Ocean Engineering 244 (2022) 110226
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Modeling asymmetrically dependent multivariate ocean data using
truncated copulas
Pengfei Ma, Yi Zhang *
Department of Civil Engineering, Tsinghua University, Beijing, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Ocean parameters
Joint distribution
Multivariate analysis
Copula
Characterizing multivariate ocean parameters is quite important for offshore engineering reliability design and
risk assessment. To fully understand ocean conditions, a robust and accurate multivariate model is essential for
the analysis and estimation of the ocean state. Therefore, advanced simulation of the ocean parameters helps to
improve practices in offshore engineering. In this work, the principle of a new type of copula, namely truncated
copula, is developed and adopted for modeling the multivariate ocean data. Unlike previous studies on modeling
asymmetric ocean data by purely mathematical fitting techniques, this study proposes a truncated method based
on physical limits to study asymmetrically dependent ocean data. The truncated copula method is contrasted
with the conventional symmetric and existing asymmetric copula from the literature using real environmental
observations for the demonstration. Various commonly used traditional copula models are modified by the
proposed truncation technique and applied to fit multivariate ocean data collected in buoys off the US coast.
Based on the fitting of ocean data, this paper compares the advantages and disadvantages of different copula
models. The properties of different copula models for data simulation and extreme value prediction are also
discussed.
1. Introduction
uncoupled from the system response. Environmental contours have been
applied in offshore, earthquake, and wind engineering (Korn Sar­
anyasoontorn and Lance Manuel, 2005; Saranyasoontorn and Manuel,
2006; Silva-González et al., 2013; van de Lindt and Niedzwecki, 2000;
Winterstein et al., 1999). In many cases, the characterization of the
natural hazards may involve several random environmental variables.
Under this condition, the research on the dependency of different
environmental factors is of great importance. In practical engineering,
coastal and offshore structures can suffer significant damage due to the
appearance of critical combinations of oceanographic variables
co-existing in disastrous weather like a tsunami and sea storms (Tian and
Zhang, 2021; Toimil et al., 2020; Xie and Chu, 2020; Zhang et al., 2018).
Therefore, it is important to determine the joint distribution of ocean
variables for the effectiveness and safety performance of marine struc­
tures. Inappropriately modeling of ocean variables can lead to inaccur­
acies in the environment contour. This ultimately leads to discrepancies
in the design load of the structure, which harm the safety and economy
of marine engineering. Especially, the bivariate distribution model of
peak wave period and maximum-significant wave height is quite
essential, which governs the sea state at a specific ocean site (Veritas,
Marine structures are enormous and possess a wide variety of cli­
matic variables, typically including wind, waves, currents, ice, tide, and
other phenomena that are often catastrophic, such as typhoons. De­
signers are typically expected to make accurate estimates of the envi­
ronmental conditions at the marine site while addressing various
environmental risks for offshore engineering, and usually, the analysis of
multivariate ocean data needs to be adopted (Zhang et al., 2015). An
accurate and robust multivariate ocean model for determining the
maximum response in offshore structure systems under given exceed­
ance probabilities is required as a foundation for producing practical
statistical results. Besides that, building environmental contours of
extreme sea-states for ocean variables requires the description of their
multivariate probability distribution. An environmental contour defines
the combinations of possible values of environmental variables that
should be considered for finding the maximum system response asso­
ciated with a given exceeding probability or return period (Mon­
tes-Iturrizaga and Heredia-Zavoni, 2015, 2016). An advantage of this
method is that contours describing the environmental hazard can be
* Corresponding author.
E-mail address: zhang-yi@tsinghua.edu.cn (Y. Zhang).
https://doi.org/10.1016/j.oceaneng.2021.110226
Received 20 June 2021; Received in revised form 5 October 2021; Accepted 18 November 2021
Available online 27 December 2021
0029-8018/© 2021 Elsevier Ltd. All rights reserved.
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
2010). Moreover, many other climatic variables remain in real nature,
including various causes of uncertainty and possible bias, both of which
affect ocean conditions (Yi and Yingyi, 2015). Uncertainties in the de­
pendencies of parameters, in particular, are one of the most influential
variables. It was known that one of the most difficult tasks is to consider
the nonlinear and asymmetric dependence between ocean variables, and
theoretical multivariate ocean models are difficult to obtain accurately
owing to the complexity between multivariate variables in the ocean
(Ewans and Jonathan, 2014; Huang and Dong, 2021a,b).
Many researchers have made their contributions to the multivariate
statistical ocean variables analysis, containing the application of a
bivariate lognormal model (M. Ochi, 1979), a bivariate logistic model
(Morton and Bowers, 1996), conditional distribution model (Bitner-­
Gregersen et al., 1989; Lucas and Guedes Soares, 2015), a Pareto dis­
tribution model (Muraleedharan et al., 2015), generalized extreme
value (GEV) distribution model (Guedes Soares and Scotto, 2004;
Mackay and Johanning, 2018). With further research on ocean vari­
ables, based on these basic models, some researchers have proposed
some improved models to analyze the distribution of ocean variables.
Scotto and Guedes Soares (2007) proposed a method that combines
Bayesian and extreme value techniques. The inference is significantly
more versatile as a result of using this approach. Petrov et al. (2013)
compared maximum entropy (MaxEnt) to models in the framework of
extreme value theory (EVT) as an effective implement to predict the
extreme values of significant wave heights. They found that the MaxEnt
is much more stable to changes in different thresholds. And De Leo et al.
(2021) proposed Non-stationary Extreme Value Analysis (NEVA), which
provides for the determination of the likelihood of extreme sea state
exceedance while considering trends in the time series data. To
construct the joint distribution model of wave heights and related wave
periods with two-peaked spectra, Huang and Dong (2020) proposed a
mixture bivariate lognormal model. Compared with traditional models,
it does a good job of describing the distribution’s bimodal existence.
However, the standard joint statistical model is no longer sufficient
when the correlation among variables becomes more complex (e.g., The
correlation coefficient is non-constant and the dependency may be
different in the upper tail region and the lower tail region.). Therefore,
in multivariate ocean data analysis, many advanced analytical methods
have been developed recently. Among the many new technologies,
Copulas has grown in popularity. Various earlier studies have shown
that a more realistic model of ocean multivariate data can be con­
structed using copula theory. De Michele et al. (2007) focuses on
multidimensional frequency analysis with copulas of maritime storm
significant wave height (H), storm period (D), the direction of the storm
(A), and storm period interarrival (I). Corbella and Stretch (2013)
started to use Archimedean copulas to simulate a multivariate sea storm.
A similar copula model for simulating the sea states are also presented
by (Antão and Guedes Soares, 2014; Montes-Iturrizaga and
Heredia-Zavoni, 2016). Besides that, more types of copula models are
also gradually being used to describe the state of the ocean. The con­
ditional mixture method, Gaussian copula, and entropy copula were
employed for multivariate statistical simulations of the storm events by
Li et al. (2018). They found that the Gaussian copula is the simplest way
to model the dependent multivariable, but the dependent form must be
Gaussian and the correlation factors can only model linear de­
pendencies, which restricts its use. The conditional mixture method
needs to choose the base copula and modeling method, which is influ­
enced by subjectivity. The entropy copula can provide similar fitting
efficiency, and its simplicity in obtaining the copula functions makes it
the most appealing approach among the others. In a word, it is now
widely acknowledged that Copula is becoming the prior choice for
modeling the statistical characteristic of ocean-dependent variables with
its ability to construct joint distributions without restricting the mar­
ginal distribution of each variable. (Jane et al., 2016; Sebastian et al.,
2017; F. Li et al., 2018; Heredia-Zavoni and Montes-Iturrizaga, 2019). In
the field of climate science, Li and Babovic (2019) used empirical copula
to restore the observed inter-site and inter-variable dependencies, the
temporal persistence, as well as inter-annual variability. Their results
show that the proposed approach can reconstruct the marginally
distributional statistics, inter-site and inter-variable dependencies, and
temporal persistence in the downscaled data for the validation period.
