Journal articles: 'Copula-based dependence'

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Yang, Jingping, Zhijin Chen, Fang Wang, and Ruodu Wang. "COMPOSITE BERNSTEIN COPULAS." ASTIN Bulletin 45, no. 2 (March 11, 2015): 445–75. http://dx.doi.org/10.1017/asb.2015.1.

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AbstractCopula function has been widely used in insurance and finance for modeling inter-dependency between risks. Inspired by the Bernstein copula put forward by Sancetta and Satchell (2004, Econometric Theory, 20, 535–562), we introduce a new class of multivariate copulas, the composite Bernstein copula, generated from a composition of two copulas. This new class of copula functions is able to capture tail dependence, and it has a reproduction property for the three important dependency structures: comonotonicity, countermonotonicity and independence. We introduce an estimation procedure based on the empirical composite Bernstein copula which incorporates both prior information and data into the estimation. Simulation studies and an empirical study on financial data illustrate the advantages of the empirical composite Bernstein copula estimation method, especially in capturing tail dependence.

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Vaz de Melo Mendes, Beatriz, and Cecília Aíube. "Copula based models for serial dependence." International Journal of Managerial Finance 7, no. 1 (February 22, 2011): 68–82. http://dx.doi.org/10.1108/17439131111109008.

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Lee, Eun-Joo, Noah Klumpe, Jonathan Vlk, and Seung-Hwan Lee. "Modeling Conditional Dependence of Stock Returns Using a Copula-based GARCH Model." International Journal of Statistics and Probability 6, no. 2 (February 13, 2017): 32. http://dx.doi.org/10.5539/ijsp.v6n2p32.

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Investigating dependence structures of stocks that are related to one another should be an important consideration in managing a stock portfolio, among other investment strategies. To capture various dependence features, we employ copula to overcome the limitations of traditional linear correlations. Financial time series data is typically characterized by volatility clustering of returns that influences an estimate of a stock’s future price. To deal with the volatility and dependence of stock returns, this paper provides procedures of combining a copula with a GARCH model which leads to the construction of a multivariate distribution. Using the copula-based GARCH approach that describes the tail dependences of stock returns, we carry out Monte Carlo simulations to predict a company’s movements in the stock market. The procedures are illustrated in two technology stocks, Apple and Samsung.

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Czado, Claudia, and Thomas Nagler. "Vine Copula Based Modeling." Annual Review of Statistics and Its Application 9, no. 1 (March 7, 2022): 453–77. http://dx.doi.org/10.1146/annurev-statistics-040220-101153.

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With the availability of massive multivariate data comes a need to develop flexible multivariate distribution classes. The copula approach allows marginal models to be constructed for each variable separately and joined with a dependence structure characterized by a copula. The class of multivariate copulas was limited for a long time to elliptical (including the Gaussian and t-copula) and Archimedean families (such as Clayton and Gumbel copulas). Both classes are rather restrictive with regard to symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by building a multivariate model using only bivariate building blocks. This gives rise to highly flexible models that still allow for computationally tractable estimation and model selection procedures. These features made vine copula models quite popular among applied researchers in numerous areas of science. This article reviews the basic ideas underlying these models, presents estimation and model selection approaches, and discusses current developments and future directions.

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Zhou, Jin Yu, Kui Zhou Sun, and Xiu Lian Li. "Reliability Modeling for Symmetric Structure Systems Based on Copulas." Advanced Materials Research 118-120 (June 2010): 319–26. http://dx.doi.org/10.4028/www.scientific.net/amr.118-120.319.

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As a new tool of statistical analysis, Copula is introduced to build reliability model for structural system consisting of identical components, by which the complex feature of failure dependence can be depicted. Aiming at symmetric structure systems, typical failure-dependence mechanism of components is discussed firstly. Considering the failure-dependence mechanism, modeling steps based on Gauss Copula and Archimedean Copulas are put forward, in which the twin stress, components strength are chosen as the basic variables and the safety margins are chosen as the analytic variables. Compared with Gauss Copula, Archimedean Copulas have powerful capability of describing the failure-dependence mechanism owing to the adjustable parameters can be determined according to the rank correlation coefficient and the information about the critical failure point. Archimedean Copula-based reliability models are applicable to non-normal situations. A numerical example is given to show that the new method is reasonable and feasible. Copula-based reliability models can give a new path for reliability analysis of complex systems with failure-dependence.

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El Hannoun, Wafaa, Salah-Eddine El Adlouni, and Abdelhak Zoglat. "Vine-Copula-Based Quantile Regression for Cascade Reservoirs Management." Water 13, no. 7 (March 31, 2021): 964. http://dx.doi.org/10.3390/w13070964.

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This paper features an application of Regular Vine (R-vine) copulas, a recently developed statistical tool to assess composite risk. Copula-based dependence modelling is a popular tool in conditional risk assessment, but is usually applied to pairs of variables. By contrast, Vine copulas provide greater flexibility and permit the modelling of complex dependency patterns using a wide variety of bivariate copulas which may be arranged and analysed in a tree structure to explore multiple dependencies. This study emphasises the use of R-vine copulas in an analysis of the co-dependencies of five reservoirs in the cascade of the Saint-John River basin in Eastern Canada. The developed R-vine copulas lead to the joint and conditional return periods of maximum volumes, for hydrologic design and cascade reservoir management in the basin. The main attraction of this approach to risk modelling is the flexibility in the choice of distributions used to model heavy-tailed marginals and co-dependencies.

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Song, Shuai, Jing Liu, Yongjiu Qian, Fang Zhang, and Gang Wu. "Dependence analysis on the seismic demands of typical components of a concrete continuous girder bridge with the copula technique." Advances in Structural Engineering 21, no. 12 (February 14, 2018): 1826–39. http://dx.doi.org/10.1177/1369433218757234.

