-
Multivariate generalized Pareto distributionin practice: models
and estimation
December 6, 2010
Pál RakonczaiProbability Theory and Statistics Department
Eotvos University, Budapest, Hungary&
Department of Mathematical StatisticsLund Institute of
Technology, Sweden
e-mail:[email protected]
Contents
1 Introduction 3
2 Models 42.1 Univariate extreme value models . . . . . . . . .
. . . . . . . . . . .42.2 Multivariate extreme value models . . . .
. . . . . . . . . . . . . .. 52.3 Multivariate threshold exceedance
models . . . . . . . . . . .. . . . 62.4 Parametric families of
dependence structures . . . . . . . .. . . . . 82.5 Construction of
new non-exchangeable models . . . . . . . . .. . . 12
3 Statistical inference on wind speed data 153.1 Computational
aspects of estimation and prediction . . .. . . . . . . 153.2
Results for bivariate observations . . . . . . . . . . . . . . . .
.. . . 153.3 Results for trivariate observations . . . . . . . . .
. . . . . . .. . . 18
4 Discussion 18
A Appendix: "classical BGPD" models fitted by ’fbvpot’ 26
1
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Abstract
Extreme values are of substantial interest in fields as diverse
as finance, environmentalscience and engineering, because they are
associated with rare but hazardous events(such as flooding,
mechanical failure or severe financial loss). There is often
interestin understanding how the extremes of two different
processes are related to each other,and, in particular, in
estimating the probability that the processes will both
simultane-ously become extreme. Analyses of the extremes of two
variables are typically basedon fitting a specific parametric
subfamily of the multivariate extreme value distributionto
component-wise maxima. An alternative approach, which potentially
uses the avail-able data much more efficiently, involves fitting
the multivariate generalised Paretodistribution (MGPD) to data that
exceed a suitably high threshold. There are several(non-equivalent)
definition of MGPD in use, one is for exceedances over a
thresholdin at least one of the components (Rootzén and Tajvidi,
2006)and another(s) for ex-ceedances which are over the threshold
in all components. Here we investigate theapplicability of the
first definition assuming different underlying dependence modelsand
compare the performance of the two substantially different type of
MGPD models.Although the dependence models are intensively used in
different extreme value mod-els, to the best of our knowledge none
of these models have been discussed and appliedfor data in the new
MGPD framework apart from the bivariate (symmetric) logisticmodel.
As there are just a few of these families, which allow asymmetry
for the differ-ent components and produce absolutely continuous
models for the new MGPD case,we introduce a general transformation
for creating such models from symmetric ones.We apply the proposed
models to wind speed data for modeling exceedances occur-ring at
locations in northern Germany, and outline methods for calculating
predictionregions as well as evaluating goodness-of-fit.
Key words: Multivariate threshold exceedances,parametric
dependence models,windspeed data.
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1 Introduction
Multivariate extreme value theory involves describing
thestatistical properties of theextreme values of two or more
processes, or, in the spatial context, of the same processat two or
more sites. It is concerned, in particular, with quantifying the
probability thatextreme values will occur simultaneously in more
than one process, or at more thanone location. Such probabilities
are of substantial interest in a wide range of fields,including the
environmental sciences, finance and internettraffic monitoring.
Evenwhen interest ultimately lies in describing the extreme values
of a single process ata single location, there will still often be
value in using data from other processes orlocations to improve the
efficiency of estimates relating tothe location and processof
interest (Ribatet al., 2007). Efficiency can sometimes also be
improved by utilizingdata that have been collected on the same
process at different temporal resolutions (e.g.daily and hourly;
Nadarajah et al., 1998), and this also requires the use of a
multivariatemodel.
A sophisticated mathematical theory to describe the
characteristics of multivariateextreme events has been developed
(e.g. Pickands, 1981 and Rootzén & Tajvidi, 2006)but there are
substantial computational and statistical challenges in applying
the re-sulting asymptotic models to data. Univariate extreme value
methods, in contrast, areroutinely used for data analysis within
finance, the environmental sciences, and a rangeof other scientific
applications (e.g. Coles, 2001). Multivariate extremes are often
mod-eled by applying block maxima methods (BMM) to the
component-wise maxima of theseries being studied (e.g. Tawn, 1988).
In environmental applications this commonlymeans focusing on the
annual maximum value of each process. If the length of theblock is
sufficiently long, and under certain other conditions, then
theoretical resultssuggest that the distribution of these
componentwise maxima can be approximated bya multivariate extreme
value distribution (MEVD). The practical application of this
re-sult typically involves selecting a particular parametricmodel
from within the MEVDclass, and then drawing statistical inferences
about the parameters of this model usingmaximum likelihood
estimation.
An important drawback of the BMM approach is the fact that it
ignores informationon whether or not the extremes of the different
processes do actually occur simultane-ously. The annual maxima of
the processes may all have arisenon the same day, forexample, but
it is also possible that they all occurred in completely different
months -the basic implementation of the MEVD model does not
distinguish between these sit-uations (although an extension of the
approach does allow some information on timingto be incorporated
into the modeling: Stephenson and Tawn, 2005). This problem canbe
avoided by modeling the sizes of all observations that exceed a
given high threshold,rather than modeling the highest value within
a particular block of time. This methodusually uses more data than
the block maxima approach, and soalso leads to efficiencygains
(i.e. more precise estimates for the quantities of interest).
Methods for analyzing multivariate threshold exceedanceswere
originally devel-oped by Tajvidi (1996), and have been further
developed for more general cases inRootzen and Tajvidi (2006).
These papers have demonstratedthat, under rather mildconditions and
given an appropriately high threshold vector, the multivariate
exceedancesof a random vector over the threshold can be
approximated by amultivariate general-
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ized Pareto distribution (MGPD). Note that the definition ofthe
MGPD in these paperssubstantially differs from the definitions
which can be found in other textbooks (ex-cept the recent book of
Beirlant et al. (2004), where this definition is mentioned butnot
discussed). The main advantage of this approach is that it includes
all observa-tions that are extreme in at least one component.
Although mathematical theory for theMGPD model has been developed,
the statistical properties of this model are not yetfully
understood - the only paper discussing the performance of the model
in practiceis, to the best of our knowledge, that of Rakonczai and
Tajvidi (2009). Similar BGPDmodels have also been used in a recent
paper by Brodin and Rootzén (2009) to studywind storm losses in
Sweden. Throughout this paper only stationary distributions
areinvestigated. A non-stationary extension of the BGPD modelwhich
allows for the pos-sibility that the characteristics of extreme
events are changing over time (or dependupon the value of some
other covariate) is discussed by Rakonczai et al. (2010).
