Introduction
There have been significant advances in the modelling procedures
available for multivariate extreme values. In particular, the research
about application of MGPD (Multivariate Generalized Pareto
Distribution) draws more and more attention. MGPD, as the natural
distribution of MPOT (Multivariate Peak Over Threshold) sampling
method, has the extreme value theory background and can retain more
extreme information from the raw data than the annual maxima of
a series.
In multivariate extreme values analysis, two sampling approaches have
been advocated, which are called Annual Maxima Series (AMS) method and
MPOT method respectively. The MGEVD (Multivariate Generalized Extreme
Value Distribution) is the natural distribution of AMS, in which the
sample is consist of the annual maxima of all components (Morton and
Bowers, 1996; Sheng, 2001; Yang and Zhang, 2013). However, Zaijin You
and Baoshu Yin (2006) propose, taking the extreme waves estimation
as an example, AMS method often ignores multiple severe storm waves
that occur in the same year, which may be much larger than the annual
largest waves in many other years. Consequently, this method may
result in underestimation of extreme variables. MGPD is the natural
distribution of MPOT method, in which the sample is consist of
independent exceedances of a suitably high threshold for all
components (Falk et al., 2004). Obviously, MPOT can
retain more independent extreme values from the raw data than AMS, and
the additional data would likely lead to greater estimation precision
(Luo and Zhu, 2014). Besides, the fluctuation of the estimation of the
extreme waves by POT is smaller than by AMS under different sample
lengths (Luo et al., 2012).
MGPD and MPOT method are widely used recently. Rootzén and Tajvidi
(2006) suggest, based on the idea of Tajvidi (1996), that MGPD should
be characterized by the following couple properties: (i) exceedances
(of suitably coordinated levels) asymptotically have a MGPD if and
only if componentwise maxima asymptotically are EVD, (ii) the MGPD is
the only one which is preserved under (a suitably coordinated) change
of exceedance levels. Morton and Bowers (1996) are based on the
response function with wave and wind speed of anchoring
semi-submersible platforms enabling to analyze extreme anchorage force
and corresponding wave height and wind speed by using logical extreme
value distribution. But the study did not use the natural distribution
MGPD of the MPOT method but MGEVD for fitting the MPOT samples. Coles
and Tawn (1994) and Bhunya et al. (2011) used the same mind too. More
details about MGPD can be found in Rootzén and Tajvidi (2005),
Tajvidi (1996), Beirlant et al. (2005) and Falk et al. (2004).
In the paper, our aim is to develop procedures of MGPD and to
demonstrate how the methodology can be exploited as part of the
analysis of extreme surges at two adjacent sites. The theory and
associated statistical methodology is presented in
Sect. 2. Fundamental to the application of MGPD is the choice of the
joint threshold and the estimation of the joint density. These aspects
included in an example are discussed in Sects. 3 and 4. Finally, the
advantage of MPOT and new possibilities based on Monte Carlo
simulation are outlined.
Methodology
MGPD theory
It is well known that MGEVD (Coles and Tawn, 1991, 1994; Beirlant
et al., 2005) arise, like in the univariate case, as the limiting
distributions of suitably scaled componentwise maxima of independent
and identically distributed random vectors. If for independent
X1,…,Xn following F, there exist vectors,
an,bn∈Rd,an>0, such that
Pmaxi=1,…,n(Xi-bn)an=Fn(anx+bn)⟶n→∞G(x)
where G(x) is a MGEVD, and F is in the domain of attraction of
G. We note this by F∈D(G).
In the paper, we stick to the MGPD definition of Falk
et al. (2004). Similarly to the relationship of GPD and GEVD in one
dimension: H(x)=1+log(G(x)),log(G(x))>-1, the distribution
function of MGPD can be deduced:
W(X)=1+log(G(x1,…,xd))=1+∑i=1dxiDx1∑i=1dxi,…,xd-1∑i=1dxi,log(G(x1,…,xd))>-1
where (x1,…,xd)=x∈U, U is a neighborhood of zero
in the negative quadrant (-∞,0)d, D is the Pickands
dependence function in the unit simplex Rd-1‾ on the
domain of definition, Rd‾={x∈[0,∞)d|∑i=1dxi=1}; G(x1,…,xn) is
a MGEVD function which marginal distribution is negative exponential
distribution (detail in René, 2007). The MGPD can use
existing multivariate extreme dependence function directly because it
is deduced from MGEV, and this enrichs greatly the expression of MGPD
for various dependence relationships. MGPD of logistic type is ease to
use and has the favorable statistical properties from Pickands
dependence function Eq. (), and widely used to hydrology,
financial and other fields.