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The seismic reliability of a bridge system is significantly affected by the dependence among typical bridge components. This study demonstrates the process of using a copula technique to describe the nonlinear dependence among component seismic demands isolated from their marginal probability distributions. A suite of 100 bridge-ground motion samples were developed with the Latin hypercube sampling approach and bin approach. Based on the incremental dynamic analysis, the tail dependence among component seismic demands at different intensity levels was analyzed with the best-fitting copula function selected by the minimum distance method. In the longitudinal direction, the dependence increased first and then decreased with the ground motion intensity, while the dependence slightly decreased in the transverse direction. At low-intensity levels, the upper tail dependence among components was strong in both directions. At high-intensity levels, the upper and lower tail dependences were weak in the longitudinal direction, while the upper and lower tail dependences were strong in the transverse direction. Compared to the linear correlation coefficient, the copula technique provides an efficient way to describe the tail dependence among component seismic demands and can be used extensively in the seismic reliability analysis of the bridge system.

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Klüppelberg, Claudia, Stephan Haug, and Gabriel Kuhn. "Copula structure analysis based on extreme dependence." Statistics and Its Interface 8, no. 1 (2015): 93–107. http://dx.doi.org/10.4310/sii.2015.v8.n1.a9.

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Ozdemir, Onur, Thomas G. Allen, Sora Choi, Thakshila Wimalajeewa, and Pramod K. Varshney. "Copula Based Classifier Fusion Under Statistical Dependence." IEEE Transactions on Pattern Analysis and Machine Intelligence 40, no. 11 (November 1, 2018): 2740–48. http://dx.doi.org/10.1109/tpami.2017.2774300.

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Liebscher, Eckhard. "Copula-Based Dependence Measures For Piecewise Monotonicity." Dependence Modeling 5, no. 1 (August 28, 2017): 198–220. http://dx.doi.org/10.1515/demo-2017-0012.

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Abstract The aim of the present paper is to develop and examine association coefficients which can be helpfully applied in the framework of regression analysis. The construction of the coeffiecients is connected with the well-known Spearman coeffiecient and extensions of it (see Liebscher [5]). The proposed coeffiecient measures the discrepancy between the data points and a function which is strictly increasing on one interval and strictly decreasing in the remaining domain.We prove statements about the asymptotic behaviour of the estimated coeffiecient (convergence rate, asymptotic normality).

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Alqawba, Mohammed, Dimuthu Fernando, and Norou Diawara. "A Class of Copula-Based Bivariate Poisson Time Series Models with Applications." Computation 9, no. 10 (October 18, 2021): 108. http://dx.doi.org/10.3390/computation9100108.

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A class of bivariate integer-valued time series models was constructed via copula theory. Each series follows a Markov chain with the serial dependence captured using copula-based transition probabilities from the Poisson and the zero-inflated Poisson (ZIP) margins. The copula theory was also used again to capture the dependence between the two series using either the bivariate Gaussian or “t-copula” functions. Such a method provides a flexible dependence structure that allows for positive and negative correlation, as well. In addition, the use of a copula permits applying different margins with a complicated structure such as the ZIP distribution. Likelihood-based inference was used to estimate the models’ parameters with the bivariate integrals of the Gaussian or t-copula functions being evaluated using standard randomized Monte Carlo methods. To evaluate the proposed class of models, a comprehensive simulated study was conducted. Then, two sets of real-life examples were analyzed assuming the Poisson and the ZIP marginals, respectively. The results showed the superiority of the proposed class of models.

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El Ktaibi, Farid El, Rachid Bentoumi, Nicola Sottocornola, and Mhamed Mesfioui. "Bivariate Copulas Based on Counter-Monotonic Shock Method." Risks 10, no. 11 (October 24, 2022): 202. http://dx.doi.org/10.3390/risks10110202.

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This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

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Fernando, Dimuthu, Mohammed Alqawba, Manar Samad, and Norou Diawara. "Review of Copula for Bivariate Distributions of Zero-Inflated Count Time Series Data." International Journal of Statistics and Probability 11, no. 6 (October 26, 2022): 28. http://dx.doi.org/10.5539/ijsp.v11n6p28.

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The class of bivariate integer-valued time series models, described via copula theory, is gaining popularity in the literature because of applications in health sciences, engineering, financial management and more. Each time series follows a Markov chain with the serial dependence captured using copula-based distribution functions from the Poisson and the zero-inflated Poisson margins. The copula theory is again used to capture the dependence between the two series. However, the efficiency and adaptability of the copula are being challenged because of the discrete nature of data and also in the case of zero-inflation of count time series. Likelihood-based inference is used to estimate the model parameters for simulated and real data with the bivariate integral of copula functions. While such copula functions offer great flexibility in capturing dependence, there remain challenges related to identifying the best copula type for a given application.  This paper presents a survey of the literature on bivariate copula for discrete data with an emphasis on the zero-inflated nature of the modelling. We demonstrate additional experiments on to confirm that the copula has potential as greater research area.

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Fernando, Dimuthu, Mohammed Alqawba, Manar Samad, and Norou Diawara. "Review of Copula for Bivariate Distributions of Zero-Inflated Count Time Series Data." International Journal of Statistics and Probability 11, no. 6 (October 30, 2022): 52. http://dx.doi.org/10.5539/ijsp.v11n6p52.

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The class of bivariate integer-valued time series models, described via copula theory, is gaining popularity in the literature because of applications in health sciences, engineering, financial management and more. Each time series follows a Markov chain with the serial dependence captured using copula-based distribution functions from the Poisson and the zero-inflated Poisson margins. The copula theory is again used to capture the dependence between the two series. However, the efficiency and adaptability of the copula are being challenged because of the discrete nature of data and also in the case of zero-inflation of count time series. Likelihood-based inference is used to estimate the model parameters for simulated and real data with the bivariate integral of copula functions. While such copula functions offer great flexibility in capturing dependence, there remain challenges related to identifying the best copula type for a given application.  This paper presents a survey of the literature on bivariate copula for discrete data with an emphasis on the zero-inflated nature of the modelling. We demonstrate additional experiments on to confirm that the copula has potential as greater research area.

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Saali, Tariq, Mhamed Mesfioui, and Ani Shabri. "Multivariate Extension of Raftery Copula." Mathematics 11, no. 2 (January 12, 2023): 414. http://dx.doi.org/10.3390/math11020414.

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This paper introduces a multivariate extension of Raftery copula. The proposed copula is exchangeable and expressed in terms of order statistics. Several properties of this copula are established. In particular, the multivariate Kendall’s tau and Spearman’s rho, as well as the density function, of the suggested copula are derived. The lower and upper tail dependence of the proposed copula are also established. The dependence parameter estimator of this new copula is examined based on the maximum likelihood procedure. A simulation study shows a satisfactory performance of the presented estimator. Finally, the proposed copula is successfully applied to a real data set on black cherry trees.

