bivariate generalized pareto distribution in...
TRANSCRIPT
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Introduction Extreme value models Applications
Bivariate generalized Pareto
distribution in practice
Pál Rakonczai
Eötvös Loránd University, Budapest, Hungary
Minisymposium on Uncertainty Modelling27 September 2011, CSASC 2011, Krems, Austria
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications
Outline
I Short summary of extreme value theory (EVT), including:I univariate/multivariate caseI representation of dependence structures within EVTI multivariate generalized Pareto distribution of type II
I Construction of new asymmetric models
I Properties and comparison with well-known models
I Applications on environmental data (wind speed / flood peak)
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Introduction
I There is often interest in understanding how the extremes oftwo different processes are related to each other
I One possible solution is using asymptotic approach from EVT
I EVT provides limits for the distribution of extremes for asingle/multivariate stationary process
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Limit for Maximum
Theorem (Fisher and Tippet, 1928)
If there exist {an} and {bn} > 0 sequences such that
P(Mn − an
bn≤ z
)
→ G (z) as n → ∞
for some non-degenerate distribution G, then
G (x) = exp
{
−(
1 + ξx − µ
σ
)−1ξ
}
,
where 1 + ξ x−µσ > 0 and in such a case random variable belongs tothe max-domain of attraction of a GEV distribution.
→ Often applied for block maxima (monthly, yearly, ...)
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Limit for Threshold Exceedances
How to model large values over a given high threshold?
Theorem (Balkema and de Haan, 1974)
Suppose, that X1, ...,Xni.i.d.∼ F , where F belongs to the
max-domain of attraction of GEV for some ξ, µ and σ > 0. Then
P(Xi − u ≤ z |Xi > u) → H(z) = 1−(
1 +ξz
σ̃
)−1ξ
as u → z+,
where σ̃ = σ + ξ(u − µ).
→ Often applied for threshold exceedances over a highquantile of the obs.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Remarks
I The limit distributions are not GaussianI Strong link between GEV and GPDI See examples below:
−2 0 2 4 6 8 10
0.00.1
0.20.3
0.4
GEV
domain
dens
ity
GumbelFrechetWeibull
2 4 6 8 10
0.00.2
0.40.6
0.81.0
GPD
domain
dens
ity
ξ= 0ξ= 0.5ξ= − 0.3
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Limit for Multivariate Maxima
I Component-wise maxima of Xi = (Xi ,1, ...,Xi ,d ) is
Mn =(
n∨
i=1
Xi ,1, ...,
n∨
i=1
Xi ,d
)
.
I If X ∼ F and there exist an and bn > 0 sequences ofnormalizing vectors, such that
P(Mn − an
bn≤ z
)
= F n(bnz+ an) → G (z),
where Gi margins are non-degenerate, then G is said to be amultivariate extreme value distribution (MEVD).
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Characterization I. - Resnick (1987)
I MEVD with unit Fréchet margins can be written as
GFréchet(t1, . . . , td ) = exp(
−V (t1, . . . , td ))
I V is called exponent measure
V (t1, . . . , td ) = ν(([0, t1]×· · ·×[0, td ])c ) =
∫
Sd
d∨
i=1
(wi
ti
)
S(dw),
I S spectral measure is a finite measure on the d -dimensionalunit simplex satisfying the equations
∫
Sd
wiS(dw) = 1 for i = 1, ..., d .
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Some properties of the characterization I.
I Total mass of S is always S(Sd ) = d
I Density not only on the interior of Sd but also on each of thelower-dimensional subspaces of Sd
I For independent margins S({ei}) = 1 for all ei vertices of thesimplex
I For completely dependent margins S((1/d , ..., 1/d)) = d
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Characterization II. (bivariate case) - Pickands (1981)
The dependence function A(t) must satisfy the following threeproperties: (P)
1. A(t) is convex2. max{(1− t), t} ≤ A(t) ≤ t3. A(0) = A(1) = 1
V (t1, t2) =( 1
t1+
1
t2
)
A( t1
t1 + t2
)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
t
A(t
)
(0,1) (1,1)
(1/2,1/2)
CONVEX
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
For those who like uniform margins better
The copula of a MEVD G is the distribution function of therandom vector (G1(Y1), ...,Gd (Yd )) and is given by
C (u) = exp(
−V( −1
log u1, . . . ,
−1
log ud
))
,u ∈ [0, 1]d .