Dr(t1,…,td-1)=∑i=1d-1tir+1-∑i=1d-1tir1/rWr(x)=1-∑i=1d(-xi)r1/r=1-‖x‖r,
where r is the correlation parameter of dependence function and r>1. xi in the interval (-1, 0) are variables of
standardization. The Bivariate Logistic GPD density function is
w(x,y)=∂W∂x∂y=(r-1)(xy)r-1[(-x)r+(-y)r]1/r-2x<0,y<0
The correlation parameter r can be evaluated by step by step method:
evaluate by using two marginal distributions firstly, and then
introduce to MGPD; By the other method, the correlation parameter r
can be evaluated by global method. r is estimated directly by using
the maximum likelihood for the density function w. The global method
evaluated results more reliable due to the final function form are to
be concerned, but the processes of evaluate are more complex. The
maximum-likelihood function is
L(r)=∑i=1nln(wr(xi,yi)).
Simulation method
The Monte Carlo Simulation method of multivariate distribution is
relatively complex, because of generating multivariate random and
relevant vectors involved. By a transformation method, the variables
become independence. And then, every variable is generated a random
vector. Finally by the inverse transformation, the random vectors of
the multivariate distribution are obtained. The simulation method was
suggested by René (2007).
Using polar coordinate to demonstrate the simulated method of MGPD
better:
Tp(x1,…,xd)=x1x1+…+xd,…,xd-1x1+…+xd,x1+…+xd=(z1,…,zd-1,c),
where Tp change vector (x1,…,xd) into polar
coordinate. C=x1+…xd and
Z=(x1/C,…,xd-1/C) are radial
component and angular component, respectively. They called Pickands
polar coordinate.
In the Pickands polar coordinate, W(X) presents different
properties. Presume that (X1,…,Xd) follow multivariate
generalized Pareto distribution W(X) and its Pickands dependence
function D exists d order differential, define the Pickands density
of H(X)
ϕ(z,c)=|c|d-1∂d∂x1,…,∂xdHTp-1(z,c)
Presume μ=∫Rd-1ϕ(z)dz>0 and constant
c0<0 existing in a neighborhood of zero, then the simulation
method of MGPD is: (1) generate uniform random numbers on unit simplex
Rd-1‾, (2) generate random vector (z1,…zd) based on the density function
f(z)=∅(z)μ of Z=(z1,…,zd-1)
in the Pickands polar coordinate combined with Acceptance–Rejection
Method, (3) generate uniform random numbers on (c0,0), and (4)
calculate vector
(cz1,…,czd-1,c-c∑i=1d-1zi)
which is random vector of satisfy the multivariate over threshold
distribution.
The c0 above is the joint threshold in MGPD method. This paper
determines the threshold by using the principle on Coles and Tawn
(1994).
The data and declustering
The data
The data used in this study are provided by The Joint Archive for Sea
Level (JASL) of UHSLC (http://uhslc.soest.hawaii.edu/home),
which consist of simultaneous hourly sea-level observations at Beihai
and Dongfang, which are on the Beibu Gulf coast in SCS (Fig. 1). The
data set used extends from June 1975 to December 1997. The data can be
used for the analysis of the extreme surge in this study, since hourly
sampling sufficiently captures the high water level. Only 0.023 %
of the data set was lost in Beihai and 0.173 % of the data in
Dongfang was not being used was due to gauge failure or others; And
the 201 578 and 201 275 hourly values are yet to be processed at
Beihai and Dongfang respectively. The amount of data can ensure enough
extreme value information of surge.
Data analysis and declustering
Beihai city is located on the coast of the Beibu Gulf, which is
a semi-closed and shallow bay. Due to special geomorphology, the
typhoon surge in Beibu Gulf is violent and might cause floods to the
city. The surge levels at a site are defined as the residuals after
removal of the astronomically induced tidal component from the
sea-level observations. The tidal component is cyclical and does not
satisfy the basic hypothesis of random variables. Tidal analysis was
undertaken using the method of Godin (1972).