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Liu, D., D. Wang, L. Wang, Y. Chen, X. Chen, and S. Gu. "POME-copula for hydrological dependence analysis." Proceedings of the International Association of Hydrological Sciences 368 (May 6, 2015): 251–56. http://dx.doi.org/10.5194/piahs-368-251-2015.

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Abstract. Hydrological multivariate analysis has been widely studied using copula-based modelling, in which marginal distribution inference is one of the key issues. The main object of this study is to discuss the applicability of the principle of maximum entropy (POME) in marginal distribution inference, thus to develop a POME-copula framework to analyse the dependence of hydrological variables. Marginal distributions are derived with the POME approach before bivariate copulas constructed with corresponding parameters estimated by the dependence of the derived margins. The proposed POME-copula has been employed in hydrological dependence analyses, with the annual maximum streamflow and water level collected from the Yangtze River, and the monthly streamflow from the Yellow River. Results show that the POME-copula method performs well in capturing dependence patterns of various hydrological variables.

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Shih, Jia-Han, Yoshihiko Konno, Yuan-Tsung Chang, and Takeshi Emura. "Copula-Based Estimation Methods for a Common Mean Vector for Bivariate Meta-Analyses." Symmetry 14, no. 2 (January 18, 2022): 186. http://dx.doi.org/10.3390/sym14020186.

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Traditional bivariate meta-analyses adopt the bivariate normal model. As the bivariate normal distribution produces symmetric dependence, it is not flexible enough to describe the true dependence structure of real meta-analyses. As an alternative to the bivariate normal model, recent papers have adopted “copula” models for bivariate meta-analyses. Copulas consist of both symmetric copulas (e.g., the normal copula) and asymmetric copulas (e.g., the Clayton copula). While copula models are promising, there are only a few studies on copula-based bivariate meta-analysis. Therefore, the goal of this article is to fully develop the methodologies and theories of the copula-based bivariate meta-analysis, specifically for estimating the common mean vector. This work is regarded as a generalization of our previous methodological/theoretical studies under the FGM copula to a broad class of copulas. In addition, we develop a new R package, “CommonMean.Copula”, to implement the proposed methods. Simulations are performed to check the proposed methods. Two real dataset are analyzed for illustration, demonstrating the insufficiency of the bivariate normal model.

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Ibragimov, Rustam. "COPULA-BASED CHARACTERIZATIONS FOR HIGHER ORDER MARKOV PROCESSES." Econometric Theory 25, no. 3 (June 2009): 819–46. http://dx.doi.org/10.1017/s0266466609090720.

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In this paper, we obtain characterizations of higher order Markov processes in terms of copulas corresponding to their finite-dimensional distributions. The results are applied to establish necessary and sufficient conditions for Markov processes of a given order to exhibitm-dependence,r-independence, or conditional symmetry. The paper also presents a study of applicability and limitations of different copula families in constructing higher order Markov processes with the preceding dependence properties. We further introduce new classes of copulas that allow one to combine Markovness withm-dependence orr-independence in time series.

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Longla, Martial. "On dependence structure of copula-based Markov chains." ESAIM: Probability and Statistics 18 (2014): 570–83. http://dx.doi.org/10.1051/ps/2013052.

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Zhuang, De Dong. "Tail Dependence Structure between Carbon Emission Allowances Returns Based on Copulas." Applied Mechanics and Materials 397-400 (September 2013): 726–30. http://dx.doi.org/10.4028/www.scientific.net/amm.397-400.726.

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This paper has focus on analyzing tail dependence structure between EUA spots returns and futures returns based on copula approach, which EUA spots negotiated on BlueNext and futures negotiated on European Climate Exchange within the European Union Emission Trading Scheme (EU ETS) during the Phase II. According to the generalized Pareto distribution (GPD) and different Copula functions, the research shows that Gumbel Copula based on the GPD marginal distribution can indicate the tail dependence structure of EUA spots returns and futures returns accurately, i.e. the dependence between upper-tails of EUA spot and Dec10 is stronger than that of lower-tails of them. In other words, EUA spots and futures are more likely to soar together than slump together during the Phase II.

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Liu, Shisong, and Shaojun Li. "Multi-model D-vine copula regression model with vine copula-based dependence description." Computers & Chemical Engineering 161 (May 2022): 107788. http://dx.doi.org/10.1016/j.compchemeng.2022.107788.

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Saminger-Platz, Susanne, Anna Kolesárová, Adam Šeliga, Radko Mesiar, and Erich Peter Klement. "New results on perturbation-based copulas." Dependence Modeling 9, no. 1 (January 1, 2021): 347–73. http://dx.doi.org/10.1515/demo-2021-0116.

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Abstract A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.

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Zhang, Xiaoqin, Hongbin Zhu, Bo Li, Ruihan Wu, and Jun Jiang. "Power Transformer Diagnosis Based on Dissolved Gases Analysis and Copula Function." Energies 15, no. 12 (June 7, 2022): 4192. http://dx.doi.org/10.3390/en15124192.

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The traditional DGA (Dissolved Gas Analysis) diagnosis method does not consider the dependence between fault characteristic gases and uses the relationship between gas ratio coding and fault type to make the decision. As a tool of the dependence mechanism between variables, a copula function can effectively analyze the correlation between variables when it cannot determine whether the linear correlation coefficient can correctly measure the correlation between variable relationships. In this paper, the edge variable of a copula function is selected from the fault characteristic gas of a transformer, and the distribution type of the edge variable is fitted at the same time. Then, Bayesian estimation with the Gaussian residual likelihood function is used to fit the parameters of a copula function and a copula function is selected to describe the optimal dependence of the fault characteristic gas of transformer. The relationship between a copula function and the state of transformer is studied. The results show that the copula function boundary with hydrocarbon gas as edge variable can divide the transformer as healthy or defective state. When the cumulative distribution probability (CDF) value of the dissolved gas in the oil in the copula function is close to 0.8, the fluctuation of its gas concentration leads to a sharp change in the probability. Therefore, the analysis of dissolved gas in oil based on a copula function can be used as a powerful technical solution for oil-immersed power transformer fault diagnosis.

