of the AL model:

Ek (ξ, η) =[(φεX

k (ξ))m

+(θεY

k (η))m] 1

m+ (1− φ)εX

k (ξ) + (1− θ)εYk (η)

The mean square error function to find optimal θ, φ and m:

Fm,θ,φ =

Nη∑j=1

Nξ∑i=1

w ′ij(

log Ek (ξi , ηj)− logAk (ξi , ηj))2−→ min,

w ′ij =wij∑∑

wijwith wij =

(log CI+ij − log CI−ij

)−2are normalized

weight factors;

Nη,Nξ are numbers of levels η and ξ, at which the empirical ACERfunction Ek (ξi , ηj) have been estimated.

k (ξ))m

+(θεY

k (η))m] 1

m+ (1− φ)εX

k (ξ) + (1− θ)εYk (η)

Fm,θ,φ =

Nη∑j=1

Nξ∑i=1

w ′ij(

w ′ij =wij∑∑

wijwith wij =

)−2are normalized

weight factors;

k (ξ))m

+(θεY

k (η))m] 1

m+ (1− φ)εX

k (ξ) + (1− θ)εYk (η)

Fm,θ,φ =

Nη∑j=1

Nξ∑i=1

w ′ij(

w ′ij =wij∑∑

wijwith wij =

)−2are normalized

weight factors;

The bivariate ACER methodBehavior of the ACER surface

We assume that the joint EVD Hxy (ξ, η) and the marginalEVD’s Fx(ξ) and Gy (η) of the

(XN , YN

)are presented through

the corresponding univariate and bivariate ACER functions.

In case of statistical independence between(

XN , YN

)the joint

EVD Hxy (ξ, η) = Fx(ξ)Gy (η). This yieldsEk (ξ, η) = εx

k (ξ) + εyk (η).

In case of statistical dependence,Hxy (ξ, η) = min

{Fx(ξ), Gy (η)

}. Therefore,

Ek (ξ, η) = max{εx

k (ξ), εyk (η)

(XN , YN

XN , YN

)the joint

k (ξ) + εyk (η).

{Fx(ξ), Gy (η)

}. Therefore,

k (ξ), εyk (η)

(XN , YN

XN , YN

)the joint

k (ξ) + εyk (η).

{Fx(ξ), Gy (η)

}. Therefore,

k (ξ), εyk (η)

To illustrate the concept let us consider stationary responses X (t)and Y (t) with the controlled correlation.

0.8 1 1.2 1.4 1.6

−3.5

−2.2

−2.4

−2.6

−2.7

−2.9

−3.2

−3.8

E20 (εx20 + εy20)

(a) Contours of the E20(ξ, η) and(εx

20(ξ) + εy20(η)

)on a log scale. ρ = 0

0.9 1 1.1 1.2 1.3 1.4

−3.6

−2.3

−2.5−2.6

−2.8−2.9

−3.1−3.3

−3.8

−4.1

E20 max{εx20, εy20}

(b) Contours of the E20(ξ, η) andmax

20(ξ), εy20(η)

}on a log scale. ρ = 1

Figure visualizes the change of the behavior of E20(ξ, η) surface asρ increases from 0 to 1.

Figure: Contour lines of the ACER surface E20(ξ, η) for increasing values of ρ. Boxesindicate the same levels on a logarithmic scale.

The bivariate ACER method: numerical resultsThe simultaneous wind speed data were analyzed. Location: Sulaand Nordøyan Fyr weather stations in the Norwegian Sea.

The bivariate ACER method: numerical resultsThe hourly maximum of the three seconds wind gust (10 m abovethe ground) were recorded during 13 years (1999 - 2012).Pearson’s ρ = 0.73, Kendall’s τ = 0.5 and Spearman’s ρ = 0.68.

The bivariate ACER method: numerical results

The data series were sectioned into 13 one-year records.

The univariate ACER functions εk were estimated:

The time dependence between the data is largely accountedfor by k = 24 since all εk converge in the far tail for k ≥ 24.

The optimized parametrical fit to the data for ε1 for was foundfor both time series.

q1 b1 a1 c1Sula 0.52 0 0.003 2.19Nordøyan 1.02 0 0.008 1.89

Surfaces Ek were estimated for different k . E1 is theuppermost; the following surfaces match in the tail for all k .

Surfaces Ek for different k plotted on a logarithmic scale.A. Naess and O. Karpa, Extreme values by the ACER method. p. 55/72

Contour plot of log10 Ek (ξ, η) for different k .

The E1 was chosen for the analysis: surfaces converge in thetail for all k ; more data is available for k = 1.

The optimized A1 fit: mA = 2.44, θ = 0.86 and φ = 0.92.Since θ andφ ≈ 1, the G1 model was considered: mG = 2.01.

20 25 30 35 40

−1−1.3

−1.5

−1.8−2

−2.3−2.7

−3.2−3.6

−4.1

E1G1A1

Level lines of three surfaces. Red boxes indicate levels of E1 on a logarithmic scale.