The first stage in an extreme value analysis is declustering: identify
a set of independence events. This is done to make adjacent elements
of the sample, which consists of the maxima of all events, to be
independent of each other. Declustering techniques have been used by
Morton and Bowers (1996) and Coles and Tawn (1991), in which the
cluster interval are 30 and 40 h respectively. But the feature
of storms and surges caused storms every place is different. The
declustering method is illustrated in an application to a sequence of
surges in Fig. 2. The duration of a typhoon surge in the Beibu Gulf is
approximately 100 h. The components of each vector are taken as the
maximum surge at each site over a 100 h event. The peak events for
both vectors in a cluster may be happen at the different
time. Dongfang was 3–5 h ahead of Beihai for the arrival of the peak
of surges from Fig. 2. Since the purpose of the case is to analysis
the relationship between the extreme surges at both sites, the
declustering method is proper because all peaks were included in these
clusters basically. The 100 h cluster interval has enough
spare to make two surge peaks at two sites in the same typhoon surge
be in a cluster. According to the above principles, the total number
of independent events N is 2016.
Constructing conditional probability functions
The extreme surge in Beibu Gulf is cause mainly by typhoons from
lower-latitude areas in SCS. Typhoons move usually through Beibu Gulf
from south to north with a small number of them moving from east to
west. The extreme surge at Dongfang, which is to the southeast of
Beihai, should be as an early warning signal for Beihai. Multivariate
extreme value analysis can be used for the warning.
In order to analyze the joint probability of the extreme surge of
Beihai and Dongfang, CP (conditional probability) distributions
can be used (Eq. 10). CP can represent the probability of
encounter between extreme surges. The joint distribution of bivariate
Pareto distribution function W(x,y) is
W(x,y)=Pr(X<x,Y<y),
where x and y represent the surge (m) in Beihai and Dongfang
respectively (the same in the paper). Wx(x) and Wy(y) are
marginal distribution of x and y respectively. Conditional
probability distributions are:
CP1:Pr(X≥x|Y≥y)=Pr(X≥x,Y≥y)Pr(Y≥y)=1-WX(x)-WY(y)+W(x,y)1-WY(y)CP2:Pr(X≤x|Y≥y)=Pr(X≤x,Y≥y)Pr(Y≥y)=WX(x)-W(x,y)1-WY(y)CP3:Pr(X≥x|Y≤y)=Pr(X≥x,Y≤y)Pr(Y≤y)=WY(y)-W(x,y)WY(y)CP4:Pr(X≤x|Y≤y)=Pr(X≤x,Y≤y)Pr(Y≤y)=W(x,y)WY(y)
Other four CP distributions can be deduced by swapping two variables.
Extreme-value analysis
In this section, the focus is on problems in extreme surges whose
solution would require advances in the methodology of the statistics
of extremes. These problems include analysis of joint threshold,
stochastic simulation, and statistics of multivariate extreme
surges. Finally, the issue of how to analyze the statistic results of
extreme surges at two locations is briefly discussed.
Marginal transformation and joint threshold
After many experiments, it is found that marginal distributions of
2016 independent events can be described by GEVD:
F(x)=P(X<x)=exp-1-ξx-μσ1/ξ,ξ≠0,
where ξ, σ and μ are three variables of GEVD. They are
estimated by using maximum likelihood estimate. This is an approach of
estimation suggested in Sect. 5.1 of Beirlant et al. (2005) and
Sect. 3.3 of Coles (2001). Figure 3 shows the probability plot and
probability plots (including the 95 % confidence intervals) of
marginal distribution before fitting MGPD.
In Sect. 2.1, the variables of MGPD must be in a neighborhood of zero
in the negative quadrant. By a suitable marginal transformation, we
can transfer a margin into a uniform margin in a neighborhood of zero
by the idea of Rene Michel (2007). To standardize the margins, the
marginal distribution of MGPD must be a negative exponential
distribution. According to Taylor expansion, we get
y=logF(xi)=log(1+F(xi)-1)≈F(xi)-1,
where F(xi) is GEVD of variable i (i=1,2),which represent
Beihai and Dongfang respectively. In Eq. (15), F(xi) closes to 1
since we care about extreme observations.