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Girard, Stéphane. "Transformation of a copula using the associated co-copula." Dependence Modeling 6, no. 1 (December 1, 2018): 298–308. http://dx.doi.org/10.1515/demo-2018-0017.

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AbstractWe investigate the properties of a new transformation of copulas based on the co-copula and an univariate function. It is shown that several families in the copula literature can be interpreted as particular outputs of this transformation. Symmetry, association, ordering and dependence properties of the resulting copula are established.

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Trimech, Anyssa. "Time-varying dependence measures: a comparative analysis through wavelet approach." International Journal of Energy Sector Management 11, no. 2 (June 5, 2017): 350–64. http://dx.doi.org/10.1108/ijesm-01-2016-0001.

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Purpose This paper aims to investigate the pattern of dependence between crude oil price and energy consumption of the most important economic sectors in the USA, over different time periods, using monthly data set from January 1986 to July 2014 and a comparative study between linear correlation versus copula correlation as a measure of dependence over the single scale and the multiscale analysis. Design/methodology/approach The proposed method is based on the multiresolution analysis which gives more extensive and detailed description of the dependence price-consumption pattern over different periods of time. Findings The empirical results show that the dependence between variables is strongly sensitive to the time varying and generally increasing with time scale. In particular, the Pearson coefficients are less than the dependence copula measures. The single-scale analysis covers many time-varying dependences which are made clear, flexible and comprehensive by the description given by the multiscale approach. It explains better the structure of relationships between variables and helps understand the variations and improve forecasts of the crude oil price and energy consumption over different time scales. Originality/value The proposed methodology offers the opportunity to construct dynamic management strategies by taking into account the multiscale nature of crude oil price and consumption relationship. Moreover, the paper uses wavelets as a relatively new and powerful tool for statistical analysis in addition to the copula technique that allows a new understanding of variable correlation. The paper will be of interest not only for academics in the field of data dependencies analysis but also for fund managers and market investors.

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Liang, Zhicheng, Junwei Wang, and Kin Keung Lai. "Dependence Structure Analysis and VaR Estimation Based on China’s and International Gold Price: A Copula Approach." International Journal of Information Technology & Decision Making 19, no. 01 (January 2020): 169–93. http://dx.doi.org/10.1142/s0219622019500445.

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Since 2013, China has become the world’s largest gold producer and consumer. To gain the corresponding global pricing power in gold, many actions have been taken by China in recent years, including the International Board at Shanghai Gold Exchange, Shanghai-Hong Kong Gold Connect and Shanghai Gold Fix. Our work studies the dependence structure between China’s and international gold price and examines whether these moves are changing the dependence structure. We use GARCH-copula models to detect the dynamic dependence and tail dependence. The research period is set to contain the Financial Crisis in 2008, the dramatical plunge of gold price in 2013 and a series of black swan events in 2016. The empirical study shows that some event driven dependence structure breaks are statistically insignificant. And the time-varying Symmetrized Joe-Clayton copula is the best copula to model the dependence structure based on AIC value. Finally, an example of applications of this dependence structure is given by estimating the VaR of an equally weighted portfolio with a simulation-based method.

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önalan, ömer. "The modeling of extreme stochastic dependence using copulas and extreme value theory: case study from energy prices." Global Journal of Mathematical Analysis 5, no. 2 (June 5, 2017): 29. http://dx.doi.org/10.14419/gjma.v5i2.7256.

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In this paper, we investigate the properties of tail dependence with an approach which is based on the copula models and extreme value theory to obtain a joint distribution function of extreme events and to quantify the dependence between random variables. To achieve this objective, we quantify the large co-movements between the random variables returns which are based on the data set daily quotes of exceeds the threshold value of random variables. In this study, stochastic dependence was modeled by the copulas which it provides a good approach for constructing multivariate probability distributions with flexible marginal’s and different forms of dependence. Choosing the right copula is very important in modeling. The multivariate distributions are easily simulated using the copulas. Finally we can describe the copula family which correctly represents the dependence. To demonstrate the usefulness of the proposed models, we confine our analysis to big price changes of energy commodity spot prices. The empirical findings demonstrated that the copula model which is combined the extreme value theory is a good approach to model the together extreme large changes.

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Lai, Yujie, and Yibo Hu. "A Study on Systematic Risks of U.S. and China Stock Markets Based on Markov Copula." Advances in Education, Humanities and Social Science Research 1, no. 1 (May 9, 2022): 154. http://dx.doi.org/10.56028/aehssr.1.1.154.

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In this paper, Markov SJC copula model is constructed based on the daily data of standard & Poor's index and Shanghai Shenzhen 300 index, and the systematic risk of American and Chinese stock market is empirically analyzed. The results show that SJC copula can well depict the systematic risk of American and Chinese stock market, the risk dependence has obvious tail asymmetry characteristics, and the probability of low risk dependence is higher than that of high risk dependence.

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Latif, Shahid, and Slobodan P. Simonovic. "Trivariate Joint Distribution Modelling of Compound Events Using the Nonparametric D-Vine Copula Developed Based on a Bernstein and Beta Kernel Copula Density Framework." Hydrology 9, no. 12 (December 7, 2022): 221. http://dx.doi.org/10.3390/hydrology9120221.

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Low-lying coastal communities are often threatened by compound flooding (CF), which can be determined through the joint occurrence of storm surges, rainfall and river discharge, either successively or in close succession. The trivariate distribution can demonstrate the risk of the compound phenomenon more realistically, rather than considering each contributing factor independently or in pairwise dependency relations. Recently, the vine copula has been recognized as a highly flexible approach to constructing a higher-dimensional joint density framework. In these, the parametric class copula with parametric univariate marginals is often involved. Its incorporation can lead to a lack of flexibility due to parametric functions that have prior distribution assumptions about their univariate marginal and/or copula joint density. This study introduces the vine copula approach in a nonparametric setting by introducing Bernstein and Beta kernel copula density in establishing trivariate flood dependence. The proposed model was applied to 46 years of flood characteristics collected on the west coast of Canada. The univariate flood marginal distribution was modelled using nonparametric kernel density estimation (KDE). The 2D Bernstein estimator and beta kernel copula estimator were tested independently in capturing pairwise dependencies to establish D-vine structure in a stage-wise nesting approach in three alternative ways, each by permutating the location of the conditioning variable. The best-fitted vine structure was selected using goodness-of-fit (GOF) test statistics. The performance of the nonparametric vine approach was also compared with those of vines constructed with a parametric and semiparametric fitting procedure. Investigation revealed that the D-vine copula constructed using a Bernstein copula with normal KDE marginals performed well nonparametrically in capturing the dependence of the compound events. Finally, the derived nonparametric model was used in the estimation of trivariate joint return periods, and further employed in estimating failure probability statistics.

