scholarly journals Trivariate copula to design coastal structures

2021 ◽  
Vol 21 (1) ◽  
pp. 239-260
Author(s):  
Olivier Orcel ◽  
Philippe Sergent ◽  
François Ropert
Keyword(s):  
Return Period ◽  
Tail Dependence ◽  
Copula Function ◽  
Structure Design ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Practical Case ◽  
Single Factor ◽  
Coastal Structures ◽  
Bivariate Copula

Abstract. Some coastal structures must be redesigned in the future due to rising sea levels caused by climate change. The design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The simplified Defra method that is currently used in particular for European coastal structures makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a single factor. These schematic correlations do not, however, represent all the complexity of the reality because of the use of this single factor. That may lead to damaging errors in coastal structure design. The aim of this paper is therefore to remedy the lack of robustness of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution functions to their univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method. In order to integrate extreme events, we also resort to the notion of tail dependence. The optimal copula parameter is estimated through the analysis of the tail dependence coefficient, the likelihood method and the mean error. The most robust copulas for our practical case with applications in Saint-Malo and Le Havre (in northern France) are the Clayton copula and the survival Gumbel copula. The originality of this paper is the creation of a new and robust trivariate copula with an analysis of the sensitivity to the method of construction and to the choice of the copula. Firstly, we select the best fitting of the bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate function satisfies the mathematical properties of the copula. We finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years. We finally conclude that the choice of the bivariate copula is more important for the accuracy of the trivariate copula than its own construction.

scholarly journals Trivariate copula to design coastal structures

2020 ◽  
Author(s):  
Olivier Orcel ◽  
Philippe Sergent ◽  
François Ropert
Keyword(s):  
Return Period ◽  
Tail Dependence ◽  
Structure Design ◽  
Likelihood Method ◽  
Dependence Structure ◽  
Practical Case ◽  
Coastal Structures ◽  
Copula Parameter ◽  
Bivariate Copula ◽  
Survival Copula

Abstract. Some coastal structures must be redesigned in the future due to rising sea levels caused by global warming. The design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The Defra method that is currently used makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a simple factor. These schematic correlations do not, however, represent all the complexity of the reality and may lead to damaging errors in coastal structure design. The aim of this paper is therefore to remedy the lack of accuracy of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution function to its univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method with a copula parameter that is obtained by the error method. In order to integrate extreme events, we also resort to the notion of tail dependence. We can select the copulas with the same tail dependence as data. In the event of an opposite tail dependence structure, we resort to the survival copula. The tail dependence parameter makes it possible to estimate the optimal copula parameter. The most accurate copulas for our practical case with applications in Saint-Malo and Le Havre (France), are the Clayton normal copula and the Gumbel survival copula. The originality of this paper is the creation of a new and accurate trivariate copula. Firstly, we select the fittest bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate function satisfies the mathematical properties of the copula. We can finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years.

scholarly journals New families of bivariate copulas via unit weibull distortion

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Fadal A.A. Aldhufairi ◽  
Jungsywan H. Sepanski
Keyword(s):  
Tail Dependence ◽  
Copula Function ◽  
Kendall’S Tau ◽  
Kendall's Tau ◽  
Density Distributions ◽  
New Family ◽  
Upper Tail Dependence ◽  
Bivariate Copula

Abstract This paper introduces a new family of bivariate copulas constructed using a unit Weibull distortion. Existing copulas play the role of the base or initial copulas that are transformed or distorted into a new family of copulas with additional parameters, allowing more flexibility and better fit to data. We present a general form for the new bivariate copula function and its conditional and density distributions. The tail behaviors are investigated and indicate the unit Weibull distortion may result in new copulas with upper tail dependence when the base copula has no upper tail dependence. The concordance ordering and Kendall’s tau are derived for the cases when the base copulas are Archimedean, such as the Clayton and Frank copulas. The Loss-ALEA data are analyzed to evaluate the performance of the proposed new families of copulas.

scholarly journals Comparison of Weibull Estimation Methods for Diverse Winds

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ferhat Bingöl
Keyword(s):  
Wind Farm ◽  
Study Data ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Estimation Methods ◽  
Weibull Function ◽  
Wind Speeds ◽  
Low Wind Speed ◽  
Low Wind Speeds ◽  
Wind Distribution

Wind farm siting relies on in situ measurements and statistical analysis of the wind distribution. The current statistical methods include distribution functions. The one that is known to provide the best fit to the nature of the wind is the Weibull distribution function. It is relatively straightforward to parameterize wind resources with the Weibull function if the distribution fits what the function represents but the estimation process gets complicated if the distribution of the wind is diverse in terms of speed and direction. In this study, data from a 101 m meteorological mast were used to test several estimation methods. The available data display seasonal variations, with low wind speeds in different seasons and effects of a moderately complex surrounding. The results show that the maximum likelihood method is much more successful than industry standard WAsP method when the diverse winds with high percentile of low wind speed occur.

scholarly journals Joint return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation

2012 ◽  
Vol 9 (5) ◽  
pp. 6781-6828 ◽  
Author(s):  
S. Vandenberghe ◽  
M. J. van den Berg ◽  
B. Gräler ◽  
A. Petroselli ◽  
S. Grimaldi ◽  
...  
Keyword(s):  
Frequency Analysis ◽  
Return Period ◽  
Three Dimensional ◽  
Distribution Functions ◽  
Return Periods ◽  
Design Variables ◽  
Joint Return ◽  
Design Values ◽  
Multivariate Frequency Analysis ◽  
Joint Return Period