On the other hand, we should also observe the shortcomings of
existing copula approaches which need to be addressed. In earlier
research, The fact that most parametric copula models, like Archime­
dean copulas, are only applicable to data with symmetric dependence
has been criticized (Genest and Favre, 2007). Unfortunately, most ocean
data possess the phenomenon of asymmetrical dependencies. Ignoring
asymmetric effects in ocean data modeling is very dangerous and un­
reasonable, like design values of loads at different return periods,
because it affects the result of the response of the marine structures after
the load is applied, and ultimately may undermine the safety of the
structure. To overcome this deficiency, Zhang et al. (2018) investigated
the theory and construction process of asymmetric copulas and
demonstrated and highlighted the features, advantages, and limitations
of asymmetric copulas through a practical case study in the paper. He
found that asymmetric copulas are more realistic and accurate in
modeling asymmetric multivariate ocean data. Besides that, the asym­
metric copula mixture model employed by Lin et al. (2020), Bai et al.
(2020), and Huang and Dong (2021b) can provide a good match to
bivariate wave data, outperforming copulas from the symmetric copula
families, Khoudraji-Liebscher, and Product families, and the traditional
conditional modeling method. However, the asymmetric copula models
built by existing methods still have two disadvantages.
1. Some asymmetric copula models need too many parameters, which
model exists the possibility of over parameterized, therefore, the
time-consuming in performing multivariate statistical parameter
estimation is extremely high, while there is a higher risk of falling
into a local optimum.
2. Although the asymmetric copula could handle the problem of
breaking wave limit mathematically to a certain extent, it only re­
duces the probability of the data appearing outside the breaking
wave limit but still allow its happening with low probabilities.
Based on these concerns, this paper proposes a truncated method to
add physical limits to the mathematical copula models to tackle this
issue.
The remainder of this paper is laid out below. Section 2 covers the
fundamentals of copulas and asymmetry measurement as well as how to
construct asymmetric and truncated copula models. Specific truncated
and non-truncated copula models for multivariate randomized preprocessed ocean data are presented in Section 3. A comparison of
truncated and non-truncated copula models is discussed in Section 4 to
develop a good understanding of using truncated copulas in multivariate
ocean data modeling. The conclusions of this paper are summarized in
Section 5.
2. Asymmetric copula and truncated copula
2.1. Definition and basic properties of copula
Copula is a strong statistical tool in modeling multivariate data and is
commonly applied in various fields, including finance and economics
(Fredheim, 2008; McNeil et al., 2015; Kielmann et al., 2021), geotech­
nical engineering (Zhang et al., 2019a, 2019b) as well as in hydrology
(F. Li et al., 2018).
Copula is a mathematical model that combines the marginal distri­
butions of individual variables which are uniformly distributed on [0,
1], depending on a particular dependency, to form a multivariate dis­
tribution. According to Sklar’s theorem (Sklar, 1959), a copula is
defined as a joint distribution with specific marginal distributions, as
given below:
2
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
Sklar’s Theorem: Let H be an n-dimensional distribution function
with marginal distributions F1 , …, Fn . An n-dimensional copula C for all
x ∈ Rn is given as
i ∈ {1, . . . ,n}, if Uij,n < uj then this pseudo-observation is regarded as
1 otherwise we take 0, which can be understood componentwise as
)
(
Ui,n = Fn,1 (Xi1 ), …, Fn,d (Xid ) , i ∈ {1, …, n},
(4)
(1)
H(x1 , …, xn ) = C(F1 (x1 ), …, Fn (xn )).
where the analytical survival copula is denoted by Cn , which is con­
structed by the pseudo-observations 1 − U1,n , …, 1 − Un,n .
The radial asymmetric dependency among ocean data can be well
handled by many traditional copula models. However, the other asym­
metric dependency in the secondary diagonal (from lower-right to
upper-left), also known as exchangeability, is quite troublesome.
The fundamental concept of asymmetry concerning the secondary
diagonal (exchangeability) in a copula model is given herein. For a given
copula C(u1 , …, un ), if ui and uj are exchangeable, then the copula C(u1 ,
…, un ) is symmetric (Genest and Nešlehová, 2013). Conversely, a copula
is called asymmetric in the secondary diagonal if it does not satisfy the
above condition. (All asymmetry in the following refers to asymmetry in
the secondary diagonal). For n = 2, a natural test statistic for the
asymmetry is given as below
∫
Snexc =
n(Cn (u1 , u2 ) − Cn (u2 , u1 ))2 dCn (u1 , u2 ).
(5)
If F1 , …, Fn are continuous, C is unique. Conversely, if F1 , …, Fn are
marginal distribution functions and C is a copula, a function H(x1 , …, xn )
with marginal distributions F1 , …, Fn is defined by Eq. (1).
A key feature of copula model is that it does not have to concern the
marginal distribution of the individual variables, which means that it
does not limit the type of marginal distribution. It should be noted that
for continuous random variables, the probability of integral trans­
formation operates, because Fi is invertible. In a word, Copula is a
multivariate cumulative distribution function with all uniform mar­
ginals that are transformed by the cumulative distribution function.
The copula method has the superiority that the dependence structure
between the marginal distribution of the individual variable could be
established based on the copula functions, which is separate from
determining the marginal distribution of individual variables itself. The
choice of the most suitable marginal distribution in copula is determined
by the statistical properties of the marginal distribution itself, inde­
pendent of the dependencies among variables. This gives the copula
models more freedom in modeling the joint relationships among pa­
rameters. In the literature, there are many different types of copulas
(Kemp et al., 1992; Nelsen, 1999; Salvadori et al., 2007; Hofert and
Pham, 2013; Joe, 2014; Hofert, 2020). Each class or family of copulas
can characterize a specific dependency. Via mathematical trans­
formations, many bivariate copulas can be extended to a multivariate
one (Nelsen, 1999).
As mentioned by Zhang et al. (2018), there is a certain amount of
copulas that are applicable for symmetrical dependencies while de­
pendencies in most multivariate variables governing the sea states are
asymmetric. Ignorance of such asymmetric dependencies in ocean data
will result in the wrong estimation of return values, which in turn affects
the reliability of the model. Therefore, a more advanced statistical
technique capable of characterizing the asymmetric dependency is
needed.
[0,1]2
The corresponding test of secondary diagonal asymmetry has been
studied in Kojadinovic and Yan (2012), Rémillard and Scaillet (2007),
Genest and Nešlehová (2013). It is observed that the measure of sec­
ondary diagonal asymmetry is calculated by the squared euclidean
distance between C and transpose CT . Therefore, a high value of this
would imply the copula to be non-exchangeable,
statistical value Sexc
n
which is deemed as asymmetric in secondary diagonal. The estimate
of asymmetry measured by Eq. (5) may be used as a measure of asym­
metry for bivariate ocean data modeling.
2.2.2. Product copulas approach
In the last few years, much of the work has made outstanding con­
tributions to the development of the construction of asymmetric copulas
(Grimaldi and Serinaldi, 2006; Mesiar and Najjari, 2014; Mazo et al.,
2015; F. Li et al., 2018; Zhang et al., 2019b). These include several
techniques that are used in multivariate data modeling for capturing
asymmetric dependencies. Not all asymmetric copulas, however, could
be applied to the real situation. When we construct a copula function
with complex dependencies, the implementation of certain asymmetric
copulas may be too complicated. This research focuses on the families of
asymmetric copulas that can be conveniently built by a variety of base
copulas, for example, Archimedean copulas. In the present analysis, we
do not present the asymmetric copulas that require a particularly com­
plex process to construct.
Following Liebscher (2008), an asymmetric copula can be con­
structed by a product of base copulas, namely Khoudraji-Liebscher
2.2. Asymmetric dependency and asymmetric copulas
The modeling of asymmetric dependence in multivariate ocean data
is quite crucial in the establishment of joint distribution models and
affects the accuracy of extreme value prediction, while the currently
commonly used copula models are deficient in this regard. In order to
address this issue, this work has gone through a few different types of
asymmetric copulas, as well as introducing a newly developed type of
copulas, namely truncated copulas.
copulas. Assume that C1 , …, Cn : [0, 1]d → [0, 1] are copulas. Let gpq :
[0, 1]→ [0, 1] for p = 1, …n, q = 1, …, d be functions which are strictly
∏
increasing. Suppose that np=1 gpq (u) = u for u ∈ [0, 1], q = 1, …, d, and
limu→0+0 gqp (u) = gpq (0) for p = 1, …, n, q = 1, …, d. Then
2.2.1. Measure of asymmetry
Asymmetry in the copula domain generally refers to asymmetry in
two directions, one in the main diagonal direction (from lower-left to
upper-right), also known as radial asymmetry, which can be measured
by the following
∫
Snsym =
n(Cn (u) − Cn (u))2 dCn (u),
(2)
n
∏
C(u1 , …, ud )product =
u ∈ [0, 1]d ,
(6)
is also a copula. The individual functions gpq ( ⋅) must fulfill the following
additional properties to ensure that this product of copulas is a copula:
where Cn is a typical empirical copula function. The empirical copula
function is also a nonparametric distribution function estimator., which
is given by
n
n ∏
d
)
(
) 1∑
(
1∑
1 Ui,n ≤ u =
1 Uij,n ≤ uj ,
n i=1
n i=1 j=1
​ for ​ uq ∈ [0, 1],
p=1
[0,1]d
Cn (u) =
)
(
Cp gp1 (u1 ), …, gpd (ud )
1. gpq (1) = 1 and gpq (0) = 0,
2. gpq is continuous on ( 0, 1],
3. When there are two individual functions gp1 q , gp2 q at least, 1 ≤ p1 ,
p2 ≤ n which are not equal to 1, then gpq (x) > x with x ∈ (0, 1),
p = 1, …, n.