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Martey, Emmanuel Nii, and Nii Attoh-Okine. "Modeling tamping recovery of track geometry using the copula-based approach." Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 232, no. 8 (February 28, 2018): 2079–96. http://dx.doi.org/10.1177/0954409718757556.

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Assessing and maintaining track geometry within acceptable limits are key components of railroad infrastructure maintenance operations. Track geometry conditions have a significant influence on rider comfort and safety. To maintain the ride quality and safety of the track, maintenance activities pertaining to track geometry, such as tamping, are performed. Tamping enhances the track geometry quality but fails to return the track geometry to an as-good-as-new condition. Majority of studies have evaluated tamping recovery using deterministic techniques, which assume that tamping recovery is dependent on the track geometry quality prior to tamping. However, they fail to capture the uncertainty of the recovery values. Probabilistic approaches are increasingly being used to account for the uncertainty but fail to model the underlying dependence between the variables, which may exhibit nonlinear dependences such as tail or asymmetric dependence. To accurately model the tamping recovery phenomenon, this research employs the copula models in combining arbitrary marginal distributions to form a joint multivariate distribution with the underlying dependence. Copula models are used to estimate the tamping recovery of track geometry parameters such as surface (longitudinal level), alignment, cross level, gage, and warp.

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Liu, Guannan, Wei Long, Xinyu Zhang, and Qi Li. "DETECTING FINANCIAL DATA DEPENDENCE STRUCTURE BY AVERAGING MIXTURE COPULAS." Econometric Theory 35, no. 4 (September 10, 2018): 777–815. http://dx.doi.org/10.1017/s0266466618000270.

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A mixture copula is a linear combination of several individual copulas that can be used to generate dependence structures not belonging to existing copula families. Because different pairs of markets may exhibit quite different dependence structures in empirical studies, mixture copulas are useful in modeling the dependence in financial data. Therefore, rather than selecting a single copula based on certain criteria, we propose using a model averaging approach to estimate financial data dependence structures in a mixture copula framework. We select weights (for averaging) by a J-fold Cross-Validation procedure. We prove that the model averaging estimator is asymptotically optimal in the sense that it minimizes the squared estimation loss. Our simulation results show that the model averaging approach outperforms some competing methods when the working mixture model is misspecified. Using 12 years of data on daily returns from four developed economies’ stock indexes, we show that the model averaging approach more accurately estimates their dependence structures than some competing methods.

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Wu, Xinyu, Meng Zhang, Mengqi Wu, and Hao Cui. "Economic Policy Uncertainty and Conditional Dependence between China and U.S. Stock Markets." Complexity 2022 (January 7, 2022): 1–9. http://dx.doi.org/10.1155/2022/8137932.

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In this paper, we investigate the impact of economic policy uncertainty (EPU) on the conditional dependence between China and U.S. stock markets by employing the Copula-mixed-data sampling (Copula-MIDAS) framework. In the case of EPU, we consider the global EPU (GEPU), the American EPU (AEPU), and the China EPU (CEPU). The empirical analysis based on the Shanghai Stock Exchange Composite (SSEC) index in China and the S&P 500 index in the U.S. shows that the tail dependence between China and U.S. stock markets is symmetrical, and the t Copula outperforms alternative Copulas in terms of in-sample goodness of fit. In particular, we find that the t Copula-MIDAS model with EPU dominates the traditional time-varying t Copula in terms of in-sample fitting. Moreover, we observe that both the GEPU and AEPU have a significantly positive impact on the conditional dependence between China and U.S. stock markets, whereas CEPU has no significant impact. The tail dependence between China and U.S. stock markets exhibits an increasing trend, particularly in the recent years.

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Syuhada, Khreshna, and Arief Hakim. "Modeling risk dependence and portfolio VaR forecast through vine copula for cryptocurrencies." PLOS ONE 15, no. 12 (December 23, 2020): e0242102. http://dx.doi.org/10.1371/journal.pone.0242102.

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Risk in finance may come from (negative) asset returns whilst payment loss is a typical risk in insurance. It is often that we encounter several risks, in practice, instead of single risk. In this paper, we construct a dependence modeling for financial risks and form a portfolio risk of cryptocurrencies. The marginal risk model is assumed to follow a heteroscedastic process of GARCH(1,1) model. The dependence structure is presented through vine copula. We carry out numerical analysis of cryptocurrencies returns and compute Value-at-Risk (VaR) forecast along with its accuracy assessed through different backtesting methods. It is found that the VaR forecast of returns, by considering vine copula-based dependence among different returns, has higher forecast accuracy than that of returns under prefect dependence assumption as benchmark. In addition, through vine copula, the aggregate VaR forecast has not only lower value but also higher accuracy than the simple sum of individual VaR forecasts. This shows that vine copula-based forecasting procedure not only performs better but also provides a well-diversified portfolio.

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Pouliasis, George, Gina Alexandra Torres-Alves, and Oswaldo Morales-Napoles. "Stochastic Modeling of Hydroclimatic Processes Using Vine Copulas." Water 13, no. 16 (August 5, 2021): 2156. http://dx.doi.org/10.3390/w13162156.

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The generation of synthetic time series is important in contemporary water sciences for their wide applicability and ability to model environmental uncertainty. Hydroclimatic variables often exhibit highly skewed distributions, intermittency (that is, alternating dry and wet intervals), and spatial and temporal dependencies that pose a particular challenge to their study. Vine copula models offer an appealing approach to generate synthetic time series because of their ability to preserve any marginal distribution while modeling a variety of probabilistic dependence structures. In this work, we focus on the stochastic modeling of hydroclimatic processes using vine copula models. We provide an approach to model intermittency by coupling Markov chains with vine copula models. Our approach preserves first-order auto- and cross-dependencies (correlation). Moreover, we present a novel framework that is able to model multiple processes simultaneously. This method is based on the coupling of temporal and spatial dependence models through repetitive sampling. The result is a parsimonious and flexible method that can adequately account for temporal and spatial dependencies. Our method is illustrated within the context of a recent reliability assessment of a historical hydraulic structure in central Mexico. Our results show that by ignoring important characteristics of probabilistic dependence that are well captured by our approach, the reliability of the structure could be severely underestimated.