The optimal surfaces G96 and A96 capture the statisticalproperties of the bivariate observations.

25 29 33 37 41

−2.9

−3.1

−3.2

−3.3−3.4

−3.5

−3.7

−3.9

−4.2

E96G96A96

Contour lines of G1 and G96 that correspond to the same returnperiod levels.

36 39 42 45 48 51

Boxes indicate return period levels in years.

The bivariate ACER method: numerical resultsWind speed (3 hours mean) vs. Significant wave height (total sea)data were analyzed. Location: Heidrun oil and gas field in theNorwegian Sea (N 65.29, E 7.32).

The bivariate ACER method: numerical resultsThe data were recorded during 54 years (1957 - 2011);8 observations per day (every three hours).Pearson’s ρ = 0.79, Kendall’s τ = 0.56 and Spearman’s ρ = 0.7.

The data series were sectioned into 18 three-years records.

12 16 20 24 28

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

(a) εk for the wind speed data.

4 6 8 10 12 14

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

(b) εk for the wave height data.

12 16 20 24 28

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

4 6 8 10 12 14

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

12 16 20 24 28

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

4 6 8 10 12 14

10−4

10−3

10−2

10−1

k=1k=2k=3k=8k=16k=24k=32

q2 b2 a2 c2Ws 0.05 0.10 1.94 · 10−4 3.14Hs 0.04 −2.27 0.02 2.23

The 100-year return level value and its 95% CI were estimated:

15 20 25 30 35

10−6

10−5

10−4

10−3

10−2

ε2(ξ)

ε2fit(ξ)

(a) εfit2 for the wind speed data; ξ1 = 14.5

6 10 14 18

10−6

10−5

10−4

10−3

10−2

ε2(η)

ε2fit(η)

(b) εfit2 for the wave height data.; η1 = 4.5.

q2 b2 a2 c2Ws 0.05 0.10 1.94 · 10−4 3.14Hs 0.04 −2.27 0.02 2.23

The 100-year return level value and its 95% CI were estimated:

15 20 25 30 35

10−6

10−5

10−4

10−3

10−2

ε2(ξ)

ε2fit(ξ)

(a) εfit2 for the wind speed data; ξ1 = 14.5

6 10 14 18

10−6

10−5

10−4

10−3

10−2

ε2(η)

ε2fit(η)

(b) εfit2 for the wave height data.; η1 = 4.5.

Surfaces Ek were estimated for different k . E1 is theuppermost; the following surfaces match in the tail for k ≥ 2.

Surfaces Ek for different k plotted on a logarithmic scale.A. Naess and O. Karpa, Extreme values by the ACER method. p. 64/72

The E2 was chosen for the analysis: surfaces for k ≥ 2 allconverge in the tail; more data is available for k = 2.

The optimized AL fit: mAL = 7, θ = 1 and φ = 0.91.Since θ andφ ≈ 1, the GL model was considered: mGL = 4.78.

18 20 22 24 26 28

−2.2

−2.4

−2.6

−2.8

−3.3−3.6

−4−4.3

Level lines that correspond to the return period levels for E2,optimized AL and GL surfaces were found.

26 28 30 3211

Red boxes indicate return period levels in years.

The optimal surfaces GL32 and AL32 capture the statisticalproperties of the bivariate observations.

20 22 24 26 286

−2.5−2.7

−2.8−3

−3.1−3.3

−3.5−3.8

−4.1

−4.3

E32GL32

Contour lines of G2 and G32 that correspond to the same returnperiod levels.

26 28 30 32

Boxes indicate return period levels in years.

The bivariate ACER method: concluding remarks

The bivariate ACER method constructs a cascade ofconditioning approximations that converge to the exactbivariate EVD given by the data;

This cascade is represented by the converged bivariate ACERsurfaces;

Plot of the ACER functions is a powerful diagnostic tool todecide on the degree of conditioning needed for extreme valueestimation;

The ACER surface is able to capture the dependence structure(spatial and time) of the considered bivariate process;

The bivariate EV copula of specific type with asymptoticallyconsistent marginals, can be used as the parametricrepresentative of the ACER surface empirically estimated fromthe given data set.

References

O. Karpa and A. Naess. Extreme value statistics of wind speeddata by the ACER method. Journal of Wind Engineering andIndustrial Aerodynamics,112:1–10, 2013.

A. Naess, O.Gaidai and O. Karpa. Estimation of extreme values bythe Average Conditional Exceedance Rate method. Journal ofProbability and Statistics,2013:15 pages, 2013.

A. Naess and O. Karpa. Statistics of extreme wind speeds andwave heights by the bivariate ACER method. In Proceedings of32nd International Conference on Offshore Mechanics and ArcticEngineering (OMAE 2013), pp. OMAE2013-10760. New York:ASME, June 2013.

A. Naess and O. Karpa. Statistics of bivariate extreme wind speedsby the ACER method. To appear in Proceedings of 11thInternational Conference on Structural Safety & Reliability(ICOSSAR 2013), New York, USA, June 2013.

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