Standard models for univariate and multivariate extremes are
presented in Section2, as well as their role and properties in MGPD
models. A new general method forconstructing asymmetric versions of
known symmetric models is also presented. InSection 3 we discuss
the practical issues involved in estimating the parameters
viamaximum likelihood, and outline a procedure for construction
prediction regions andmodel validation techniques. Additionally
some useful conclusions are summarised inSection 4.
2 Models
In this section – after a short introduction to univariate
extreme value modells – wepresent the key models that are currently
used for modeling maxima of multivariate ob-servations and
simultaneously discuss their applicability within models for
multivariatethreshold exceedances. The presented models1 are
available in the ’mgpd’ package ofR, which has been produced for
providing a tool for the statisticians to apply the pro-posed
methods and models and for illustrating the practicalapplications
on wind speeddata.
2.1 Univariate extreme value models
Univariate extreme value theory provides the limit resultsfor
the distribution of ex-tremes of a single process: either the
maximum of observations or the distribution ofexceedances of
observations over a high threshold. Maxima (daily, weekly, yearly
etc.)can be modeled using the generalized extreme value
distribution (EVD), which has adistribution function (df.) of the
form
G(x) = exp
{
−(
1 + ξx− µ
σ
)−1ξ
}
, (1)
1These models are different from those which are available inthe
’evd’ package ofR, the main differ-ences are discussed later
on.
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where1 + ξ x−µσ > 0, µ ∈ R is called the location parameter,σ
> 0 the scale pa-rameter andξ ∈ R the shape parameter.
Similarly, threshold exceedances (e.g. overa high quantile of the
observations) can be approximated by the generalized
Paretodistribution (GPD), which has a cdf. of the form
H(x) = 1 −(
1 + ξx
σ
)− 1ξ
, (2)
where1 + ξ xσ > 0 andσ > 0. Both of the limit
distributions are strongly linked in thesense that, as the
threshold tends to the right endpoint of the underlying
distribution,the conditional distribution of the exceedances
convergesto the GPD if and only if thedistribution of the maximum
converges to the EVD.
2.2 Multivariate extreme value models
Analogously to the univariate case it can be shown if the
componentwise maxima hasa limit distribution, it is necessarily a
multivariate extreme value distribution (MEVD).Additionally, all of
its univariate margins are univariateEVD-s. There are
severaldifferent (but equivalent) way of characterising the
underlying dependence structure.One of these is shown by Resnick
(1987) assuming unit Fréchetmargins
GFréchet(t1, . . . , td) = exp(
−V (t1, . . . , td))
, (3)
with
V (t1, . . . , td) = ν(([0, t1] × · · · × [0, td])c) =
∫
Sd
d∨
i=1
(witi
)
S(dw), (4)
whereS is a finite measure on thed-dimensional simplexSd = {w ∈
Rd : |w| = 1},which satisfies
∫
Sd
wiS(dw) = 1 for i = 1, ..., d,
whereV andS are called exponent measure and spectral measure
respectively. Inparticular, the total mass ofS is alwaysS(Sd) = d.
If G is absolutely continuous,then we can reconstruct the densities
ofS from the derivatives ofV . In this contextit is better using
"densities" instead of "density" asS can have a density not only
onthe interior ofSd but also on each of the lower-dimensional
subspaces ofSd. E.g. ifd = 2, the unit simplex is partitioned into
two vertices and the interior of the intervalor if d = 3 into three
vertices, three edges and the interior of the triangle. Hence,even
ifG is absolutely continuous, the spectral measureS can put
positive mass to thevertices; for instance, when the margins ofG
are independent thenS({ei}) = 1 for alli = 1, . . . , d. More
details about the characterization and its propertiescan be found
inthe recent textbooks as e.g. Chapter 3 in Kotz and Nadarajah
(2000) or Chapter 8 inBeirlant et al. (2004).
Further relevant formulas for maximum likelihoood inference are
the following.Coles and Tawn (1991) found how to compute the
spectral densities of all subspaces
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from the partial derivatives ofV . E.g for the interior the
spectral densitys can beexpressed as
∂V
∂t1 · · · ∂td
(
t1, . . . , td
)
= −(
d∑
1=1
ti
)−(d+1)
s( t1∑
ti, . . . ,
td∑
ti
)
.
Another characterisation of MEVD, due to Pickands (1981)
ispossible by the so-calleddependence function. In the bivariate
setting the dependence functionA(t) must sat-isfy the following
three properties, which we denote by (P).
i. A(t) is convex
ii. max{(1 − t), t} ≤ A(t) ≤ t
iii. A(0) = A(1) = 1.
The lower bound in (ii.) corresponds to the complete dependence,
whereas the up-per bound corresponds to independence. The exponent
measure and its mixed partialderivative can be written as
V (t1, t2) =( 1
t1+
1
t2
)
A( t1t1 + t2
)
∂V
∂t1∂t2(t1, t2) =
( 1
t1+
1
t2
)
×(
A′′(ζ)ζ′t1ζ′t2 +A
′(ζ)ζ′′t1t2
)
−A′(ζ) ×(ζ′t2t21
+ζ′t1t22
)
,
where
ζ =t1
t1 + t2, ζ′t1 =
t2(t1 + t2)2
, ζ′t2 =−t1
(t1 + t1)2, ζ′′t1t2 =
t1 − t2(t1 + t2)3
.