Many dependence models between extreme variables have been suggested:
Logistic, Bilogistic, Dirichlet, etc. However, it appears that the
choice of dependence model is not usually critical to the accuracy of
the final model (Morton and Bowers, 1996). So the simple bivariate
Logistic GPD was selected. The MGPD model of the paper is based on
multivariate extreme value distribution, the joint threshold can be
calculated by the method in Sect. 2.2. The joint threshold is
c0=-0.28, and there are 218 groups of combination of Beihai and
Dongfang over c0. Figure 4a shows that the samples of over
threshold value. In the left subfigure, c0=-0.28 is a curve, and
c in right side of the curve are greater than c0. The converted
data is shown in the right subfigure and c0=-0.28 is a line.
The correlation parameter r of dependence function is estimated by
the maximum-likelihood method and r=2.14992. Having obtained
estimates for all of the parameters, the joint extreme pdf was
constructed, as illustrated in Fig. 4b.
Comparison of stochastic simulation results
According to the simulation method in Sect. 2.2, we generate
the enormous simulation data. The section will compare the CP results
from the two approaches: simulation and directly solve. Figure 5 shows
the data of stochastic simulation by N=10 000 and N=100 000. The
simulation results are in basic conformity with the observations, and
this represents that the MGPD simulation method is worked. The scatter
diagrams show the result of simulation directly, however they need
further quantitative analysis to show the differences of them
objectively.
A couple of CP are to be used in the paper: CP1 P(X>x|Y>y)
and CP4 P(X<x|Y<y) which means CP1 is the probability of the
surge in Beihai over x under the surge in Dongfang over y and CP4
is the probability of the surge in Beihai less than x under the
surge in Dongfang less than y, respectively. As it's showed in
Fig. 6, the relative difference value of the simulation and the
analytic solution are related to the simulation times N. it is
obvious from all subplots in Fig. 6 that relative difference value is
reduced with the increase of simulation times. When the simulation
times up to 1.5×106, the maximum relative error value of
simulation results and calculation results is 5.258 %, which
appears satisfactory.
We conducted also runtime experiments. 1×104, 5×104, 10×104, 1×106, 1.5×106
random vectors were generated on a desktop with an Intel Core i7
Processor with 3.4 GHz. Their runtimes are 3, 24, 82, 9649 and
22 106 s respectively.
For estimating M-year surge of Beihai and Dongfang, the
Poission–Gumbel distribution is used. The Poission–Gumbel
distribution is
F(x)=P(X<x)=e-λ[1-G(x)],
where G(x) is Gumbel distribution.
Based on the results by simulation times 1.5×106, Tables 2
and 3 show the value of CP1 and CP4. The tables represent 5 groups'
calculation and stochastic simulation results of CP1 and CP4 on
different combination of M-year surges at Dongfang and Beihai. It is
found that two results are closely. For instance, the
analytic solution of P(X>x50|Y>y10) is 12.94 % and its
simulation result is 14.06 %. Their relative error is 8.6 %,
which is the maximum of all relative errors for CP1; The
analytic solution of P(X<x5|Y<y50) is 99.35 % and its
simulation result is 94.13 %. Their relative error is 5.25 %,
which is the maximum of all relative errors for CP4.
Warning of extreme surges
The relations between extreme surges at Beihai and Dongfang can be
analyzed by CP. Because the peak surge at Dongfang occurred earlier
than one at Beihai, we use CP1: P(X>x|Y>y) as a standard for
warning of extreme surge at Beihai. From Table 2, we can know that
when 50 year surge is appeared at Dongfang, the probability of
greater 50 year surge is 94.55 % at Beihai.
According to long-term records of surges, we can build the
relationship between extreme surges at Beihai and
Dongfang. Additionally, because of the special geographical relation
of two places, the peak surge at Dongfang is a precursory signal for
the prediction of the probability of the largest surge's occurrence at
Beihai. So we can predict the probability of different surges at
Beihai ahead, and then take preventive measures in order to prevent
society and people from suffering some pains.