(3)
where Ui,n = (Ui1,n , . . . , Uid,n ), i ∈ {1, . . . , n}, are the pseudoobservations of original random variables and inequalities Ui,n ≤ u,
3
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
⌣
linear convex combinations of C k (.). For the base copula C, many copula
families or classes may be specified. For example, a general bivariate
copula C(u1 , u2 ) according to Eq. (7) can be expressed as
Table 1
The three most applicable individual function.
Individual function
Parameters
Range of values
1. gpq (u) = uθpq
n
∑
θpq ε[0, 1]
2. gpq (u) = uθpq e(u−
p=1
n
∑
1)αpq
p=1
n
∑
3. g1q (u) = exp(θq −
̅
√⃒̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⃒
⃒ ⃒
⃒lnu⃒ + θ2q )
p=1
θpq = 1
θpq = 1
αpq = 0
θq for q ε {1, …,
d}
⌣
C1 (u1 , u2 ) = C(1, u2 ) − C(1 − u1 , u2 ) = u2 − C(1 − u1 , u2 )
.
⌣
C2 (u1 , u2 ) = C(u1 , 1) − C(u1 , 1 − u2 ) = u1 − C(u1 , 1 − u2 )
θpq ε(0, 1)
αpq ∈ ( − ∞, 1), θpq ≥ −
As a result, the constructed asymmetric copula by linear combination
could be given as
αpq
θq ≥
1
2
⌣
2.2.4. Skewed copula
The skewed copula is also a widely used method to construct
asymmetric copula models for data with asymmetric dependency.
Skewed multivariate distributions inspired this approach, which extends
the original distribution to the skewed copula. The gist of this method is
to transform a multivariate distribution to an asymmetric multivariate
distribution by introducing a skewness parameter (Kollo et al., 2013).
The skewed Gaussian copula is one of the most commonly used version.
An n-dimensional skewed Gaussian copula is given by
(
(u1 , …, un ; μ, Σ, β) = Fn, ​ skew ​ F1,− 1​ skew ​ (u1 ; μ1 , 1, β1 ), …, F1,− 1​ skew ​ (un ; μn , 1, βn ); μ, Σ, β),
cation of Khoudraji-Liebscher copulas has been widely used in hydrol­
ogy, such as in terrestrial hydrology (Durante and Salvadori, 2010;
Michele et al., 2013) and marine hydrology (Salvadori et al., 2013,
2015; Y. Li et al., 2019; Chen et al., 2019; Aghatise et al., 2021).
bution SN(ui , 1, βi ). The n-dimensional skew-normal distribution Fn,skew
∑
has a mean parameter μ, covariance matrix , and shape parameter β.
Typically, we make mean values equal to zero. As a consequence, the
asymmetric property is solely determined by the shape parameters.
While β = 0, skewed Gaussian copula is the non-skewed standard
Gaussian copula. As β becomes larger, the skewness of the corresponding
distribution grows. Therefore, the shape parameters could be used to
describe the asymmetric properties of the marginals and the multivar­
iate distribution.
2.3. Truncated copulas
In order to better improve the fitting ability of copula for asymmetric
⌣
Ck (u1 , …, un ) = C(u1 , …, uk− 1 , 1, uk+1 , …, un ) − C(u1 , …, uk− 1 , 1 − uk , uk+1 , …, un ),
where C (.) is the base copula. The variable uk in Eq. (7) is not
exchangeable with other variables. This kind of copula is referred to
flipped copula as proposed by Salvadori et al. (2007).
Equation (8) can be used to construct LCC copulas to capture the
asymmetric characteristic in multivariate variables:
n
∑
⌣
k=0
where pk is a weighting factor for 0 ≤ pk ≤ 1 and
⌣
n
∑
k=0
(7)
data and dealing with the physical limitations in ocean data modeling,
this paper proposes a new type of copulas by using the truncation
technique. The truncation technique has been widely used in the trun­
cated multivariate normal distributions, see Arnold et al. (1993), Gupta
and Chaudhary (1993), Horrace (2005), Rodriguez-Yam et al. (2004),
Wilhelm and Manjunath (2010). The reason for introducing the trun­
cation method is that the nature parameters are often bounded by
certain physical limits. Therefore, the statistical domain of these pa­
rameters has to be “truncated” in order to offset the infeasible region.
This strategy can help us to truncate the non-feasible space from the
original space to obtain the space that we want. Therefore, the combi­
nation of this truncation technique and copula modeling should help to
(8)
pk Ck (u1 , …, un ),
(11)
− 1
where F1,skew
(.) is the inverse of the univariate skewed-normal distri­
2.2.3. Linear convex combinations (LCC) copula
Linear convex combination (LCC) of copulas and their transformed
forms is another algebraic way to build an asymmetric copula. From the
point of view of symmetry of functions, the results built by direct linear
convex combinations of symmetric copula functions are still symmetric
copulas, for the same type of base copula functions. To deal with this
problem, Wu (2014) has derived a method to modify the base copulas to
include asymmetric characteristics. In the methodology raised by him, a
new asymmetric copula can be derived as
CLCC ​ (u1 , …, un ) =
(10)
It can capture the asymmetric dependency in the binary data by
adjusting the weights of the base copulas in Eq (10).
As shown in Eq. (6), we can observe that the copulas C1 , …, Cn could
be specified in different classes or groups of parametric copulas.
Regarding functions gpq , three alternatives for construction of asym­
metric copulas are given by Liebscher (2008). Table 1 shows the three
most applicable individual functions. It can be realized the construction
needs to determine the number of base copulas as well as the parameters
in the individual functions gpq (uq ). The Khoudraji-Liebscher copulas
could be conveniently employed in R (Hofert et al., 2020). The appli­
Gaussian ​
⌣
CLCC ​ (u1 , u2 ) = p0 C(u1 , u2 ) + p1 C1 (u1 , u2 ) + p2 C2 (u1 , u2 ).
g2q (u) = uexp( − θq +
̅
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
⃒ ⃒
⃒ ⃒
⃒lnu⃒ + θ2q )
Cskew−
(9)
pk = 1. When k
= 0, C 0 = C(u1 , …, un ). As a result, an asymmetric copula is built by
4
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
Fig. 1. Scatterplot of the significant wave height (WVHT) and average wave period (APD) in the original domain (left) and copula domain (right).
improve the modeling of multivariate ocean data.
In this work, the truncation technique is applied to copula functions.
The reason why the truncated method is used in the copula domain
instead of the marginal distribution of individual variables is that the
infeasible domain subject to physical limits usually can not be simply
truncated from marginal domains. For example, a typical asymmetric
data set (WVHT-significant wave height (m), APD- average wave period
(s)) in the original domain and copula domain are plotted in Fig. 1. The
highlighted red circle indicates the infeasible domain due to the
breaking wave limit. It can be observed from Fig. 1 that the non-feasible
region is quite irregular which could not be only truncated from either
marginal. It needs a way to truncate the region from the multivariate
domain rather than an individual domain (marginal domain).