2.3 Multivariate threshold exceedance models
The multivariate extension of the GPD models has also been
intensively investigatedin the last decades and as a result,
different definitions runup different careers. Basedon its longer
history, the multivariate GPD models using exactly the same
construc-tion as the MEVD model has, became more popular, we refer
it asthe "classicalMGPD" , which is also implemented in theevd
package ofR. It concentrates on thoseexceedances which are greater
than the threshold in every single components. Its uni-variate
margins are GPD distributions, which are linked by adependence
model. Usingthe following transformation for the GPD margins
t̃i = t̃i(xi) =−1
logHξi,σi(xi)=
−1
log{
1 −(
1 + ξixσi
)− 1ξi
}
,
where1 + ξ xσ > 0 andσ > 0 the MGPD is the form of
H̃(x1, . . . , xd) = H̃Fréchet(t̃1, . . . , t̃d) = exp(
−V (t̃1, . . . , t̃d))
, (5)
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wheret̃i ≥ 0 andV is defined analogously to (4).The other
definition was proposed by Rootzén and Tajvidi (2006). Let Y =
(Y1, ..., Yd) denote a random vector,u = (u1, ..., ud) be a
suitably high thresholdvector andX = Y − u = (Y1 − u1, ..., Yd −
ud) be the vector of exceedances. De-fine the multivariate
generalized Pareto distribution (MGPD) for theX exceedances bysomeG
multivariate extreme value distribution (MEVD) with non-degenerate
marginsas in the paper of Rootzén and Tajvidi (2006)
H(x1, . . . , xd) =1
logG(
0, . . . , 0) log
G(
x1, . . . , xd)
G(
x1 ∧ 0, . . . , xd ∧ 0) , (6)
where0 < G(0, . . . , 0) < 1 and∧ denotes the minimum. By
to this definition, themodel – which we shall simply call MGPD –
provides a model for observations thatare extreme in at least one
component. On the other hand it is obvious that the distribu-tion
is uniquely determined by the underlying MEVD model. Inthe related
literaturethere are several equivalent approaches of
characterisingan MEVD assuming differentmarginal transformations
and different dependence structure representation. Through-out this
paper we assume unit Fréchet margins having df. asΦ(x) = exp(−x−1)
andto achieve it, we apply the following marginal
transformations
ti = ti(xi) =−1
logGξi,µi,σi(xi)= (1 + ξi(xi − µi)/σi)
1/γ1 , (7)
with 1 + ξi(xi − µi)/σi > 0 andσi > 0 as in (1) fori = 1,
..., d. In the follow-ing, the definition in (6) is called MGPD and
(5) is called the "classical MGPD". It isimportant to notice that
even though the dependence structure for two MGPD modelsin is the
same, the models are substantially different. Again, the most
conspicuousdifference between the MGPD and "classical MGPD" is that
(6)gives (parametric)probabilistic inference about the exceedances
which are above the threshold at least inone component, in contrast
to the another one modeling thoseones, which are higherthan the
threshold in every components. For illustrations on real
observations comparethe upper and lower panels of Fig.7 where the
two substantially different approachesof constructing BGPD models
are displayed assuming symmetric logistic family asdependence
structure for both cases. Further difference between (6) and (5) is
in pa-rameter interpretation. Namely, that theξi, µi andσi
parameters of (6) can not be anymore considered as marginal
parameters for the model, as the1/ logG(0, . . . , 0) termremains
in the distribution even ifd− 1 components ofx converges to
infinity. More-over ifH1(x) = H(x,∞) thenH1 is not a one
dimensional GPD. Although it is shownin Rootzén and Tajvidi (2006)
that ifX1 has a distributionH1 then conditional distri-bution
ofX1|X1 > 0 is GPD. Similar properties hold for all marginal
distributionsregardless of the dimension of the MGPD. Further, the
MGPD density can be obtainedas
h(x) =∂H
∂x1 . . . ∂xd(x) =
∂
∂x1 · · · ∂xd
(
1 −logG(x)
logG(0)
)
(8)
=
∏di=1 t
′i(xi)
V(
t1(0), . . . , td(0)) ×
∂V
∂t1 · · · ∂td
(
t1(x1), . . . , td(xd))
.
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Although the logarithm in (6) cancels the exponent in (3), the
MGPD densityh(x)cannot be computed immediately from the spectral
density, as the constant termG(0)still containsV as well.
Additionally, for the MGPD models, it is reasonableto requirethat
there will be no positive probability mass put on the boundary of
the distribution,otherwise the model will not stay absolutely
continuous, which is rarely realistic andcauses further
complications for the maximum likelihood estimation.
Lemma 1. Let H be a MGPD represented by an absolutely continuous
MEVD Gwith spectral measure S. The distribution H is absolutely
continuous if and only ifS(Sd) = S(int(Sd)) = d, i.e all mass is in
the interior of the simplex.
Corollary. In such cases the MGPD decays to zero when reaching
the boundaries
limxi→x
+i
H(x1, . . . , xd) = 0 for anyxj > 0, j 6= i,
wherex+i = µxi − σxi/ξxi is the finite lower endpoint of thei-th
univariate GEVmargins of the underlying MEVD model.Proof. It is
easy to see that the subspaces of theSd unit simplex represent the
bound-aries of the space containing the unit Fréchet
coordinates.E.g. if d = 2 then the point(w,w − 1) ∈ int(S2)
represents they = 1−ww x line in the unit Fréchet scale andlimiting
cases are{1, 0} and{0, 1} representing thex andy axes,
respectively. Thisterminology remains the same for higher
dimensions as well.Therefore when calcu-lating the distribution
function for any point of the interior on the unit Fréchet
scale,the masses originated from the faces will always be
cumulated, which is equivalent tocausing a jump in the function and
so its continuity gets broken.
In the next subsections we summarize, which specific parametric
cases includesuch a model. As we shall see, that there are just a
few such models, in section 2.5. wepropose a general approach to
construct asymmetric models from symmetric ones.
2.4 Parametric families of dependence structures
Since MGPD models are defined by the underlying MEVD model, and
practicallyMEVD models are defined by the dependence structures
(apart from margins), the mostpopular parametric families for MEVD
models have been considered as foot-stones forthe MGPD models. The
most important characteristics of these models are summarizedin
Table 1. The list is not exclusive but covers a rather wide range
of families. Althoughthese models are intensively used in different
extreme value models, to the best of ourknowledge none of these
models have been discussed and applied for data in the
MGPDframework apart from the bivariate (symmetric) logistic model.
As characterizationwe give both the exponent measure function and
the spectral density for the models.Additionally we discuss which
parameter setting allows non-exchangeability and putsall mass in
the interior of [0,1] providing absolutely continuous models within
theBGPD framework by Lemma 1. More details about these models can
be found in therespective papers indicated, giving the first
appearance ofthe related models. Graphicalillustration of the
bivariate distribution function for some of the models can be
foundin Fig 5.
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Model Non-Exchangeable All mass inint(Sd) Complications
Sym. Logistic − X −Asym. Logistic X − −
Sym. Neg. Logistic − X −Asym. Neg. Logistic X − −
Bilogistic X X not explicitNeg. Bilogistic X X not explicit
Gen.Sym.Logistic − X −Dirichlet X X only s(w) is givenMixed −
unlessψ = 1 −
Asym. Mixed X − −
Table 1: Summary of bivariate dependence models. Asymmetric
logistic, negativelogistic and mixed models are considered
excluding their symmetric case, because theirdifferent continuity
properties. Absolutely continuous BGPD models can be obtainedif all
mass is in the interior of the[0, 1], these cases are highlighted
by bold fonts. Thenon-exchangeable cases are the bilogistic,
negative bilogistic and Dirichlet models.