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Rusyda, Hasna Afifah, Achmad Zabar Soleh, Lienda Noviyanti, Anna Chadidjah, and Fajar Indrayatna. "Utilization Copula in Determination of Shallot Insurance Premium Based on Regional Harvest Results." EKSAKTA: Journal of Sciences and Data Analysis 20, no. 2 (October 1, 2020): 160–66. http://dx.doi.org/10.20885/eksakta.vol1.iss2.art11.

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Abstract: Shallot is one of the highest-yielding horticultural crops in Indonesia and has the tendency to increase the profits of farmers in Indonesia. But until now in Indonesia there is no insurance for horticultural crops other than corn, whereas the shallot farmers face various sources of risk such as weather changes, pest attacks, or other technical factors that ultimately lead to uncertainty of agricultural yields (revenue risk). To overcome this loss, insurance companies can make products based on shallot yields and shallot market prices. Therefore it is essential to grasp the distribution of risk variables (shallot yields and shallot market prices) that interact simultaneously, not separate from one another. Omitting dependencies among risk variables can cause biased risk estimation. Copula can model the non-linear dependencies and can identify the structure of the dependencies between variables. The suitable copula for modeling yield and price risk of shallot is simulated to compute the premium. Result show that clayton copula is suitable for dependence modelling between risk variables.

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Bildirici, Melike, and Özgür Ömer Ersin. "Regime-Switching Fractionally Integrated Asymmetric Power Neural Network Modeling of Nonlinear Contagion for Chaotic Oil and Precious Metal Volatilities." Fractal and Fractional 6, no. 12 (November 27, 2022): 703. http://dx.doi.org/10.3390/fractalfract6120703.

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This paper aims at analyzing nonlinear dependence between fractionally integrated, chaotic precious metal and oil prices and volatilities. With this respect, the Markov regime-switching fractionally integrated asymmetric power versions of generalized autoregressive conditional volatility copula (MS-FIAPGARCH-copula) method are further extended to multi-layer perceptron (MLP)-based neural networks copula (MS-FIAPGARCH-MLP-copula). The models are utilized for modeling dependence between daily oil, copper, gold, platinum and silver prices, covering a period from 1 January 1990–25 March 2022. Kolmogorov and Shannon entropy and the largest Lyapunov exponents reveal uncertainty and chaos. Empirical findings show that: i. neural network-augmented nonlinear MS-FIAPGARCH-MLP-copula displayed significant gains in terms of forecasts; ii. asymmetric and nonlinear processes are modeled effectively with the proposed model, iii. important insights are derived with the proposed method, which highlight nonlinear tail dependence. Results suggest, given long memory and chaotic structures, that policy interventions must be kept at lowest levels.

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Dohi, Tadashi, and Hiroyuki Okamura. "Failure-Correlated Opportunity-based Age Replacement Models." International Journal of Reliability, Quality and Safety Engineering 27, no. 02 (October 4, 2019): 2040008. http://dx.doi.org/10.1142/s0218539320400082.

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In this paper, we extend the existing opportunity-based age replacement policies by taking account of dependency between the failure time and the arrival time of a replacement opportunity for one-unit system. Based on the bivariate probability distribution function of the failure time and the arrival time of the opportunity, we focus on two opportunity-based age replacement problems and characterize the cost-optimal age replacement policies which minimize the relevant expected costs, with the hazard gradient, which is a vector-valued bivariate hazard rate. Through numerical examples with the Farlie–Gumbel–Morgenstern bivariate copula and the Gaussian bivariate copula having the general marginal distributions, we investigate the dependence of correlation between the failure time and the opportunistic replacement time on the age replacement policies.

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Kumar, Pranesh. "Statistical Dependence: Copula Functions and Mutual Information Based Measures." Journal of Statistics Applications & Probability 1, no. 1 (March 1, 2012): 1–14. http://dx.doi.org/10.12785/jsap/010101.

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DETTE, HOLGER, KARL F. SIBURG, and PAVEL A. STOIMENOV. "A Copula-Based Non-parametric Measure of Regression Dependence." Scandinavian Journal of Statistics 40, no. 1 (February 20, 2012): 21–41. http://dx.doi.org/10.1111/j.1467-9469.2011.00767.x.

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Fernandez, Viviana. "Copula-based measures of dependence structure in assets returns." Physica A: Statistical Mechanics and its Applications 387, no. 14 (June 2008): 3615–28. http://dx.doi.org/10.1016/j.physa.2008.02.055.

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Xie, Yuan-tao, Juan Yang, Chong-guang Jiang, Zi-yu Cai, and Joshua Adagblenya. "Incidence, Dependence Structure of Disease, and Rate Making for Health Insurance." Mathematical Problems in Engineering 2018 (August 12, 2018): 1–13. http://dx.doi.org/10.1155/2018/4265801.

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In order to analyze the two goals under the national strategy of “Healthy China”, this paper attempts to solve the problem of coverage rate and guarantee level of health insurance, as well as the rational allocation of full life cycle health insurance resources. This paper uses pair copula to model the dependence of different disease incidence and proposes an actuarial model for rate making in health insurance based on the dependence captured by pair copula. These are far more accurate than any other model and more proper for covering a basket of several different diseases. The data for the paper was drawn from the experience incidence table of major diseases (malignant tumor, acute myocardial infarction, and stroke sequelae) from the ages 0-65 years in the Chinese life insurance industry. Extending the hypothesis of independence in actuarial modeling, the authors comprehensively use a hierarchical copula theory to extract the dependence structure of risk variable in insurance. The classification rate making technology and survival analysis method in traditional actuarial pricing were also considered. This paper applied the generalized linear model, which is commonly used in nonlife insurance pricing for empirical study of health insurance rate making. The authors discovered that the incidence of major diseases and the single premium rate calculated by the generalized linear model under HAC dependence structure were both significantly different from that calculated by the Manchester United method without dependency. The authors also stated that the rate based on the generalized linear model under HAC dependence structure was a bit different from that without dependency but both were generally the same as that of Care Expert in PICC Health. The underestimation or overestimation of systematic risks and the distortion of the rate system can be eliminated if we combine risk dependence into modeling.