The truncated copula model has been investigated by some pio­
neering works. Juri and Wüthrich (2002), Charpentier and Segers
(2007) have concentrated on the determination of bivariate truncated
copula for a given truncation point. Both references demonstrated that if
the initial copula is Archimedean, the truncated copula is the same type
as well. This is mainly because the truncation point lies on the diagonal
line in the copula domain, i.e. copula domains are equally divided in the
marginals. An extension study was done by Hofert (2020) for consid­
ering a fixed d-dimensional vector t = (t1 , . . . , td ) ∈ (0, 1]d as a right
truncation point. He pointed out that the thresholds do not have to be
the same for each variables. This makes it possible to create an arbitrary
shaped truncated domain in copula. The truncated space also has its
physical meaning in ocean data modeling. In marine engineering, a
wave with a definite period has an upper limit of its wave height due to
the breaking wave limit. Therefore, the modeling of dependency be­
tween average wave period and significant wave height should count on
this physical limitation. This means that in the copula domain we need
to remove the space that are not feasible due to physical limits. The
truncated copula model is capable to offset these infeasible space in
copula domain according to physical limits. The derivation of the
truncated copula is shown below.
∮
∮
c(U)dU = 1,
(12)
after shifting the term, we can get
∮
∮
c(U)dU = 1 −
c(U)dU,
(13)
c(U)dU +
feasible ​ space
infeasible ​ space
feasible ​ space
infeasible ​ space
divide both sides of the equation by 1 −
neously
∮
feasible ​ space
1−
∮
∮
c(U)dU
c(U)dU
infeasible ​ space
=∮
feasible ​ space
c(U)dU
feasible ​ space
c(U)dU
∮
infeasible space c(U)dU
= 1,
simulta­
(14)
finally, a d-variate truncated density copula distribution function can be
formulated for U as:
ctruncated (U) = ∮
c(U)dU
​ ; ​ U ∈ [0, 1]d , U ∈ Ωfeasible .
c(U)dU
(15)
feasible ​ space
∮
For ease of expression, we can denote the definite integral 1/
feasible space
c(U)dU as truncated coefficient αfeasible , and the truncated
density copula could be given as
ctruncated (U) = αfeasible c(U)dU ​ ; ​ U ∈ [0, 1]d , U ∈ Ωfeasible .
(16)
For the bivariate case, when the feasible space Ωfeasible is defined as
[0, t1 ] × [0, t2 ] for u1 and u2 , the truncated bivariate copula density
function can be further expressed as
ctruncated (u1 , u2 , t1 , t2 ) = ∫ t1 ∫ t2
0
0
c(u1 , u2 )
; u ∈ Ωfeasible .
c(u1 , u2 )du2 du1
The cumulative copula distribution function is given by
∫ U1 ∫ U2
c(u1 , u2 )du2 du1
Ctruncated (U1 , U2 , t1 , t2 ) = ∫0 t1 ∫0t2
; ​ u, U ∈ Ωfeasible .
c(u1 , u2 )du2 du1
0
0
2.3.1. Definition of the truncated copula
Let U* = [U1 , U2 , …, Ud ] ∼ C for a d-dimensional copula C and let
(17)
(18)
Therefore, it can be observed the truncated copula is a special con­
ditional distribution. The original density is magnified by the inverse of
probabilities in the feasible space. Examples of truncated Archimedian
copulas with different dependencies are derived and presented in Ap­
pendix A.
d
Ω ∈ (0, 1] for a d-dimensional space in copula domain with d ≥ 2.
Now, let U = [U1 , U2 , …, Ud ] be the truncation of U* by the Ω ∈ Rd . For
the original copula function having density c(.), and the copula domain
is divided into feasible space Ωfeasible and infeasible space Ωinfeasible ac­
cording to the actual physical limits in the modeling. Then the truncated
copula can be defined as follows.
From the probability theory, we can get
2.3.2. Simulation algorithm
Having determined the expression for the truncated copula, a
generator of random vectors with truncated density copula functions
remains a matter of concern. It is critical to obtain samples from given
5
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
densities by Monte Carlo simulation. Drawing samples from the density
copula and rejecting those that are outside the feasible space Ωfeasible is a
simple way to generate them. This rejection sampling method can be
inefficient in some cases, especially for higher dimensions and small
feasible space for random vectors. However, our focus in this work is to
explore the applicability of truncated copulas in asymmetric ocean
datasets rather than to improve the sampling efficiency of truncated
copulas, so more efficient sampling methods such as Gibbs sampling are
not considered. Therefore the rejection sampling has been adopted and
the procedure of sampling is the following.
3.1. Data pre-processing
The aim of data pre-processing is to obtain relatively independent
and identically distributed continuous random variables, where the
filtered data can be considered as the same statistical model.
The first thing is to determine the time interval of the data that can be
considered as the identically distributed statistical model. Data parti­
tioning is firstly performed. A particular period of data is selected to test
its stationarity. For a short period with a relatively calm climate, the
observed ocean data is believed to be quasi-stationary and identically
distributed. The ocean data for this analysis has been chosen to cover the
most extreme time from October 2018 to March 2019. Fig. 3 illustrates
how the mean and standard deviation of WVHT and APD have changed
over time. The statistical tests are performed to examine if the data in
separate months vary significantly from one another in Table 2. Here,
ANOVA is used to test the difference of the data of WVHT and APD in
different months. The p-values in the statistical test for WVHT and APD
are 0.264 and 0.343, which means that the hypothesis of the bivariate
ocean data in different months possesses statistically consistent mean
values that can not be rejected. The data from October to March are
assumed to be identically distributed. Therefore, the dataset (WVHT,
APD) from these six-months in 2018 and 2019 is used in the analysis.
From Table 2 we can observe that there are significant statistical vari­
ations between bivariate ocean data. Individual statistical characteris­
tics within the ocean parameters WVHT and APD must be investigated
separately.
Serial correlation is the next issue that needs to be concerned before
modeling. Ignoring serial correlation can result in the overestimation of
extreme events (Mackay et al., 2021). The autocorrelation functions
(ACF) for WVHT and APD are plotted in Fig. 4. The figure shows the time
series have strong correlations between 0 and 40 h. Therefore, the serial
correlations could not be ignored. In this work, we adopt the method
proposed by Vanem (2016). There are two forms of serial dependence in
ocean data. The first is a short-term dependency due to the process of
physical wave formation, and the second is long-term dependency
influenced by seasonal variations. The first dependency can be
addressed by subsampling the data, that is by interval sampling from
hourly data. It can be done by resampling every 3 h from the hourly data.
This results in 1454 reduced sample datasets in each case. As a result, the
seasonal effect is removed according to Eq. (19). While Xi is the original
ocean data of WVHT or APD belonging to week j, j = 1, …, 52, the
pre-processed sample Yi is then obtained using Eq (19), where μj is the
weekly average for week j, σj is the weekly standard deviation for week j,
M is the total mean.
1. For i = 1,…, n, do: Repeat sampling U ∼ C, in R Code: U <- rCopula
(n,copula). The method for drawing random samples from different
copula functions can be found in Hofert et al. (2018). Until U ∈
Ωfeasible for the truncated copula, then set Xi = U.
2. Return the pseudo-observations of X1 ,…, Xn for a sufficiently large n,
see (GENEST et al., 1995).
According to this algorithm, pseudo samples can be obtained from
the truncated copula, which can be used for further data analysis.
3. Ocean data analysis
To demonstrate the superiorities of truncated copulas over other
non-truncated copula models, a comparative analysis is conducted using
ocean data from the National Data Buoy Center, US (NDBC, 2018). The
data were gotten at an ocean site in the East Hatteras, 150 NM east of
Cape Hatters (34.724◦ N 72.317◦ W, No. 41001 Buoy) with a water depth
of 4566 m. The analysis will use hourly reported ocean data in the year
2018–2019. (2018/01/01 01:00–2019/12/31 23:50). This work has
studied two ocean parameters: significant wave height (WVHT), average
wave period (APD). The former is in meters and the latter in seconds.
Due to the limitation of the beaking wave limit, non-breaking waves are
collected. As seen in Fig. 3, the ocean data record indicates a strong
seasonal difference. Extreme weather occurs more often in winter than
in summer.
As shown in Fig. 2, entire copula modeling process can be mainly
divided into two main parts, data pre-processing and copula modeling.
Yi =
X i − μj
σj
+M
(19)
After pre-processing of the bivariate ocean data, it is clear from Fig. 5
that the pre-processed bivariate ocean data are free from serial depen­
dence. This proves that the two steps of resampling and removing sea­
sonal effects effectively removed the long-run and short-run serial
correlation from the original data.