Most of the bivariate cases are straightforward to extend
tohigher dimensions, butthese extensions rarely result valid MGPD
models with the full parameterisation. Asan example, the
multivariate logistic model provides a richset of model
parameters,but due to Lemma 1 the absolute continuity holds only in
a verylimited parametersetting (it coincides with the symmetric and
non-exchangeable case). The multivari-ate negative logistic or
nested logistic models behave similarly. An alternative,
themultivariate extension of Coles-Tawn model (Dirichlet model)
provides a flexible non-exchangeable model if such a model is
needed, but it requiressophisticated numericalintegration tools for
its fitting, as only the spectral density is given explicitly. For
thedetailed description of the above models see Section 2.4.1 and
2.4.2.
2.4.1 Bivariate models
• Asymmetric logistic model (Tawn, 1988b).
VLog(x, y) = (1 − ψ1)/x+ (1 − ψ2)/y + ((ψ1/x)α + (ψ2/y)
α)1/α
andsLog(w) = (α − 1)ψα1 ψα2 (w(1 − w))
α−2((ψ2w)α + (ψ1(1 − w))
α)1/α−1,whereα ≥ 1 and0 ≤ ψ1;ψ2 ≤ 1. It allows exchangeability
unlessψ1 = ψ2.In the special case ifψ1 = ψ2 = 1, it is
calledsymmetric logistic model.This is the only case when the model
has all its mass in the interior, otherwiseS({0}) = 1 − ψ2
andS({1}) = 1 − ψ1. (See the upper left panel of Fig 5.)
• Asymmetric negative logistic model (Joe, 1990). The negative
logistic model issimilar in structure to the logistic
VNeglog(x, y) = 1/x+ 1/y − ((ψ1/x)α + (ψ2/y)
α)−1/α
andsNeglog.(w) = −sLog.(w), whereα > 0 and0 < ψ1;ψ2 ≤ 1.
AnalogouslyS({0}) = 1 − ψ2, S({1}) = 1 − ψ1 and soψ1 = ψ2 = 1 gives
the only
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symmetric version with all its mass in the interior. We referto
it assymmetricnegative logistic model. (See the upper right panel
of Fig 5.)
• The bilogistic model (Smith, 1990). The bilogistic model isan
the asymmet-ric extension of the logistic model, where the spectral
measureS does not putpositive mass on the boundary points. Its
exponent measure function is
VBilog(x, y) =
∫
[0,1]
max
{
(ψ1 − 1)z−1/ψ1
ψ1x,(ψ2 − 1)(1 − z)
−1/ψ2
ψ2y
}
dz,
whereψ1;ψ2 > 1. A disadvantage is though, that there is only
an implicitformula for its spectral density on (0, 1) in terms of
the rootof an equation as
sBilog(w) =(1 − ψ1)(1 − q)q
1−ψ1
(1 − w)w2((1 − q)ψ1 + qψ2),
whereq = q(x, y;ψ1, ψ2) is the root of the equation
(1 − ψ1)x(1 − q)ψ2 − (1 − ψ2)yq
ψ1 = 0.
(See the lower left panel of Fig 5.) For valid input parameters
the equation hasa unique root in [0,1], what makes the numerical
root finding quite handy in thiscase.
• Negative bilogistic model (Coles and Tawn, 1994). The negative
bilogistic modelhas the same exponent measure function as the
bilogistic model except that in thiscaseψ1;ψ2 < 0. The spectral
density issNegbilog(w) = −sBilog(w) and similarlyS({0}) = S({1}) =
0.
• Tajvidi’s generalized symmetric logistic model
(Tajvidi,1996).
VTajvidi(x, y) =
(
( 1
x
)2α
+ 2(1 + ψ)( 1
xy
)α
+(1
y
)2α)
12α
,
where1 ≤ α and1 < ψ ≤ 2α− 2.
• Dirichlet model (Coles and Tawn, 1991). The Dirichlet modelis
non-exchangeablelike the two bilogistic models above, and has all
probability mass of the spectraldensity confined to the
interior.
VCT(x, y) = 1/x(1 − β(q;ψ1 + 1, ψ2)) + 1/yβ(q;ψ1, ψ2 + 1)
and
sCT(w) =ψψ11 ψ
ψ22 Γ(ψ1 + ψ2 + 1)
Γ(ψ1)Γ(ψ2)
wψ1−1(1 − w)ψ2−1
(ψ1w + ψ2(1 − w))1+ψ1+ψ2
whereq = ψ1y/(ψ1y + ψ2x), ψ1, ψ2 > 0, β is a normalized
incomplete betafunction. (See the upper right panel of Fig 5.)
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• Polynomial model (Klüppelberg and May, 1999).
VPoli(x, y) = 1/x+ 1/y −m∑
k=2
ψk
m−k∑
l=0
(
m− k
l
)
xl+k−1ym−k−l−1
(x + y)m−1
andsPoli(w) = m(m− 1)ψmwm−2 + (m− 1)(m− 2)ψm−1wm−3 + ...+
2ψ2,whereψ2 > 0,
∑mk=2 ψk ≥ 0, 0 ≤
∑mk=2(k − 1)ψk ≤ 1 and
∑mk=2 k(k −
1)ψk ≥ 0. The spectral measureS on the sides on the boundary
points isS({0}) = 1 −
∑mk=2 ψk andS({1}) = 1 −
∑mk=2(k − 1)ψk.
A special case is theasymmetric mixed model (Tawn, 1988b).
VAsyMix(x, y) = 1/x+ 1/y − xy(x+ y)−2((ψ1 + ψ2)/x+ (ψ1 +
2ψ2)/y)
andsAsyMix(w) = ψ1w3 +ψ2w2−(ψ1 +ψ2)w+1,whereψ1 ≥ 0,ψ1 +ψ2 ≤ 1,ψ1
+ 2ψ2 ≤ 1 andψ1 + 3ψ2 ≥ 0. Whenψ1 = 0, the asymmetric mixed
modelreduces to thesymmetric mixed model, in such a caseS({0}) = 1
− ψ2 andS({1}) = 1 − ψ2. Consequentlyψ2 = 1 is the only symmetric
case when allmass is in the interior of [0,1].