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Aminuddin Jafry, Nurul Hanis, Ruzanna Ab Razak, and Noriszura Ismail. "Authors: Nurul Hanis Aminuddin Jafry ; Ruzanna Ab Razak ; Noriszura Ismail." Journal of Social Sciences Research, SPI6 (December 26, 2018): 646–52. http://dx.doi.org/10.32861/jssr.spi6.646.652.

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Studies on dependence between stock markets are important because of their implications on the process of decision-making in investment. Many previous studies measure the dependence between markets using static copula. However, in recent years, time-varying copula has been used as an alternative for measuring dependence due to its capability of capturing time-varying dependence between markets. This study uses both static and time-varying copulas to measure the dependence structure between Malaysia and major stock markets (US, UK and Japan) based on the sample data from year 2007 Q1 until year 2017 Q3. The results reveal that the best model for all pairs of indices is the time-varying SJC copula, which also reveals that the Malaysia-US pair has the weakest dependence structure compared to other pairs. In terms of lower and upper tails, the Malaysia-UK and the Malaysia-Japan pairs have the strongest dependence structure respectively. Evidence from this research suggests that diversifications involving Malaysia and US stock markets are not effective.

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Ma, Huizi, Lin Lin, Han Sun, and Yue Qu. "Research on the Dependence Structure and Risk Spillover of Internet Money Funds Based on C-Vine Copula and Time-Varying t-Copula." Complexity 2021 (August 24, 2021): 1–11. http://dx.doi.org/10.1155/2021/3941648.

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Internet money funds (IMFs) are the most widely involved products in the Internet financial products market. This research utilized the C-vine copula model to study the risk dependence structure of IMFs and then introduces the time-varying t-copula model to analyze the risk spillover of diverse IMFs. The results show the following: (1) The risks of Internet-based IMFs, bank-based IMFs, and fund-based IMFs have obvious dependence structure, and the degree of risk dependence among different categories of IMFs is significantly different. (2) There are risk spillover effects among diverse IMFs, and their risk dependence relationship is characterized by cyclical feature. (3) The risk spillover effect among diverse IMFs is pronounced, and dynamic risk dependence between IMFs is characterized by synchronization.

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Mensah, Prince Osei, and Anokye M. Adam. "Copula-Based Assessment of Co-Movement and Tail Dependence Structure Among Major Trading Foreign Currencies in Ghana." Risks 8, no. 2 (June 1, 2020): 55. http://dx.doi.org/10.3390/risks8020055.

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This paper examines the joint movement and tail dependence structure between the pair of foreign exchange rates (EUR, USD and GBP) against the GHS, using daily exchange rates data expressed in GHS per unit of foreign currencies (EUR, USD and GBP) between the time range of 24 February 2009 and 19 December 2019. We use different sets of both static (time-invariant) and time-varying copulas with different levels of dependence and tail dependence measures, and the study results reveal positive dependence between all exchange rates pairs, though the dependencies for EUR-USD and GBP-USD pairs are not as strong as the EUR-GBP pair. The findings also reveal symmetric tail dependence, and dependence evolves over time. Notwithstanding this, the asymmetric tail dependence copulas provide evidence of upper tail dependence. We compare the copula results to DCC(1,1)-GARCH(1,1) model result and find the copula to be more sensitive to extreme co-movement between the currency pairs. The afore-mentioned findings, therefore, offer forex market players the opportunity to relax in hoarding a particular foreign currency in anticipation of domestic currency depreciation.

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Bahraoui, Tarik, Taoufik Bouezmarni, and Jean-François Quessy. "Testing the symmetry of a dependence structure with a characteristic function." Dependence Modeling 6, no. 1 (December 1, 2018): 331–55. http://dx.doi.org/10.1515/demo-2018-0019.

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AbstractThis paper proposes competing procedures to the tests of symmetry for bivariate copulas of Genest, Nešlehová and Quessy (2012). To this end, the null hypothesis of symmetry is expressed in terms of the copula characteristic function that uniquely determines the copula of a given bivariate population with continuous marginal distributions. Then, test statistics based on L2 weighted distances computed from an empirical version of the copula characteristic function are proposed. Their asymptotic behavior is derived under the null hypothesis as well as under general alternatives. In particular, it is established that these rank statistics behave asymptotically as first-order degenerate V-statistics under the null hypothesis and this large-sample representation is exploited in order to provide suitably adapted multiplier bootstrap versions for the computation of p-values. The simulations that are reported show that the new tests are more powerful than the competing methods based on the empirical copula introduced by Genest, Nešlehová and Quessy (2012).

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Yao, Can Zhong, Bo Yi Sun, and Ji Nan Lin. "A study of correlation between investor sentiment and stock market based on Copula model." Kybernetes 46, no. 3 (March 6, 2017): 550–71. http://dx.doi.org/10.1108/k-10-2016-0297.

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Purpose This paper aims to capture tail dependence between sentiment index and Shanghai composite index (SCI) by proposing a sentiment index based on text mining. Design/methodology/approach Online text mining and the Copula model were used in this study. Findings First, the paper finds herding effect in the expression of investors’ sentiment from online text data, and the usage occurrence frequency of most vocabulary is less correlative with SCI. Second, given these two features, the paper uses weighted divide-and-conquer algorithm to construct a sentiment index. Finally, because of multivariate non-Gaussian joint distribution between them, the paper uses the Copula model to detect their tail dependences, and finds that both upper and lower tail dependences could have a significant influence between positive sentiment and SCI, with a higher probability on the upper one. Additionally, only the upper tail dependence exhibits the significant influence between negative sentiment and SCI. Originality/value This paper proposes a framework of constructing investment sentiment index with the weighted conquer-and-divide algorithm, and characterizes tail dependence between sentiment index and SCI. The implication can measure the environment of investment market of China and provide an empirical ground for bandwagon effect and bargain shopper effect.