It is worth noting that, as shown in Fig. 6, the key properties of the
dependency structure between WVHT and APD are maintained in the
pre-processed results. The pre-processed bivariate ocean data are pre­
pared for further analysis by modeling with copula models.
In modeling the bivariate data using copulas, as a first step, the
marginal distribution of each variable has to be determined. A set of
candidate distributions is selected to fit the individual pre-processed
ocean data. These include Normal, Exponential, Weibull, Rayleigh,
Gamma distributions, Extreme value, and Lognormal distributions. The
model parameters for each distribution are estimated by the maximum
likelihood method. To choose the best models, the corrected Akaike
Information Criterion (AICc) is chosen as the goodness of fit measure.
The estimated statistics for each model are summarized in Table 3. It
Fig. 2. Flow chart of the entire analysis of copula modeling.
6
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
Fig. 3. Box plot of WVHT and APD in two years.
shown in Fig. 6. It can be observed that the dependency between the
bivariate data is quite nonlinear and there is a relatively strong de­
pendency in the upper and lower tails.
By adopting the most suitable marginal distribution models as given
in Table 3, the bivariate pre-processed data (WVHT, APD) are converted
to the pseudo-observations in the copula domain. A probability density
contour plot of (WVHT, APD) as pseudo-observations is presented in
Fig. 7. Asymmetric dependency structures can also be observed obvi­
ously in the density contour of the pseudo-observations. The data
(WVHT, APD) concentrate at both the minimum and maximum
extremes.
Finally, before the multivariate ocean data analysis, the issue of
repeated observations also known as ties needs to be catered in the preprocessed data set (Genest et al., 2011; Bücher and Kojadinovic, 2016).
According to the fundamentals of the probability theorem, the proba­
bility of sampling a specific value from a continuous joint distribution is
zero, in other words, ties should not appear. In reality, however, repe­
titions in the collected data set resulting from the occurrence of a
genuinely continuous random phenomenon are not unusual due to a
lack of measuring accuracy and rounding. The repetition rate in the
pre-processed data can affect the quality of multivariate analysis. If too
many ties occur, the data is deemed not random and continuous (Genest
Table 2
Statistical results of WVHT and APD
Amount of data
Mean
Skewness
Kurtosis
Std. Deviation
WVHT
4684
2.363
1.321
2.014
1.191385
APD
4684
6.313218
1.255
2.296
1.142158
shows that the best models for WVHT and APD are Lognormal and
Gamma distributions, respectively. In the Kolmogorov-Smirnov test, the
p-values are 0.244 and 0.158, which indicates that the best-fit model for
each ocean variable has passed the goodness-of-fit test at a significance
level of 5%.
To investigate the dependency between the bivariate dataset, several
dependency measure principles including Kendall’s tau, Spearman’s
rho, and Pearson correlation coefficient are employed and calculated in
Table 4. Meanwhile, the measure of the asymmetric dependency named
tests of exchangeability as given as Eq. (5) is also calculated. The p-value
for testing whether the data is symmetrically dependent or not is also
provided. The calculated p-values for testing the asymmetric de­
pendency of (WVHT, APD) is 0.0004, which indicates the bivariate data
are asymmetrically dependant. The scatter plot of (WVHT, APD) is
Fig. 4. Autocorrelation function of WVHT (left) and APD (right) for the selected period—initial data.
Fig. 5. ACF of WVHT (left) and APD (right) for the selected period—pre-processed.
7
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Ocean Engineering 244 (2022) 110226
Fig. 6. Scatterplots of the original data (left), the resampled data (middle), and the pre-processed data (right).
Table 3
AICc statistical results for parameter estimation of the marginal distribution.
Normal
Exponential
Weibull
Rayleigh
Gamma
Extreme value
Lognormal
WVHT
26295.55
28470.77
23343.49
23470.23
21737.75
35634.45
20822.09a
APD
28798.51
52066.11
30574.83
40045.91
33732.17
28985.09
a
27495.83a
The best distribution model.
et al., 2014; Joe, 2014). Furthermore, if ties exist, copula parameter
estimation will be subjected to a greater bias comparing to the absence
of ties (Hofert et al., 2018). Therefore it is important for us to know the
situation of ties in the dataset. A summary of the ties in pre-processed
data is given in Table 5. It can be seen that several repeated samples
existed in the pre-processed data. As a consequence, prior to multivar­
iate modeling, another statistical treatment is used to remove ties in
pre-processed ocean data.
Adding random components to every observation is a clear and
realistic way to deal with this problem (Michele et al., 2013; Salvadori
et al., 2014). According to Eq. (20), a random term is introduced into the
bivariate ocean observations.
Table 4
Summary of statistical dependency between (WVHT, APD) (p-value of the
asymmetric dependency test is provided in the bracket).
Data set
Amount
of data
Kendall’s
tau
Spearman’s
rho
Pearson
correlation
Measure of
asymmetry
(WVHT,
APD)
1454
0.4055
0.5593
0.6354
2.5088
(0.0004)
WVHTiR = WVHTi + ΔWVHT αi ​ and ​ APDRi = APDi + ΔAPD βi ​ , i = 1, …, n
(20)
where n is the number of data, ΔWVHT and ΔAPD are the data resolutions
for WVHT and APD, α and β are random factors chosen from the uniform
distribution between 0 and 1. The resolutions are ΔWVHT = 0.01 m and
ΔAPD = 0.01s for the pre-processed ocean data. As a result, through the
transformation according to Eq. (20), the pre-processed ocean data is
randomized. WVHTR and APDR could describe continuous variables that
were previously recorded as discrete.
The influences of the ties to both the joint and univariate marginals
statistical properties of bivariate ocean data are investigated. Fig. 8
presents the cumulative distribution function (CDF) plot of the nonrandomized and randomized pre-processed ocean data with adopted
parametric models. The discrepancies between the CDFs of nonrandomized pre-processed data and randomized ocean data are negli­
gible. Both scenarios were well-fitted to the adopted models according
to the p-value obtained by the KS test, which is a statistical tool used to
determine whether two samples belong to the identical distribution
(Chakravarti et al., 1967). As seen in Table 6, the marginal distribution
model parameters are estimated for randomized ocean data and
compared to non-randomized one. The p-values implies that the adopted
marginal distributions can complement the randomized data well. The
scatter plots of (WVHT, APD) in Fig. 9 also proves that the randomized
ocean data has a high consistency with the non-randomized one. The
randomized ocean data and the non-randomized ocean data have almost
identical statistical properties. The calculated p-values are all greater
Fig. 7. Empirical contour plot of (WVHT, APD) in the copula domain.
Table 5
Percentage of ties in the bivariate data.
Percentage of repeated observations
WVHT
APD
(WVHT,APD)
18.9%
19.1%
6.9%
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Ocean Engineering 244 (2022) 110226
Fig. 8. Plots of CDF of non-randomized and randomized ocean data.
than 5% which means the fitted parametric models are accepted. As a
result, the bivariate data after randomization can represent continuous
random variables for the next step of the analysis.
In conclusion, after data pre-processing and randomization, a
bivariate dataset that is relatively independent and identically distrib­
uted, and continuously randomized are obtained. This guarantees the
reliability of the further statistical analysis.
Table 6
The comparison of estimated parameters of models between non-randomized
and randomized ocean data (p-values of the KS tests between non-randomized
and randomized pre-processed data are noted in the bracket).
WVHT
APD
Non-randomized
data
μ = 0.7735
σ = 0.4569
k = 37.5182
θ = 0.1663
Randomized data
μ = 0.7716
σ = 0.4530 (p-value =
k = 37.0141
θ = 0.1663 (p-value =
0.774)
0.809)
3.2. Copulas modeling
In modeling the bivariate data, the traditional symmetric copulas,
asymmetric copulas as discussed in Section 2, and truncated copulas are
all applied.
In this paper, the Archimedean copulas are used as the base function
for constructing the asymmetric copulas and the truncated copulas. This
includes the Gumbel, Clayton, and Frank copulas. The truncated copula
is constructed by the truncation of all the copula models as mentioned
above. The following five types of copula models will be concerned:
Fig. 9. Comparision
( WVHT, APD)
of
Non-randomized
and
randomized
data
I. One-parameter Archimedean copulas: The typical symmetric
copulas from the Archimedean family are considered in the first
category, which are Gumbel, Clayton, and Frank copula.