2.4.2 Multivariate models
• Logistic models and variations (Tawn, 1990). LetB be the set
of all nonemptysubsets of{1, . . . , d} and letB1 = {b ∈ B : |b| =
1}. Thed-dimensionallogistic modelcan be represented by
VLogistic(t1, . . . , td) =∑
b∈B
(
∑
j∈b
(ψj,bti
)αb
)1/αb
,
where the dependence parametersαb’s are in (0, 1] for all b ∈
B\B1 and theasymmetry parametersθj,b’s are in[0, 1] for all b ∈ B
andj ∈ b. The constraints
∑
b∈B(i)
θj,b = 1, (9)
for j = 1, . . . , d ensure that the univariate margins are of
the correct form, wherewe defineB(j) =
{
b ∈ B : j ∈ b}
. As we pointed out previously in the bivari-ate case, for
getting an absolutely continuous model, both asymmetry parame-ters
must be equal to 1. Consequently, as every bivariate margins must
satisfythe same criteria, all asymmetry parameters must necessarily
be equal to 1, andso actually almost all terms assuming different
dependenceparameters for anysubsets of components is cancelled from
the formula apart from the one whichcaptures alld components
together. Following the above logic the only realisticlogistic
model within an MGPD is the form of
VLogistic(t1, . . . , td) =
(
d∑
i=1
(ψiti
)α)1/α
,
11
-
having only one parameter to capture the dependence among the
dimensions. Ofcourse we should also emphasize, that this simplicity
appears only in the depen-dence structure and the correct choice
for the marginal parameters gives moreflexibility to the model.
Moreover we should notice that marginal parametersof the underlying
MEVD model are not marginal parameters in the usual sensefor the
MGPD model, because of having thelogG(0, ..., 0) constant term in
itsdefinition (6).
• Dirichlet model (Coles and Tawn, 1991) The spectral densityof
the model is
sDirichlet(w) =Γ(
(∑d
i=1 ψi) + 1)
(
∑di=1 ψiwi
)p+1
d∏
i=1
ψiΓ(ψi)
d∏
i=1
(
ψiwi∑di=1 ψiwi
)ψi−1
,
whereψi > 0 for i = 1, . . . , d. Although the corresponding
exponent measureV is complicated, it is obtainable by using
numerical integration.
2.5 Construction of new non-exchangeable models
From the second and third column of Table 1 we can see that
there is a lack of easilycomputable non-exchangeable models,
especially if all probability mass is required tobe put on the
interior of theS2, ensuring the absolute continuity of the model.
Thebilogistic and negative bilogistic models are available, but
without having an explicitformula for thier exponent measures, or
the Dirichlet model, in which case there isexplicit formula only
for the spectral density. In order to solve this problem, herewe
propose a methodology which allows to construct new dependence
models withextra asymmetry parameter(s) from any valid models. As
the result of this method wemay obtain more flexible
non-exhangeable models defining a new class for
absolutelycontinuous MGPD-s. Because of its mathematical
simplicitywe illustrate the methodin the bivariate setting using
dependence function, but thesame methodology can beextended to the
higher dimensional cases as well. The algorithm is fairly
simple:
i. take a differentiablebaseline dependence modelfrom Table 1
(except asym. lo-gistic or asym. negative logistic) and switch
characterisation form from exponentmeasure to dependence function
byA(t) = V ( 11−t ,
1t );
ii. take a strictly monotonictransformation Ψ(x) : [0, 1] → [0,
1], such thatΨ(0) = 0, Ψ(1) = 1;
iii. construct anew dependence modelfrom the baseline modelAΨ(t)
= A(Ψ(t));
iv. check the constraints:A′Ψ(0) = −1,A′Ψ(1) = 1 andA
′′Ψ ≥ 0 (convexity).
For the construction of a feasible transformation we can assume
e.g. that it has a formof
Ψ(t) = t+ f(t)
hence(A(Ψ(t)))′ = A′(Ψ(t))Ψ′(t) = A′(Ψ(t)) × [1 + f ′(t)].
(10)
12
-
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
tψ2(1 − t)ψ2
t
ψ2 = 2ψ2 = 2.25ψ2 = 2.5ψ2 = 2.75
0.0 0.2 0.4 0.6 0.8 1.0−
0.2
−0.
10.
00.
10.
2
δt(tψ2(1 − t)ψ2)
t
ψ2 = 2ψ2 = 2.25ψ2 = 2.5ψ2 = 2.75
0.0 0.2 0.4 0.6 0.8 1.0
−1.
0−
0.5
0.0
0.5
1.0
1.5
2.0
δ2t(tψ2(1 − t)ψ2)
t
ψ2 = 2ψ2 = 2.25ψ2 = 2.5ψ2 = 2.75
Figure 1: Examples of functionsf(t) in (11) and its derivatives,
assuming fixedψ1 = 1and differentψ2 parameters
Obviously by choosing the second term ofΨ(t) such thatf ′(0) = f
′(1) = 0, theA′Ψ(0) = −1 andA
′Ψ(1) = 1 constrains are fulfilled for the new model if and only
if
A′(0) = −1 andA′(1) = 1 is true for the baseline model. The only
problem which canoccur, is that the new dependence functionAΨ is
not necessarily convex. In generalwe found that the following
functional form leads to valid models
fψ1,ψ2(t) = ψ1(t(1 − t))ψ2 , for t ∈ [0, 1], (11)
whereψ1 ∈ R and andψ2 ≥ 2 are asymmetry parameters. (See Fig 1
for illustration.)If ψ1 = 0 we get back the baseline model. The
convexity inequation below providesthe valid range for the
asymmetry parameterψ1, assuming fixed baselineA(t) andψ2we
obtain
(A(Ψ(t)))′′ = A′′(t+ f(t))(1 + 2f ′(t)) +A′(t+ f(t))f ′′(t) ≥
0.
Graphical illustration of some valid model parameterisation can
be found in Fig 2.Later when using these transformation we put the
"Ψ−" prefix before the name of thebaseline dependence model, as
e.g.Ψ-logistic model. The difference caused by thenew asymmetry
parameters can be seen in Fig 3 for 3Ψ-logistic BGPD models.
13
-
0.0 0.2 0.4 0.6 0.8 1.0
−1.
00.
00.
51.
0
δtAαLog=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
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
−1.
00.
00.
51.
0
δtAαNegLog=1.2(Ψ(t))
t
Psi−NegLogψ1 = − 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
Figure 2: First and second derivatives ofΨ-logistic
andΨ-negative logistic depen-dence functions, where the
transformationΨ(t) = t+ fψ1,ψ2(t) has fixedψ2 = 2 anddifferentψ1
parameters.
−4 −2 0 2 4 6
−4−2
02
46
Transformed logistic BGPD density
0.00
0.05
0.10
0.15
0.20
0.25
0.30
ψ1 = 0.3ψ2 = 2
−4 −2 0 2 4 6
−4−2
02
46
Transformed logistic BGPD density
0.00
0.05
0.10
0.15
0.20
0.25
ψ1 = 0ψ2 = 2
−4 −2 0 2 4 6
−4−2
02
46
Transformed logistic BGPD density
0.00
0.05
0.10
0.15
0.20
0.25
0.30
ψ1 = − 0.3ψ2 = 2
Figure 3: BGPD density plots usingΨ-logistic dependence models
with fixedψ2 = 2andψ1 = −0.3/0/0.3 parameters
14
-
3 Statistical inference on wind speed data
We illustrate the practical application of the proposed methods
using data on windspeeds that have been collected at four locations
in northern Germany over the pastfive decades (1957-2007):
Hannover, Bremerhaven, Schleswig and Fehmarn. The en-tire period
covers 18061 days, and 17926 complete observations are actually
availableduring this period (the remaining 135 values have missing
coordinates). The observa-tions are considered as being stationary,
but non-stationarity can also be handled e.g.by choosing
time-dependent model parameters as in Rakonczai et al. (2010).