Such a copula necessarily satisfies the stability property
C s(u) = C (us1, . . . , usd )
and conversely if this property is satisfied than we get a copula of aMEVD.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Logistic Dependence (Gumbel copula)
A(t) = (tα + (1− t)α)1/α
C (u, v) = exp(−((− log u)α + (− log v)α)1/α)
Copula distribution function
u
v
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
α = 2.5
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
Spectral density
t
(α−1)(
t(1−t))
(α−2) (t
α +(1−
t)α )(1
α−2)
α = 1.5α = 2α = 2.5α = 3α = 3.5
0.0 0.2 0.4 0.6 0.8 1.0
0.40.5
0.60.7
0.80.9
1.0
Dependence function
t(tα
+(1−t)
α )(1α)
α = 1.5α = 2α = 2.5α = 3α = 3.5
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Logistic BEVD and its asymmetric extension
A(t) = (1− Φ1)t + (1− Φ2)(1− t) + ((Φ1t)α + (Φ2(1− t))
α)1/α
0 ≤ Φ1,Φ2 ≤ 1
→ S({0, 1}) = 1− Φ2,S({1, 0}) = 1− Φ1
−2 0 2 4 6
−20
24
6
x
y
0.00
0.05
0.10
0.15
0.20
0.25
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.1
6
symmetric, dep=2.5
−2 0 2 4 6
−20
24
6x
y0.00
0.05
0.10
0.15
0.02 0.04
0.06
0.08
0.1 0.1
2 0.14
asy = c(0.5,0.9)
−2 0 2 4 6
−20
24
6
y
0.00
0.05
0.10
0.15
0.02
0.04
0.06 0.08
0.1
0.12
asy = c(0.5,0.5)
−2 0 2 4 6
−20
24
6
y
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.01
0.02
0.03
0.04 0.05 0.06
0.07
0.08
0.0
9
0.1
0.11
asy = c(0.5,0.1)
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
MGPD of type II
I Problem: taking maxima hides the time structure within theperiod..
I Might be better investigating all observations over a givenhigh threshold
Let Y = (Y1, ...,Yd ) denote a random vector, u = (u1, ..., ud ) ahigh threshold vector and so X = Y − u the vector of exceedances.The distribution of X � 0 exceedances can be well approximatedby the multivariate generalized Pareto distribution (Rootzén andTajvidi, 2006)
H(x1, . . . , xd ) =−1
logG(
0, . . . , 0) log
G(
x1, . . . , xd)
G(
x1 ∧ 0, . . . , xd ∧ 0) ,
where 0 < G (0, . . . , 0) < 1.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
What is type I. then?
I If we claim exceedance in all of the coordinates (BGPD I), weusually get simpler models, with nice properties (marginals areunivariate GPD etc.)
I But we may use less data
Logistic bgpd by ’evd’
Fehmarn m/s
Hann
over
m/s
0 5 10 15 20 25
05
1015
2025
30
density curves
No Inference!
No Inference!No Inference!
0 5 10 15 20 25
05
1015
2025
30
Logistic bgpd by ’mgpd’
Fehmarn m/s
Hann
over
m/s
density curves
No Inference!
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Remarks
I H1(x) = H(x ,∞) is not a one dimensional GPD, only theconditional distribution of X1|X1 > 0 is GPD.
I The ξi , µi , σi parameters cannot be considered as the ithmarginal parameters any more
I Motivating problem: asymmetric logistic model does not leadto an absolutely continuous MGPD..
TheoremLet H be a MGPD represented by an absolutely continuous MEVD
G with spectral measure S. The MGPD H is absolutely continuous
if and only if S(int(Sd )) = d holds, i.e all mass is put on theinterior of the simplex.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Overview of parametric dependence models
Model Asym. Density Complications
Sym. Logistic − X −Asym. Logistic (Tawn, 1988) X − −Ψ-logistic∗ X X convexity constr.Sym. Neg. Logistic − X −Asym. Neg. Logistic (Joe, 1990) X − −Ψ-negative logistic∗ X X convexity constr.Bilogistic (Smith, 1990) X X not explicitNeg. Bilogistic (C. & T., 1994) X X not explicitTajvidi’s (Tajvidi, 1996) − X −C-T (C. & T., 1991) X X only s(w) is givenMixed (Tawn, 1988) − if ψ = 1 −Asym. Mixed X − −
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Construction of absolutely continuous BGPDs
1. take an absolutely continuous baseline dependence model;
2. take a strictly monotonic transformationΨ(x) : [0, 1] → [0, 1], such that Ψ(0) = 0, Ψ(1) = 1;
3. construct a new dependence model from the baseline modelAΨ(t) = A(Ψ(t));
4. check the constraints: A′Ψ(0) = −1, A′
Ψ(1) = 1 and A′′
Ψ ≥ 0(convexity).