II. Product copulas approach: Follow the rules given in Section
2.2.2, the asymmetric copulas are formed by the product of
copulas. Gumbel-Frank, Gumbel-Clayton, and Clayton-Frank are
selected as the base copulas in formulating the asymmetric cop­
ulas defined in Eq. (6).
III. Linear convex combinations (LCC) copulas: The third category
of asymmetric copulas are constructed by the linear convex
combinations as discussed in Section 2.2.3. The base copulas in
Eq. (10) are chosen from Gumbel, Clayton, and Frank copulas.
IV. Skewed copula: The skewed Gaussian copula is the fourth
category in constructing an asymmetric copula as introduced in
Section 2.2.4.
V. Truncated copula: In formulating the truncated copulas, the
above four copula types are truncated according to the breaking
wave limit for (WVHT, APD). To find the truncated space, it is
necessary to know why the bivariate data (WVHT, APD) are
for
9
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
asymmetrically dependant because of this physical limit. Wave
breaking represents a crucial nearshore phenomenon that in­
corporates many environmental and engineering factors. This has
led to the progress of many breaking onset criteria, including
kinematic criteria based on a maximum value of the ratio uc / v,
which compares horizontal particle velocity at the crest of the
wave uc to its phase velocity v, which is measured in the wave
propagation orientation. Recently, Varing et al. (2021) used the
Fully Nonlinear Potential Flow (FNPF) model proposed by Grilli
and Subramanya (1996) to investigate numerically the validity of
this criterion in capturing breaking onset for solitary and
quasi-regular two-dimensional shallow-water waves, which is
proofed to be more accurate than current criteria in the detection
of wave breaking initiation. Besides that, there are many other
methods used to study breaking onset criteria. To facilitate the
coupling of mathematical and physical expressions, the wave
breaking criterion raised by Goda (2010) is used in this paper. He
gave the relationship among the relative water depth db , the
breaking wave height Hb , and the slope of the bank tanβ , by
experiment, as shown in Eq. (21):
{
[
)]}
Hb
πdb (
4
1 + 15tan3 β
= A 1 − exp − 1.5
,
(21)
L0
L0
Fig. 10. Plot of breaking wave limit curve and the bivariate ocean data.
where β is the angle between the seafloor and the horizontal plane, L0 is
the length of the deep-sea wave, db is the water depth when the wave has
broken, A is the coefficient modified by experiment and equals 0.17.
Then Li et al. (2000) and Li and Li (1993) amended Eq. (21) by arguing
that the coefficient A should be taken as 0.15. According to the linear
Apply F1 on both sides of the inequality in Eq. (25). Since F1 is nondeceasing, the inequality relationship does not change. Therefore, the
truncated space can be expressed as
{
{
[
]}}
(
)
πdb
4
2
− 1
3
u1 ≤ F1 1.17F2 (u2 ) A 1 − exp − 1.5
1 + 15tan β
.
1.17F2− 1 (u2 )2
2
wave theory, we have L0 = 1.56T , where T is the average period of
waves as denoted as APD in this work. Li (1983); Lu and Xu (1999); Ochi
and Tsai (1983) suggested that the irregular wave breaking wavelength
is shorter than that calculated by the linear wave and its wavelength is
0.75 times of the linear wavelength. Therefore, in this work, we use the
(26)
For simplicity, the above equation is denoted as u1 ≤ g(u2 ), the
truncated copula can then be derived as follows
∫ u2 ∫ u1
c(u1 , u2 )du1 du2
Cright− truncated (u1 , u2 ) = ∫ 10 ∫ g(u0 2 )
​ , ​ u1 ≤ g(u2 ) ∪ u1 , u2
c(u1 , u2 )du1 du2
0
0
2
formula L0 = 1.17T to describe the relationship between L0 and T.
Thus, the relationship between the average period and the breaking
wave height can be derived as
{
[
)]}
πdb (
4
2
3β
Hb = 1.17T A 1 − exp − 1.5
1
+
15tan
.
(22)
2
1.17T
where c(u1 , u2 ) is the copula density function fitted to the bivariate data.
The truncated space can be estimated by integrating the copula domain
based on the inequality equation given in Eq. (26). In this work, the
aforementioned copula model types (I-IV) are all utilized to construct
the truncated copulas. Equation (27) is used as the general equation for
constructing the corresponding truncated copula model. For the ease of
application, the truncation factors αfeasible , considering the breaking
wave limit, are estimated for Gumbel, Frank, and Clayton copulas and
presented in Appendix A.
So for a significant wave height (WVHT) and average period (APD),
the following inequality is satisfied
{
[
)]}
π db (
4
3β
WVHT ≤ 1.17APD2 A 1 − exp − 1.5
1
+
15tan
,
(23)
1.17APD2
the physical limit about significant wave height and its breaking wave
limit can be shown in Fig. 10.
Once the physical constraint relationship between WVHT and APD is
determined, a truncated copula could be constructed. Let the cumulative
marginal probability distribution function for WVHT and APD be
F1 (xWVHT ) and F2 (xAPD ).
xWVHT ⋅ = ⋅F1− 1 (u1 ), xAPD ⋅ = ⋅F2− 1 (u2 ), u1 , u2 ∈ [0, 1]2 ,
(27)
∈ [0, 1]2 ,
3.3. Goodness-of-fit tests
(24)
In order to compare the performance of the selected I-Ⅳ asymmetric
models and truncated copulas, the corrected Akaike Information Crite­
rion (AICc) is applied for judging the goodness-of-fit,
where u1 and u2 are the transformed variables in the copula domain. By
relating Eq. (24) to Eq. (23), the following formula is derived
{
[
]}
(
)
π db
4
3β
1
+
15tan
F1− 1 (u1 ) ≤ 1.17F2− 1 (u2 )2 A 1 − exp − 1.5
.
1.17F2− 1 (u2 )2
AICc = − 2l(p) + 2p +
2(p + 1)(p + 2)
,
n− p− 2
(28)
where n is the total number of data samples, p is the number of estimated
parameters in the model, and l(p) is the maximum log-likelihood for the
(25)
10
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
Table 7
The estimated parameters, total log-likelihood, and AICc for the bivariate ocean data.
Copula Type
I. One parameter Archimedea-n
copula
II. Product copulas approach
Parameter estimate
Total loglikelihood
No. of
parameter
AICc
Gumbel
γ = 1.582
− 10980
5
21970.02a
Clayton
γ = 0.64
− 11728
5
23466.02
Frank
γ = 4.207
− 11103
5
22216.02
Gumbel-Clayton Type1
γ1 = 3.911, γ2 = 6.15;
θ11 = 0.961, θ12 = 0.390;
θ21 = 0.039, θ22 = 0.610
− 10468
10
20956.06
γ1 = 3.778, γ2 = 35.482; θ11 = 0.983, θ12 = 0.418; θ21 =
0.017, θ22 = 0.582
− 10715
10
21450.06
− 10214
10
20448.06a
Gumbel-Clayton Type2
γ1 = 28.507, γ2 = 14.011; θ11 = 0.012, θ12 = 0.557; θ21 =
0.988, θ22 0.443
γ1 = 2.507, γ2 = 7.011;
θ11 = 0.771, θ12 = 0.690;
θ21 = 0.229, θ22 = 0.310;
α11 = − 0.129, α12 = -0.592;
α21 = 0.129, α22 = 0.592
− 10317
14
20662.1
Gumbel-Frank Type2
γ1 = 3.007, γ2 = 37.011;
θ11 = 0.971, θ12 = 0.734;
θ21 = 0.029, θ22 = 0.266;
α11 = − 0.025, α12 = -0.457;
α21 = 0.025, α22 = 0.457
− 10409
14
20846.1
Frank-Clayton Type2
γ1 = 25.318, γ2 = 32.946;
θ11 = 0.205, θ12 = 0.717;
θ21 = 0.795, θ22 = 0.283;
α11 = − 0.724, α12 = -0.157;
α21 = 0.724, α22 = 0.157
− 10045
14
20111a
− 11002
8
22020.04
γ1 = 4.9 07, γ2 = 32.061; θ1 = 0.862, θ2 = 0.797
− 11981
8
23978.04
− 10850
8
Gumbel-LCC
γ = 1.601; p0 = 0.981,
p1 = 0.009, p2 = 0.010
− 10975
8
21966a
Clayton-LCC
γ = 0.618; p0 = 0.999,
p1 = 1.443e-07, p2 = 0.001
− 11721
8
23459.04
γ = 4.1292; p0 = 0.333,
p1 = 0.333, p2 = 0.333
− 11994
8
24004.04
Gumbel-Frank Type1
Frank-Clayton Type1
Gumbel-Clayton Type3
Gumbel-Frank Type3
Frank-Clayton Type3
III. linear convex combination-s
(LCC) copulas
Frank-LCC
γ1 = 4.105, γ2 = 4.352; θ1 = 0.782, θ2 = 0.557;
γ1 = 6.507, γ2 = 12.011; θ1 = 0.653, θ2 = 0.998
21716a
IV. Skewed copula
Skewed-Gaussian
β1 = − 0.589, β2 = 5.035;
β = [ − 5.431, 3.966]
− 10888
8
21792.04
V. Truncated copula
Truncated Gumbel
αfeasible = 1.023
− 10934
5
21879
Truncated Frank-Clayton
Type1
αfeasible = 1.012
− 10190
10
20400
Truncated Frank-Clayton
Type2
αfeasible = 1.002
− 10041
14
20111
Truncated Frank-Clayton
Type3
αfeasible = 1.003
− 10846
8
21708
Truncated Gumbel-LCC
αfeasible = 1.022
− 10931
8
21878
Truncated SkewedGaussian
αfeasible = 1.663
− 9853
8
19722a
a
γ = 1.582
γ1 = 28.507, γ2 = 14.011; θ11 = 0.012, θ12 = 0.557; θ21 =
0.988, θ22 = 0.443
θ11 = 0.205, θ12 = 0.717;
θ21 = 0.795, θ22 = 0.283;
α11 = − 0.724, α12 = -0.157;
α21 = 0.724, α22 = 0.157
γ1 = 6.507, γ2 = 12.011; θ1 = 0.653, θ2 = 0.998
γ = 1.601; p0 = 0.981,
p1 = 0.009, p2 = 0.010
β1 = − 0.589, β2 = 5.035;
β = [ − 5.431, 3.966]
The best model in each type.