Afterchoosing a relatively high quantile (95%) as threshold level
the main aim is model-ing threshold exceedances occurring
simultaneously at anypairs or triple of measuringstations. The
bivariate data and the threshold levels are displayed in Fig 6. The
windspeed data for the above cities are rather strongly correlated,
due to the relatively shortdistance between them (with Kendall’s
correlations in the range0.45-0.55 for pair-wise comparisons
between the four stations), and previous studies have suggested
thatBGPD models can have reasonable performance for this level of
association (Rakon-czai and Tajvidi, 2010).
3.1 Computational aspects of estimation and prediction
The parameters can be estimated by numerical maximum likelihood
method and ina Bayesian way by using MCMC simulations with
conditional resampling algorithm.The effective parameterisation of
the MCMC algorithm seemed to be more compli-cated and required
longer time for computation, so we found the maximum
likelihoodmethod preferable. Although the maximum likelihood
estimator is complicated to de-rive analytically it is fairly easy
to find by numerical optimisation using the Nelder-Mead algorithm
(as implemented inR using theoptim function). Some complicationsmay
arise as usually theµi, σi parameters show high correlation within
the estimatedMGPD models. Our general finding is that the effect of
settingone of these param-eters (e.g.µ1) to be zero has a
negligible effect upon the parameter estimates forξiandα, and upon
the maximum value of the log-likelihood, suggesting that the
modelswith and without fixedµ1 are effectively equivalent. Fixingµ1
to be zero removes thestrong correlations between the remaining
parameters. Another problem can be that inthe parameterisation of
the MGPD there are no automatic constrains providing posi-tivity in
the original scale of the observations. Hence, theoretically, it
can happen thatthe fitted model gives positive probability on
negative regions even for wind speed databeing non-negative by its
nature. This discrepancy can be avoided by assuming themargins
having finite left endpoints not less than the theoretical minimum
for the givenapplications. Similar assumptions have been used in
Drydenand Zempléni (2006).
3.2 Results for bivariate observations
The large number of model parameters (7-9), which are beyondthe
interpretable com-plexity (location and scale parameters of the
underlying MEVD’s of MGPD are not anymore location and scale
parameters for the MGPD margins) requires some
transparentillustration. Hence we used bivariate prediction regions
as an alternative for visualising
15
-
the estimates. A prediction region (Hall and Tajvidi, 2004)for a
given probabilityγis a region bounded by a horizontal level curve
of the bivariate density over which theintegral of the density
equalsγ. The dependence parameters for 4 pairs of locations canbe
found in Table 2 and the prediction regions of some models are
displayed in Fig 4.In the new (Ψ) models the asymmetry parameters
seem to be non-zero, providing someevidence of asymmetry in the
data, whereas the dependence parameters are very closeto those
estimated for the baseline models. The effect of asymmetry
parameters canbe clearly seen in the top right panel of Fig 4,
where theΨ-NegLog model has a rele-vant torsion in its regions to
the left. Similar torsion occurs for the regions calculatedfrom the
Coles-Tawn dependence model. It is still difficult to make
difference basedon visually very similar regions, hence a formal
test would be very useful. The test wepropose in this case can be
easily performed as a simple byproduct of the predictionregion
method. It is known for any region that how many observations are
expectedto be in/out of it, as well as between two neighboring
regions, so by comparing thesetheoretical frequencies with the
realisation we can perform aχ2 test. The results forBremerhaven and
Schlesswig can be found in Table 3. The critical value for the
testis 11.07 (p = 0.95, df = 5), hence the Coles-Tawn(TCT = 9.1)
andΨ-Neglog mod-els (TΨ-NegLog = 10.7) may be accepted. In general
the new asymmetric (Ψ) modelsare significantly better than their
symmetric baseline versions,TLog = 21.5 reduces toTΨ-Log = 16.9
andTNegLog = 14 reduces toTΨ-NegLog = 10.7. The Tajvidi’s
general-ized symmetric logistic model(TTajvidi’s = 11.5) is the
best among the symmetric ones.
5 10 15 20 25 30
510
1520
25
NegLog
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
Regionsγ = 0.99γ = 0.95γ = 0.9
5 10 15 20 25 30
510
1520
25
Psi−NegLog
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
Regionsγ = 0.99γ = 0.95γ = 0.9
5 10 15 20 25 30
510
1520
25
Coles−Tawn
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
Regionsγ = 0.99γ = 0.95γ = 0.9
5 10 15 20 25 30
510
1520
25
Tajvidi’s
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
Regionsγ = 0.99γ = 0.95γ = 0.9
Figure 4: Prediction regions at Bremerhaven and Schleswig for 4
different models(negative logistic,Ψ-negative logistic, Coles-Tawn
and Tajvidi’s generalizedsymmet-ric)
16
-
Hannover and Schleswig Bremerhaven and FehmarnModels Dependence
Dependence
Log α = 1.66 α = 1.985Ψ-Log α = 1.662, ψ1 = 0.207, ψ2 = 2 α =
2.049, ψ1 = −0.423,ψ2 = 2NegLog α = 0.94 α = 1.257Ψ-NegLog α =
0.963, ψ1 = 0.354, ψ2 = 2 α = 1.287, ψ1 = −0.249,ψ2 = 2Mix ψ1 = 0,
ψ2 = 1 ψ1 = 0, ψ2 = 1C-T ψ1 = 1.42, ψ2 = 0.69 ψ1 = 1.12, ψ2 =
2.39Tajvidi’s ψ1 = 1.84, ψ2 = 0.37 ψ1 = 2.24, ψ2 = 0.44Bilog ψ1 =
0.63, ψ2 = 0.55 ψ1 = 0.43, ψ2 = 0.57Negbilog ψ1 = 0.8, ψ2 = 1.37 ψ1
= 1.02, ψ2 = 0.6
Bremerhaven and Schleswig Fehmarn and SchleswigModels Dependence
Dependence
Log α = 2.06 α = 1.95Ψ-Log α = 2.072, ψ1 = 0.334, ψ2 = 2 α =
1.992, ψ1 = 0.369, ψ2 = 2NegLog α = 1.34 α = 1.22Ψ-NegLog α =
1.303, ψ1 = 0.231, ψ2 = 2 α = 1.226, ψ1 = 0.315,ψ2 = 2Mix ψ1 = 0,
ψ2 = 1 ψ1 = 0, ψ2 = 1C-T ψ1 = 2.23, ψ2 = 1.25 ψ1 = 2.2, ψ2 =
1.06Tajvidi’s ψ1 = 2.13, ψ2 = 0.1 ψ1 = 2.26, ψ2 = 0.59Bilog ψ1 =
0.54, ψ2 = 0.42 ψ1 = 0.56, ψ2 = 0.45Negbilog ψ1 = 0.59, ψ2 = 0.93
ψ1 = 0.62, ψ2 = 1.05
Table 2: Some fitted models. The dependence parameters are
listed for 4 pairs oflocations. It is difficult to compare the
results by the parameter values, the differencescan be shown by
quantiles or prediction regions (see Fig 4).