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Wang, Mao-Xin, Duruo Huang, Gang Wang, Wenqi Du, and Dian-Qing Li. "Vine Copula-Based Dependence Modeling of Multivariate Ground-Motion Intensity Measures and the Impact on Probabilistic Seismic Slope Displacement Hazard Analysis." Bulletin of the Seismological Society of America 110, no. 6 (June 30, 2020): 2967–90. http://dx.doi.org/10.1785/0120190244.

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ABSTRACT Multivariate normality of logarithmic intensity measures (IMs) is conventionally assumed in earthquake engineering applications. This article introduces a vine copula approach as a useful tool for multivariate modeling of IMs. This approach provides a flexible way to decompose a joint distribution into individual marginal distributions and multiple dependences characterized by a cascade of bivariate copulas (pair-copulas), whereas the conventional multivariate normality can be considered as a special case of the vine copula model. Based on the Next Generation Attenuation-West1 database and various combinations of ground-motion prediction equations (GMPEs), the optimal dependence structures among peak ground acceleration, peak ground velocity, and Arias intensity, as well as that for spectral accelerations at four periods, are identified. The joint normality assumption for the two vector sets of logarithmic IMs is examined from the perspective of copula theory. The results illustrate that the normality assumption is generally adequate for bivariate IMs but may not be optimal for multivariate IMs. Using the same set of GMPEs (developed by the same researchers) may improve the joint normality for logarithmic IMs. Furthermore, the impact of dependence structures among IMs on probabilistic seismic slope displacement hazard analysis is explored. The results indicate that using the same Pearson correlation coefficients but different dependence structures for IMs produces different hazard results and this difference is generally enlarged with increasing hazard levels. As hazard difference from different dependence structures is generally not significant, the multivariate normality distribution for logarithmic IMs is judged to be an acceptable assumption in engineering practice. Alternatively, engineers may make a choice between the joint normal distribution and the vine copula tool depending on the specific situation because of the better generality of the latter.

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AghaKouchak, Amir. "Entropy–Copula in Hydrology and Climatology." Journal of Hydrometeorology 15, no. 6 (December 1, 2014): 2176–89. http://dx.doi.org/10.1175/jhm-d-13-0207.1.

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Abstract The entropy theory has been widely applied in hydrology for probability inference based on incomplete information and the principle of maximum entropy. Meanwhile, copulas have been extensively used for multivariate analysis and modeling the dependence structure between hydrologic and climatic variables. The underlying assumption of the principle of maximum entropy is that the entropy variables are mutually independent from each other. The principle of maximum entropy can be combined with the copula concept for describing the probability distribution function of multiple dependent variables and their dependence structure. Recently, efforts have been made to integrate the entropy and copula concepts (hereafter, entropy–copula) in various forms to take advantage of the strengths of both methods. Combining the two concepts provides new insight into the probability inference; however, limited studies have utilized the entropy–copula methods in hydrology and climatology. In this paper, the currently available entropy–copula models are reviewed and categorized into three main groups based on their model structures. Then, a simple numerical example is used to illustrate the formulation and implementation of each type of the entropy–copula model. The potential applications of entropy–copula models in hydrology and climatology are discussed. Finally, an example application to flood frequency analysis is presented.

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Sugimoto, T., A. Bárdossy, G. S. S. Pegram, and J. Cullmann. "Investigation of hydrological time series using copulas for detecting catchment characteristics and anthropogenic impacts." Hydrology and Earth System Sciences Discussions 12, no. 9 (September 10, 2015): 9157–203. http://dx.doi.org/10.5194/hessd-12-9157-2015.

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Abstract. Global climate change can have impacts on characteristics of rainfall-runoff events and subsequently on the hydrological regime. Meanwhile, the catchment itself changes due to anthropogenic influences. In this context, it can be meaningful to detect the temporal changes of catchments independent from climate change by investigating existing long term discharge records. For this purpose, a new stochastic system based on copulas for time series analysis is introduced. While widely used time series models are based on linear combinations of correlations assuming a Gaussian behavior of variables, a statistical tool like copula has the advantage to scrutinize the dependence structure of the data in the uniform domain independent of the marginal. Two measures in the copula domain are introduced herein: 1. Copula asymmetry is defined for copulas and calculated for discharges; this measure describes the non symmetric property of the dependence structure and differs from one catchment to another due to the intrinsic nature of both runoff and catchment. 2. Copula distance is defined as Cramér-von Mises type distance calculated between two copula densities of different time scales. This measure describes the variability and interdependency of dependence structures similar to variance and covariance, which can assist in identifying the catchment changes. These measures are calculated for 100 years of daily discharges for the Rhine rivers. Comparing the results of copula asymmetry and copula distance between an API and simulated discharge time series by a hydrological model we can show the interesting signals of systematic modifications along the Rhine rivers in the last 30 years.

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Mirbagherijam, Mohammad, Mohammad Nabi Shahiki Tash, Gholamreza Zamanian, and Amir Safari. "Aggregation of underwriting risks in insurance industry of Iran using vine copula." Risk Governance and Control: Financial Markets and Institutions 5, no. 4 (2015): 149–61. http://dx.doi.org/10.22495/rgcv5i4c1art4.

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In this paper, the underwriting risks of the insurance industry of Iran were aggregated using various vine copula classes and historical data of loss ratios which corresponds to each business line. The estimated economic capital (EC) for the entire insurance industry considerably varies across different risk measures and vine copula models. In addition, less than the risk-based capital (RBC) charge assessed based on the standard model of RN69 and amounted to 96,943,391 million of Iran Rials. Therefore, it was concluded that using the Vine copula method and allowing symmetry and tail dependence for pairs of business lines’ risks in the risk aggregation process leads to overestimation of the RBC risk charge, as compared to the estimated results of simple and linear aggregation methods of such standard model. Furthermore, the choice of dependency structure and risk measures have a paramount effect on the aggregate economic capital. Highlights: Estimated aggregated economic capital varies across different risk measures and vine copula models; Selecting the appropriate copula model is an important consideration in risk aggregation process; Using the Vine copula method in the risk aggregation leads to overestimation of the RBC risk charge; The estimated economic capital is less than RBC risk charge calculated under standard model of RN69.

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Journal articles on the topic 'Copula-based dependence'

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