Abstract. Most of the hydrological and hydraulic studies refer to the notion of a return period to quantify design variables. When dealing with multiple design variables, the well-known univariate statistical analysis is no longer satisfactory and several issues challenge the practitioner. How should one incorporate the dependence between variables? How should the joint return period be defined and applied? In this study, an overview of the state-of-the-art for defining joint return periods is given. The construction of multivariate distribution functions is done through the use of copulas, given their practicality in multivariate frequency analysis and their ability to model numerous types of dependence structures in a flexible way. A case study focusing on the selection of design hydrograph characteristics is presented and the design values of a three-dimensional phenomenon composed of peak discharge, volume and duration are derived. Joint return period methods based on regression analysis, bivariate conditional distributions, bivariate joint distributions, and Kendal distribution functions are investigated and compared highlighting theoretical and practical issues of multivariate frequency analysis. Also an ensemble-based method is introduced. For a given design return period, the method chosen clearly affects the calculated design event. Eventually, light is shed on the practical implications of a chosen method.

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

Seismic fragility models of a bridge system based on copula method

2021 ◽  
pp. 875529302110525
Author(s):  
Libo Chen ◽  
Caigui Huang ◽  
Haiqiang Chen ◽  
Zhenfeng Zheng
Keyword(s):  
Seismic Performance ◽  
Marginal Distribution ◽  
Fragility Curves ◽  
Distribution Functions ◽  
System Level ◽  
Likelihood Method ◽  
Seismic Fragility ◽  
Distribution Model ◽  
Copula Models ◽  
Bridge System

Seismic fragility assessment widely uses a probabilistic measure for assessing the seismic performance of structural components or systems. This article proposes a copula-based seismic fragility (CBSF) method to derive the system-level damage probabilities of reinforced concrete bridges and to consider the correlation among seismic demands of components. First, the marginal distribution functions of the random variables are calibrated, and three Archimedean copula models are considered. Subsequently, the relevant parameters of the copula models are estimated using the semi-parametric maximum likelihood method. Next, the damage probabilities of a bridge system are calculated based on the joint distribution model with the most suitable copula model and the calibrated marginal distribution functions. Finally, the CBSF method is used to estimate the damage probability of a simply supported box girder bridge. The performance of the CBSF method is validated by a comparison of fragility curves obtained using the CBSF method and the probabilistic seismic demand analysis (PSDA) method. The comparative results demonstrate that the fragility curves obtained by the CBSF method are better than those obtained using the PSDA method. The proposed CBSF model can serve as a tool for assessing the seismic performance of structures and estimating the economic loss of existing bridge systems.

scholarly journals On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 79 ◽  
Author(s):  
Vadim Semenikhine ◽  
Edward Furman ◽  
Jianxi Su
Keyword(s):  
Risk Model ◽  
Tail Dependence ◽  
Random Variable ◽  
Copula Function ◽  
Background Risk ◽  
Aggregate Risk ◽  
Multivariate Gamma Distribution ◽  
Multivariate Probability Distribution ◽  
Dependent Risks ◽  
Main Message

One way to formulate a multivariate probability distribution with dependent univariate margins distributed gamma is by using the closure under convolutions property. This direction yields an additive background risk model, and it has been very well-studied. An alternative way to accomplish the same task is via an application of the Bernstein–Widder theorem with respect to a shifted inverse Beta probability density function. This way, which leads to an arguably equally popular multiplicative background risk model (MBRM), has been by far less investigated. In this paper, we reintroduce the multiplicative multivariate gamma (MMG) distribution in the most general form, and we explore its various properties thoroughly. Specifically, we study the links to the MBRM, employ the machinery of divided differences to derive the distribution of the aggregate risk random variable explicitly, look into the corresponding copula function and the measures of nonlinear correlation associated with it, and, last but not least, determine the measures of maximal tail dependence. Our main message is that the MMG distribution is (1) very intuitive and easy to communicate, (2) remarkably tractable, and (3) possesses rich dependence and tail dependence characteristics. Hence, the MMG distribution should be given serious considerations when modelling dependent risks.

Lifetime assessment method for products with dependent competing failures based on copula theory

2013 ◽  
Vol 228 (9) ◽  
pp. 1622-1633 ◽  
Author(s):  
Xiangpo Zhang ◽  
Jianzhong Shang ◽  
Xun Chen ◽  
Chunhua Zhang ◽  
Yashun Wang
Keyword(s):  
Statistical Inference ◽  
Failure Modes ◽  
Likelihood Estimation ◽  
Copula Function ◽  
Assessment Method ◽  
Distribution Functions ◽  
Life Testing ◽  
Copula Theory ◽  
Dependent Relationship ◽  
Margin Distribution

Based on copula theory and methods, we construct the dependent relationship between the margin distribution functions of the competing failure modes and their joint distribution function through copula function. With the dependent relationship, we study statistical inference method of the life testing with dependent competing failure modes, and found the maximum likelihood estimation (MLE) model for the parameter estimations to evaluate the lifetime of the products. The results and analysis of case studies prove that sample size, proportion of censored samples, proportion of failure samples with masked failure mode, and copula model types have great impact on the accuracy of the lifetime assessment of the products with dependent competing failure modes. And with appropriate test data and right copula modes, method developed in this paper has very good accuracy for the lifetime assessment with dependent competing failure modes. It provides an effective and accurate way to solve the problems of statistical inference of life testing with dependent competing failure modes, and also an accurate way of lifetime assessment for products.

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