GoF with χ2-test: Bremerhaven and Schleswigγ-range 0.99<
0.95-0.99 0.9-0.95 0.75-0.9 0.5-0.75
-
3.3 Results for trivariate observations
There are 3D-logistic and negative logistic models estimated for
the three triplets ofstations. The shape parameters and dependence
parameter are summarized in Table4. By investigating the estimated
marginal quantile curves(Table 5) we found thatthere is a
significant overestimation of the high quantiles by the logistic
models. Therates for the negative logistic are closer to their
nominal level. Further improvementscan be possibly obtained by
allowing non-exchangeability in the dependence structure.To this
end our proposed modification in Section 2.5 is the most promising,
as fit-ting the Dirichlet model gets significantly more complicated
than in the bivariate case.The main difficulty is that the spectral
density must be numerically integrated overthe interior of the unit
simplex. The 3-dimensional modeling based on the proposedasymmetric
extension for the above models is in progress.
Hannover, Bremerhaven and FehmarnModels ξ1 ξ2 ξ3 DependenceLog
-0.146 -0.027 0.038 2.157NegLog 0.093 0.110 0.137 0.770
Hannover, Fehmarn and SchleswigModels ξ1 ξ2 ξ3 DependenceLog
-0.099 -0.026 0.015 2.031NegLog 0.088 0.098 0.101 0.817
Bremerhaven, Fehmarn and SchleswigModels ξ1 ξ2 ξ3 DependenceLog
-0.083 0.031 -0.006 2.189NegLog 0.075 0.157 0.111 0.927
Table 4: Fitted 3D models with logistic or negative
logisticdependence structures.
4 Discussion
We have presented several models for multivariate exceedances
and illustrated the prac-tical application of these models using a
real dataset on wind speed. AnR library forfitting the proposed
model,mgpd, has been developed, and will, in due course, be
sub-mitted toCRAN. For a visual (and empirical) model evalution we
suggested to comparethe cover rate of different prediction regions.
While this is a valuable tool, the de-velopment of more formal
tests, like the ones known for copula modeling (Rakonczaiand
Zempléni, 2008) would definitely be a further improvement.
Alternative methodsfor comparing models, assessing the statistical
significance of individual terms in themodel, and assessing
predictive performance should also beexplored.
The suggested asymmetric models are promising, as with their
help one becomesable to fit non-exchangeable MGPD families without
heavy computational burden. Ourresults have shown that they are
real competitors of the previously known such modelsin terms of
coverage probabilities.
18
-
Hannover, Bremerhaven and Fehmarn3D Logistic 3D Negative
Logistic
Quant. H(x, y,∞) H(x,∞, z) H(∞, y, z) H(x, y,∞) H(x,∞, z) H(∞,
y, z)0.99 0.000 0.000 0.001 0.009 0.009 0.0080.95 0.004 0.003 0.005
0.030 0.026 0.0340.90 0.012 0.009 0.018 0.058 0.049 0.068
Hannover, Fehmarn and Schleswig3D Logistic 3D Negative
Logistic
Quant. H(x, y,∞) H(x,∞, z) H(∞, y, z) H(x, y,∞) H(x,∞, z) H(∞,
y, z)0.99 0.000 0.001 0.001 0.014 0.015 0.0110.95 0.004 0.005 0.005
0.030 0.032 0.0270.90 0.011 0.014 0.019 0.054 0.064 0.046
Bremerhaven, Fehmarn and Schleswig3D Logistic 3D Negative
Logistic
Quant. H(x, y,∞) H(x,∞, z) H(∞, y, z) H(x, y,∞) H(x,∞, z) H(∞,
y, z)0.99 0.002 0.002 0.002 0.003 0.008 0.0040.95 0.007 0.009 0.008
0.033 0.034 0.0230.90 0.024 0.026 0.022 0.071 0.077 0.043
Table 5: GoF for 3D models by marginal quantile curves. Thereis
a significant over-estimation of the high quantiles by the logistic
models in general. The rates for thenegative logistic are closer to
their nominal level. Further improvement could be ob-tained by more
complex non-exchangeable models.
There are a number of ways in which this work could be developed
further. Themost obvious extension would be assuming time-dependent
model parameters or pa-rameters depending on multiple explanatory
variables. There are no additional con-ceptual difficulties
involved in extending the model in thisway, but the
computationalissues involved in maximising the likelihood function
would probably be even morepronounced than for the current model.
This may motivate theconsideration of alter-native methods of
inference, such as Markov chain Monte Carlo (e.g., in the context
ofextreme value modeling, Fawcett & Walshaw, 2008).
The possible presence of residual dependence (caused by
theclustering, which im-plies that a single extreme event may be
associated with multiple extreme values) mayalso be investigated.
Residual dependence occurs in the extremes of many environmen-tal
(and financial) time series, and methods that ignore thiswill tend
to under-estimatestandard errors and other measures of uncertainty.
The block bootstrap, or extensionsthereof, may provide a strategy
for constructing confidenceintervals in a way that ac-counts for
residual dependence.
Acknowledgements
The European Union and the European Social Fund have provided
financial support tothe project under the grant agreement no. TÁMOP
4.2.1./B-09/KMR-2010-0003.