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
An example for Ψ(x)
I Feasible transformation if
Ψ(t) = t + f (t) ⇒ (A(Ψ(t)))′ = A′(Ψ(t))× [1 + f ′(t)]
so choosing f such that f ′(0) = f ′(1) = 0, the A′Ψ(0) = −1and A′Ψ(1) = 1 are fulfilled.
I
fψ1,ψ2(t) = ψ1(t(1− t))ψ2 , for t ∈ [0, 1],
where ψ1 ∈ R and ψ2 ≥ 1 are asymmetry parameters.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Graphical examples
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
δ2tAαLog=2(Ψ(t))
t
Psi−Logisticψ1 = − 0.3ψ1 = − 0.15ψ1 = 0ψ1 = 0.15ψ1 = 0.3
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
δ2tAαNegLog=1.2(Ψ(t))
t
Psi−NegLogψ1 = − 0.3ψ1 = − 0.15ψ1 = 0ψ1 = 0.15ψ1 = 0.3
1.4 1.6 1.8 2.0 2.2
−0
.4−
0.2
0.0
0.2
0.4
Constraints of convexityfor psi−logistic models
α
ψ1
ψ2 = 1.8ψ2 = 2ψ2 = 2.2
0.9 1.0 1.1 1.2 1.3 1.4 1.5
−4
−2
02
4
Constraints of convexityfor psi−negative logistic models
α
ψ1
ψ2 = 1.8ψ2 = 2ψ2 = 2.2
Figure: Upper panels: second derivatives of Ψ-logistic and Ψ-negativePál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Remarks
I Of course f and or the baseline dependence function can bechosen differently
I This transformation has particular impact beyond the BGPDmodels as well: applicable for BEVDs, BGPDs of type I or forcopulas of any bivariate distributions
I Although the transformations Ψ was defined now for thebivariate case only, it can be extended for the higherdimensional cases as well as e.g. in the trivariate case it canhave a form of
Ψ(t1, t2) = (t1+ψ1,1(t1([1−t2]−t1))ψ2,1 , t2+ψ2,1(t2([1−t1]−t2))
ψ2,2).
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Further models - polinomial transformations
fa,p(t) =64a(p5 − p4 − 2p3 − 2p2 + 5p − 2)p5
(p + 1)2(p − 1)2(−1 + 2p)(1 + 2p2 − 3p)t6
+−32a(5p6 − 2p5 − 13p4 − 12p3 + 15p2 + 6p − 6)p4
(p + 1)2(p − 1)2(−1 + 2p)(1 + 2p2 − 3p)t5
+32a(4p7 + 3p6 − 14p5 − 15p4 + 23p2 − 8p − 2)p3
(p + 1)2(p − 1)2(−1 + 2p)(1 + 2p2 − 3p)t4
+−32a(p7 + 4p6 − 5p5 − 10p4 − 5p3 + 12p2 + 2p − 4)p3
(1 + 2p2 − 3p)(p + 1)2(4p − 1 + 2p3 − 5p2)t3
+32a(p6 − 3p4 − p2 + 4p − 2)p3
(1 + 2p2 − 3p)(p + 1)2(4p − 1 + 2p3 − 5p2)t2.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
Further models - polinomial transformations
−4 −2 0 2 4
−4−2
02
4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
−4 −2 0 2 4
−4−2
02
4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
−4 −2 0 2 4
−4−2
02
4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Univariate Extremes Multivariate Extremes New models
BEVD case
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Wind data
I Daily windspeed values at two locations in northern Germany(Bremerhaven and Schleswig) over the past five decades(1957-2007)
I Originally hourly windspeed was recorded, but in order toreduce the serial correlation within the series, we calculatedthe maxima of the original observations for each day.
