We use a novel statistical approach-MGPD to analyze the joint probability distribution of storm surge events at two sites and present a warning method for storm surges at two adjacent positions in Beibu Gulf, using the sufficiently long field data on surge levels at two sites. The methodology also develops the procedure of application of MGPD, which includes joint threshold and Monte Carlo simulation, to handle multivariate extreme values analysis. By comparing the simulation result with analytic solution, it is shown that the relative error of the Monte Carlo simulation is less than 8.6 %. By running MGPD model based on long data at Beihai and Dongfang, the simulated potential surge results can be employed in storm surge warnings of Beihai and joint extreme water level predictions of two sites.
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.
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
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:
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:
In the Pickands polar coordinate,
The
The data used in this study are provided by The Joint Archive for Sea
Level (JASL) of UHSLC (
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
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
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.
After many experiments, it is found that marginal distributions of
2016 independent events can be described by GEVD:
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
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
The correlation parameter
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
A couple of CP are to be used in the paper: CP1
We conducted also runtime experiments.
For estimating
Based on the results by simulation times
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:
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.
The primary theme of this paper concerns how recent developments about MGPD can be applied to marine disaster forecasting. The paper not only develops the process of determining joint threshold and simulation, but makes some analyses contributed to warning of extreme surges by MGPD. The MGPD is the nature distribution of MPOT method, which can dig up more extreme information from the raw data. The model based on multivariate extreme value theory which is well-founded. The intrinsic properties of all extreme variables are also into consideration. The method of determining the joint threshold was introduced to MGPD in the paper. The Monte Carlo simulation of MGPD was used for the conditional probability of two extreme surges, and the accuracy was verified to be acceptable.
Warning of extreme surges will be more reliable if we can build the relationship among extreme surges at three or more sites. In the paper, the theory about MGPD and its simulation is derived for multidimensional variables. The methodology could be extrapolated to higher dimensional space. So difficulties of solving procedure for MGPD can not restrict its application under the condition of high dimensionality. A potentially better warning approach is possible based on Monte Carlo simulation. Once the long-term (such as thousands of years) sea state data has been simulated, several ocean environment factors can be assessed quickly by the law of large numbers.
This work was supported by the “Strategic Priority Research Program” of the Chinese Academy of Sciences (XDA11010302).
Parameters of marginal distribution.
Comparision of the results of CP1.
RP: return periods, a: analytic solution, s: simulation results, D: Dongfang, B: Beihai (the same below).
Comparision of the results of CP4.
Location of two stations in Beibu Gulf, SCS.
Declustered surge.
Fitting testing of marginal distribution.
Value over threshold and data of stochastic simulation.
The variance of the relative error under the different number of simulation