19
-
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21
-
−4 −2 0 2 4
−4
−2
02
4
Logistic
x
y
0.0
0.2
0.4
0.6
0.8
0.6 0.4
0.2
0.005
−4 −2 0 2 4
−4
−2
02
4
Negative Logistic
x
y
0.0
0.2
0.4
0.6
0.8
0.6 0.4
0.2
0.1
0.005
−4 −2 0 2 4
−4
−2
02
4
Bilogistic
x
y
0.0
0.2
0.4
0.6
0.8
0.6 0.4
0.2
0.1 0.005
−4 −2 0 2 4
−4
−2
02
4
Dirichlet
x
y
0.0
0.2
0.4
0.6
0.8
0.6 0.4
0.2
0.1 0.005
Figure 5: Examples. BGPD distribution functions with sameµ1 = µ2
= 0, σ1 =σ2 = 1 andξ1 = ξ2 = 0.1 parameters assuming different
dependence structures. Inthe logistic and negative logistic modelsα
= 0.5, in the bilogistic modelψ1 = 1.25,ψ2 = 2 and in the Dirichlet
modelψ1 = 0.6, ψ2 = 0.2.
22
-
0 5 10 15 20 25 30
05
1020
30
Hannover(m/s)
Sch
lesw
ig(m
/s)
0 5 10 15 20 25 30
05
1020
30
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0 5 10 15 20 25 30
05
1020
30
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
0 5 10 15 20 25 30
05
1020
30
Fehmarn(m/s)
Sch
lesw
ig(m
/s)
Figure 6: Example. Observations of daily wind speeds over the
95% threshold levelfor the period 1957-2007. Those exceedances that
exceed thethreshold in both com-ponents are distinguished by blue
colour.
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!
Figure 7: "classical BGPD" and BGPD models fitted to exceedances
above the95% marginal quantiles. The shape and dependence
parameters are (ξ̂1, ξ̂2, α̂) =(−0.001,−0.06, 1.49) and(ξ̂1, ξ̂2,
α̂) = (0.18, 0.16, 1.59) for the left and right
panelrespectively.
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0 5 10 15 20 25 30
05
1020
30
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0 5 10 15 20 25 30
05
1020
30
Bremerhaven(m/s)
Sch
lesw
ig(m
/s)
0 5 10 15 20 25 30
05
1020
30
Fehmarn(m/s)
Sch
lesw
ig(m
/s)
5 10 15 20 25 30 0 5
1015
2025
0 5
1015
2025
Bremerhaven
Fehm
arn
Sch
lesw
ig
Figure 8: Exceedances in 3D. As only one component must be over
a threshold, the 2Dmargins can contain observations which are below
both of their thresholds. In thesecases the third component exceeds
its threshold.
24
-
15 20 25
1015
20
Margins of 3D Logistic
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0.9
0.95
0.99
15 20 25
1015
20
Margins of 3D Logistic
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0.2
0.4
0.6
0.8
0.9 0.95
0.99
15 20 25
1015
20
Margins of 3D NegLog.
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0.9 0.95
0.99
15 20 25
1015
20
Margins of 3D NegLog.
Bremerhaven(m/s)
Fehm
arn(
m/s
)
0.2
0.4
0.6
0.8
0.9 0.95
0.99
Figure 9: Examples. 2D margins of the 3D logistic and negative
logistic models forBremerhaven, Fehmarn and Schleswig. The two
models are rather different, the lo-gistic model leads to
relevantly higher quantile curves than the negative logistic.
Thecomparison with the observation shows that there is a
strongoverestimation in the lo-gistic and a slight overestimation
in the negative logisticmodel.
25
-
A Appendix: "classical BGPD" models fitted by ’fb-vpot’
In Section 2.3 there are two different approaches describedfor
modelling multivariateexceedances. Later on, the difference between
the "classical BGPD" in (5) and BGPDin (6) is illustrated in Fig 7.
Further details about "classical BGPD" are presented here,making
the results comparable with those we got for the otherdefinition.
The parameterestimates, produced byfbvpot routine of theevd
package, are summarised in Table6. Similarly to the results in
Table 2 there is a slight asymmetry present in the data. Inthis
"classical" case there is parametric inference only onthe upper
right quarter ofR2.The density plots are presented in Fig 10.
Hannover and SchleswigModels σ1 ξ1 σ2 ξ2 DependenceLog 2.359
-0.059 1.529 0.025 α′ =0.635NegLog 2.366 -0.056 1.517 0.028 α′
=0.858Bilog 2.391 -0.068 1.505 0.035 ψ1 =0.694ψ2 =0.559Negbilog
2.389 -0.065 1.499 0.038 ψ1 =0.872ψ2 =1.517C-T 2.388 -0.065 1.500
0.037 ψ1 =1.123ψ2 =0.562
Bremerhaven and FehmarnModels σ1 ξ1 σ2 ξ2 DependenceLog 2.381
-0.072 1.654 0.001 α′ =0.535NegLog 2.361 -0.062 1.651 0.004 α′
=1.160Bilog 2.352 -0.063 1.661 0.003 ψ1 =0.489ψ2 =0.575Negbilog
2.333 -0.052 1.656 0.004 ψ1 =1.030ψ2 =0.712C-T 2.323 -0.047 1.653
0.005 ψ1 =1.018ψ2 =1.756
Bremerhaven and SchleswigModels σ1 ξ1 σ2 ξ2 DependenceLog 2.291
-0.060 1.518 0.023 α′ =0.514NegLog 2.293 -0.054 1.507 0.028 α′
=1.239Bilog 2.346 -0.073 1.484 0.034 ψ1 =0.581ψ2 =0.430Negbilog
2.337 -0.066 1.478 0.039 ψ1 =0.581ψ2 =1.071C-T 2.315 -0.062 1.470
0.040 ψ1 =2.268ψ2 =1.032
Fehmarn and SchleswigModels σ1 ξ1 σ2 ξ2 DependenceLog 1.653
-0.002 1.534 0.026 α′ =0.552NegLog 1.654 0.001 1.520 0.030 α′
=1.101Bilog 1.682 -0.010 1.496 0.044 ψ1 =0.630ψ2 =0.450Negbilog
1.675 -0.007 1.488 0.049 ψ1 =0.617ψ2 =1.269C-T 1.671 -0.006 1.485
0.050 ψ1 =2.087ψ2 =0.803
Table 6: "classical BGPD" models fitted byfbvpot in evd.
Prediction regions arenot available in default, although the
density plot is given. (See Fig 10 below.) In thelogistic and
negative logistic modelsα = 1/α′.
26
-
15 20 25 30
1216
20
Logistic Density
Bremerhaven
Sch
lesw
ig
15 20 25 30
1216
20
Negative Logistic Density
Bremerhaven
Sch
lesw
ig
15 20 25 30
1216
20
Negative Bilogistic Density
Bremerhaven
Sch
lesw
ig
15 20 25 30
1216
20
Coles−Tawn Density
Bremerhaven
Sch
lesw
ig
Figure 10: "classical BGPD" contour plots
27