I The data for the above cities are rather strongly correlated,due to the relatively short distance between them (Kendall’scorrelation is 0.57) and also can be considered asasymptotically dependent
I 95% quantiles have been used as threshold level (13.8m/s,10.3m/s)
I Observations are considered as being stationary,non-stationarity can also be handled e.g. by choosingtime-dependent model parameters
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Wind data
10 15 20 25
510
1520
Exceedances (m/s)
Bremerhaven
Schl
eswi
g
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Copula sample
Bremerhaven
Schl
eswi
g
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Parameter estimation
I We found that maximum likelihood methods perform well ingeneral
I However there is strong pairwise correlation between theparameters µi , σi , i = 1, 2
I This can be extinguished in practice by keeping e.g. onelocation parameter fixed during the optimization process
I Estimation within this subclass gives stable convergenceresults
Prediction regions (Hall and Tajvidi,2000) have been computed -the smallest region containing an observation with a given (usuallyhigh) probability..
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Prediction regions for the 4 best models(having the smallest χ2-statistics)
The effect of asymmetry parameters - relevant torsion in theregions regions:
5 10 15 20 25 30
510
1520
25
NegLog
Bremerhaven(m/s)
Schle
swig(
m/s)
Regionsγ= 0.99γ= 0.95γ= 0.9
5 10 15 20 25 30
510
1520
25
Psi−NegLog
Bremerhaven(m/s)
Schle
swig(
m/s)
Regionsγ= 0.99γ= 0.95γ= 0.9
5 10 15 20 25 30
510
1520
25
Coles−Tawn
Bremerhaven(m/s)
Schle
swig(
m/s)
Regionsγ= 0.99γ= 0.95γ= 0.9
5 10 15 20 25 30
510
1520
25
Tajvidi’s
Bremerhaven(m/s)
Schle
swig(
m/s)
Regionsγ= 0.99γ= 0.95γ= 0.9
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Goodness-of-fit: Bremerhaven és Schleswigγ 0.99 0.95-0.99 0.9-0.95 0.75-0.9 0.5-0.75 0.5 χ2
Exp. 12.6 50.3 62.8 188.6 314.2 628.5 -
Log 7 30 66 216 350 588 21.5Ψ-L 9 33 60 215 349 591 16.9NL 11 31 66 215 334 600 14.0Ψ-NL 11 34 63 210 337 602 10.7Mix 12 46 71 247 331 550 30.3C-T 12 34 65 209 330 607 9.1Tajv. 11 35 62 212 340 597 11.5BL 6 29 62 220 354 586 25.6NBL 13 32 61 220 337 594 15.5
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Results
I C-T (χ2CT = 9.1) and Ψ-Neglog models (χ2Ψ-NegLog = 10.7)
perform the best
I In general the new asymmetric (Ψ) models are better thantheir symmetric baseline versions, χ2Log = 21.5 reduces to
χ2Ψ-Log = 16.9 and χ2NegLog = 14 reduces to χ
2Ψ-NegLog = 10.7
I The difference between the log-likelihood values of the Neglogand Ψ-NegLog models is around 10, which shows a significantimprovement in the likelihood ratio as well
I The Tajvidi’s generalized symmetric logistic model(χ2Tajvidi’s = 11.5) is the best among the symmetric ones.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Danube water level data - 100 years of daily flood peaks
Fitted BGPD models for flood data from Budapest and Tass(threshold was chosen at about the 95% quantiles. Black regioncovers 95%, green: 90%, blue: 75%).
Logistic
−400 −200 0 200 400
−400
−200
020
040
0
Psi−Logistic
−400 −200 0 200 400
−400
−200
020
040
0
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
R package available
I Graphs and computations have been made by the add-onpackage mgpd of R
I Wind data are also availabe
I Updated version is coming up very soon (mgpd v2.0)
Acknowledgements: The European Union and the EuropeanSocial Fund have provided financial support to the project underthe grant agreement no. TÁMOP 4.2.1./B-09/KMR-2010-0003.
Pál Rakonczai Bivariate generalized Pareto distribution
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Introduction Extreme value models Applications Wind data Flood data Software
Thank you for your attention!
Pál Rakonczai Bivariate generalized Pareto distribution
IntroductionExtreme value modelsUnivariate ExtremesMultivariate ExtremesNew models
ApplicationsWind dataFlood data Software