11
Ocean Engineering 244 (2022) 110226
P. Ma and Y. Zhang
Table 8
gof
The Sn values in each model before and after truncation.
Copulas
Gumbela
Clayton
Frank
Gumbel-Clayton Type1
Gumbel-Frank Type1
Frank-Clayton Type1a
before Truncated
After Truncated
Copulas
before Truncated
After Truncated
Copulas
before Truncated
After Truncated
0.433
0.265
Gumbel-Clayton Type2
0.285
0.226
Gumbel-LCCa
1.338
0.239
2.207
1.414
Gumbel-Frank Type2
0.221
0.201
Clayton-LCC
2.317
1.437
0.513
0.465
Frank-Clayton Type2a
0.197
0.184
Frank-LCC
1.507
0.386
0.292
0.230
Gumbel-Clayton Type3
0.586
0.317
Skewed-Gaussian
0.415
0.170
0.228
0.192
Gumbel-Frank Type3
0.618
0.421
0.199
0.185
Frank-Clayton Type3a
0.387
0.254
a
The best model in each type.
model. The AIC approach is to find the model that best explains the data
but contains the least number of parameters. A model with a lower AIC
value is considered to be better. In the situation of the small data sample,
AIC transforms into AICc. As n rises, AICc converges to AIC.
According to the conclusions from a large number of simulations
obtained by Genest et al. (2009), the most useful and cogent form of this
goodness-of-fit test is based on the Cramér-von Mises statistic, which is
given by
∫
n
∑
( (
)
(
))2
Sngof =
Cn Ui,n − Cθn Ui,n , (29)
n(Cn (u) − Cθn (u))2 dCn (u) =
[0,1]d
where Cn (.) is the empirical copula and θn is the estimator of θ estimated
gof
from the data. The lower statistical value Sn represents the closer this
fitted asymmetric copula model is to the original data, indicating that
the model fits better.
4. Results and discussion
The parameters, total log-likelihood, and the value of AICc for each
type of copula model have been calculated and shown in Table 7. The
gof
goodness-of-fit statistic Sn as mentioned in Eq. (29) are computed and
i=1
Fig. 11. Scatterplots of simulated data from the best non-truncated copulas in different types for (WVHT, APD). (Red line is the breaking wave limit in copula
domain.). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
12
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Ocean Engineering 244 (2022) 110226
Fig. 12. Scatterplots of simulated data from the best truncated copulas in different types for (WVHT, APD).(Red line is the breaking wave limit in copula domain.).
(For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
presented in Table 8. A total number of 5000 data points were simulated
optimal model is the truncated skewed-Gaussian copula according to
Tables 7 and 8. The αfeasible in Table 7 is the inverse of the integral of the
probability density of the feasible space in the copula domain according
to Eq. (16). In the same type of asymmetric copula models, a larger value
of αfeasible implies a weaker ability to capture the asymmetry caused by
breaking wave limits. In other words, the probability in the infeasible
space is not close to zero in these copulas. Therefore, it can be observed
that the skewed-Gaussian copula is less capable of modeling the asym­
metric dependency in this bivariate case.
gof
in the calculation of Sn . The best models in each category, which
include various individual functions, are noted in the table.
For the non-truncated copulas fitted to the randomized preprocessed
data set (WVHT, APD), Tables 7 and 8 all show that the best asymmetric
copula model is the Frank-Clayton Type 2 model. As compared to the
symmetric copulas, nearly all asymmetric copulas have a lower AICc
gof
value and Sn in the results. However, this does not indicate that all nontruncated asymmetric copulas are a better choice. Both the selection of
base copula and the way of constructing the asymmetric copula affect
the performance of the asymmetric copula model. For example, when
using products of copulas to create an asymmetric copula, the combi­
nation of Gumbel and Frank copulas performs worse than other com­
binations, regardless of which individual functions are used, as can be
observed in Tables 7 and 8. Furthermore, the asymmetric copula con­
structed by-products and the skewed-Gaussian copula outperform the
other two forms of copulas, as can be seen from the comparison in Ta­
bles 7 and 8.
gof
From Table 8, the statistical value Sn
of the truncated Gumbel
gof
copula is 0.265, which is smaller than the Sn of the non-truncated
asymmetric Gumbel-Clayton copula of 0.292 and slightly larger than
gof
the Sn of the non-truncated asymmetric Frank-Clayton Type 1 copula of
gof
0.199. It is worth mentioning that the Sn of the truncated Gumbel
copula is already much smaller than that of LCC copulas. However, most
of the asymmetric copula models are better than the symmetric copula
model in fitting bivariate ocean data before the application of the
truncated method. From the engineering point of view, a model that is
more computationally efficient while satisfying accuracy is preferred.
The performance of the truncated symmetric copula model such as the
truncated Gumbel copula performs better than some asymmetric models
gof
For the truncated copula, both the AICc and the statistical values Sn
show that the performance of symmetric and asymmetric copula models
was greatly improved by the truncated method in Tables 7 and 8. The
13
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Ocean Engineering 244 (2022) 110226
Fig. 13. The contour plot between empirical copula and best-fitted copula models for (WVHT, APD)
14
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
Fig. 14. Estimated 1- year return period value from different copula models.
such as model III in Table 8, furthermore many asymmetric models still
need many parameters and there is a risk of over-parameterization.
Besides that, some of the parameters are not necessary to be applied.
Thus, in this situation, the truncated symmetric copulas could be a better
model with the premise of satisfying accuracy.
In the final part of this work, a brief comparison of the performance
of truncated copula and non-truncated copula in predicting extreme
values is presented. This can be done by comparing simulated data from
the constructed models with the original data. The simulated data for
(WVHT, APD) in the copula domain are plotted in Fig. 11 and Fig. 12.
The nonlinear dependencies in the simulated data for (WVHT, APD)
are well presented in Fig. 11. The data simulation matches the AICc
result consistently, which obtains that the Frank-Clayton product copula
is the best choice for modeling (WVHT, APD) in non-truncated copulas.
As for the asymmetric dependency presented in Fig. 11, although the
asymmetric copula model captures the asymmetry between the bivariate
ocean data very well, there are still some points that lie in the infeasible
space as shown in Fig. 11. This is exactly the key drawback of the nontruncated copula. In truncated copula models, instead, it is observed that
a clear truncated red line has divided the domain into two parts and no
simulated data point is within the infeasible space.
The contour plots of the empirical data and simulated data in the
original domain, as seen in Fig. 13, provide even clearer views. The
contour lines are derived from kernel density plots for the joint distri­
bution of (WVHT, APD) given by the optimal copula models in each
type. The contour lines with low values reflect extremes at different tails.
From the comparison of all these plots in Fig. 13, it can be seen the
simulated data from best-fitted copula models in each type has high
similarity to the original data even at for the extremes. The simulated
data basically incorporate the empirical data at the same level of
exceedance probability, especially in the upper tail area, which is a
further evidence that our model does not result in an underestimation of
the results. Furthermore, it can be observed that the truncated skewedGaussian copula model has the best performance.
To further demonstrate the advantage and importance of using
truncated copula model on the offshore structural design, this paper
compares the difference among the 1-year return period value estimated
by the symmetric, asymmetric and truncated copula models (Gumbel,
Frank-Clayton type 2, and skewed Gaussian copula) Since the data
represent 3-h sea states, exceedance probability, γ, for a T-year return
period is calculated as
1
1
γ= =
.
n 365.25 × 24/3 × T
(30)
Therefore, the 1-year return period value is corresponding to an
exceedance probability of 3.42 × 10− 4 . The corresponding results are
shown in Fig. 14. It can be observed that the truncated copula does not
have inaccurate estimations as that from non-truncated copula (e.g. The
return period value of (WVHT, APD) has crossed the breaking wave
limit). It removes incorrect estimates in the infeasible space, and reduces
extreme value estimates in the upper tail domain, thus further improves
the accuracy of the return value estimation. This is of great significance
for the safety and economy of marine engineering.
In summary, it can be concluded that the truncated copula could
effectively model the bivariate ocean data (WVHT, APD). However, it
should be noticed the results can only be interpreted for the investigated
dataset. If we want to study the long-term sea conditions of the area, the
effect of long-term seasonality should be considered in statistical
modeling. The ocean data analysis in this work is only applicable for the
studied geographic ocean area for the specific water depth. If the loca
15
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
tion changed, the truncated space derived based on the breaking wave
limits according to the actual geographical situation has to be recon­
sidered. Additionally, it is also worth noting that, in comparison to
conventional copulas, truncated copulas are much more versatile. As
long as the physical limits (relationship) or constraint functions are
known, the truncated copula model can be easily constructed not limited
to the bivariate ocean data been studied herein.
Finally, it is realized the results of this study could be useful to design
engineers, marine exploiters, or shipowners who work on the open sea.
By offsetting the zero probability space (infeasible space) can analyze
the return period of extreme values more consistent with the real situ­
ation. This is important for engineers to determine the return period of
extreme values of marine parameters when designing marine structures.
More accurate load data can help us to optimize the design of the
structure and thus reduce the cost of the whole construction project.
Furthermore, the empirical results of this study will assist researchers in
more specifically exploiting truncated copula features.
shows the truncated copula models are the most prominent candidate.
This study also found that truncated copula models are more effective at
estimating extreme values than non-truncated copula models. The re­
turn period value obtained from conventional copula model is inaccu­
rate and could be improved by the truncation technique. From the
perspective of modeling accuracy and efficiency, truncated copulas are
expected to have a lot of potential for use in the risk management of
offshore and coastal systems. More applications of the truncated copula
model are to be further explored in the future.
CRediT authorship contribution statement
Pengfei Ma: conceived of the presented idea. verified the analytical
methods. investigate truncated copula supervised the findings of this
work, All authors discussed the results and contributed to the final
manuscript. Yi Zhang: conceived of the presented idea, developed the
theory and performed the computations, investigate truncated copula
supervised the findings of this work.
5. Conclusion
Declaration of competing interest
In this work, a new type of copula, named truncated copula, is
developed and applied to model ocean parameters. The conventional
way of constructing asymmetric copulas as well as the methods for
measuring asymmetric dependencies are discussed. Based on the trun­
cation strategy, the formulation of a truncated copula is introduced. In
an example analysis, the truncated copulas were compared to nontruncated copulas on modeling a bivariate ocean dataset obtained
from the East Hatteras. Different types of copula models covering a wide
range of asymmetric models were tested on the pre-processed random­
ized ocean data. The truncated copula models were discovered to be
more accurate in modeling ocean data. The key advantage of the trun­
cated copula is that the physical limits among the variables can be well
mathematically characterized. Consequently, the goodness-of-fit test
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge the financial support from Na­
tional Natural Science Foundation of China under project number of
Grand No. 51908324&52111540161. The support from Tsinghua Uni­
versity Initiative Scientific Research Program (20213080003) is also
greatly appreciated.
Appendix A. Example of truncated Archimedean copulas with breaking wave limit
This section presents the derivation of basic truncated one parameter Archimedean copula with determined truncated domain Ωfeasible due to
breaking wave limits as mentioned in Section 3.2. Engineers could look up the table and use it directly in ocean data modeling and design.
Truncated one parameter Archimedean copulas with truncated domain Ωfeasible
Based on the previous definition of the truncated copula density function in Eq (16). The bivariate truncated Archimedean copula equation with
the feasible space Ωfeasible can be derived as follows.
1. For the Gumbel family
∫
U1
∫
U2
Ctruncated ​ gumbel (U1 , U2 ) = αfeasible ​ gumbel
0
cgumbel (u1 , u2 )du2 du1 ; u, U ∈ Ωfeasible ,
(31)
cclayton (u1 , u2 )du2 du1 ; u, U ∈ Ωfeasible ,
(32)
0
2. For the Clayton family
∫
U1
∫
Ctruncated ​ clayton (U1 , U2 ) = αfeasible ​ clayton
0
U2
0
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Ocean Engineering 244 (2022) 110226
3. For the Frank family
∫
U1
∫
Ctruncated ​ frank (U1 , U2 ) = αfeasible ​ frank
U2
(33)
cfrank (u1 , u2 )du2 du1 ; u, U ∈ Ωfeasible .
0
0
The αfeasible can be given as
αfeasible = ∮
1
; u ∈ Ωfeasible .
c(u)du
Ωfeasible
(34)
The truncation factor αfeasible for different correlation coefficients (Kendall’s tau) in these copulas is estimated in Table A.1.
The value of the parameter αfeasible can be easily used to construct the truncated copula based on a given value of correlation coefficient according to
Eq. (16). It is worth noting that the feasible space Ωfeasible is derived from the breaking wave limit according to Eq. (23). When the geographical
location changes, the water depth and the slope of the bank in Eq. (23) would also need to change. The variation of the breaking wave limit with the
change of water depth and slope of the bank is shown in Fig. A.1 . For using the truncated copula developed in this study, one should realize such
physical limit changes and ensure the feasible space is accurately captured in the copula model.
Table A.1
αfeasible with different correlation coefficients
Copula
Kendall’s tau
Parameter θ of Copula
αfeasible
Gumbel
τ = 0.1
τ = 0.2
τ = 0.3
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
τ = 0.9
θ = 1.111
θ = 1.250
θ = 1.429
θ = 1.667
θ =2
θ = 2.5
θ = 3.333
θ =5
θ = 10
1.072
1.044
1.033
1.018
1.008
1.003
1.001
1.000
1.000
Clayton
τ = 0.1
τ = 0.2
τ = 0.3
τ = 0.4
τ = 0.5
τ = 0.6
τ = 0.7
τ = 0.8
τ = 0.9
θ
θ
θ
θ
θ
θ
θ
θ
θ
= 0.222
= 0.5
= 0.857
= 1.333
=2
=3
= 4.667
=8
= 18
1.033
1.028
1.033
1.028
1.021
1.018
1.014
1.007
1.004
Frank
τ
τ
τ
τ
τ
τ
τ
τ
τ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
θ
θ
θ
θ
θ
θ
θ
θ
θ
= 0.907
= 1.861
= 2.917
= 4.161
= 5.736
= 7.930
= 11.412
= 18.192
= 38.281
1.077
1.055
1.040
1.028
1.015
1.011
1.005
1.002
1.001
=
=
=
=
=
=
=
=
=
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Ocean Engineering 244 (2022) 110226
Fig. A.1. The variation of the breaking wave limit with the change of slope of the bank in different water depth
18
P. Ma and Y. Zhang
Ocean Engineering 244 (2022) 110226
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