Copula density estimation by total variation penalized likelihood with linear equality constraints

Qu, Leming; Yin, Wotao

doi:10.1016/j.csda.2011.07.016

Cited by 18 publications

(12 citation statements)

References 35 publications

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“…With the suitable specification of marginal probabilities and dependence structures, the Gaussian copula can be derived with the principle of maximum entropy [29]. The entropy copula can also be interpreted as the approximation of the copula, for which other types of copula approximation schemes exist [95,97,134,135], such as shuffle of min copula [136,137], Bernstein copula [138,139], checkerboard copula [31,134] or others based on splines or kernels [140][141][142]. Moreover, the entropy based bivariate copula can also be integrated in the vine structure to derive the vine copula [32], which is particularly attractive in modeling the flexible dependence in higher dimensions when parametric copulas fall short in this case [88,132].…”

Section: Discussionmentioning

confidence: 99%

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

Hao

Singh

2015

Entropy

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Abstract:Entropy is a measure of uncertainty and has been commonly used for various applications, including probability inferences in hydrology. Copula has been widely used for constructing joint distributions to model the dependence structure of multivariate hydrological random variables. Integration of entropy and copula theories provides new insights in hydrologic modeling and analysis, for which the development and application are still in infancy. Two broad branches of integration of the two concepts, entropy copula and copula entropy, are introduced in this study. On the one hand, the entropy theory can be used to derive new families of copulas based on information content matching. On the other hand, the copula entropy provides attractive alternatives in the nonlinear dependence measurement even in higher dimensions. We introduce in this study the integration of entropy and copula theories in the dependence modeling and analysis to illustrate the potential applications in hydrology and water resources.

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Section: Discussionmentioning

confidence: 99%

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

Hao

Singh

2015

Entropy

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“…A copula is a multivariate probability distribution with uniform marginals. It has emerged as an useful tool for modeling stochastic dependencies allowing the separation of dependence modeling from the given marginals [57]. Based on this formulation we obtain pdf representations for the joint response-response and response-excitation at different time instants.…”

Section: Correlation Structure Between Two-time Statisticsmentioning

confidence: 99%

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Joo

Sapsis

2016

Probabilistic Engineering Mechanics

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We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics, based on copula functions that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to the unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-offreedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations.

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“…There are also spline-or wavelet-based approximation methods, but most of them are only discussed in the two-dimensional case. Likewise, in [12], the authors discuss a penalized nonparametrical maximum likelihood method in the two-dimensional case. A detailed survey of literature about nonparametrical copula density estimation can be found in [6].…”

Section: Copula Density Estimation As An Inverse Problemmentioning

confidence: 99%

Nonparametric Copula Density Estimation Using a Petrov–Galerkin Projection

Uhlig

Unger

2015

Innovations in Quantitative Risk Management

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Nonparametrical copula density estimation is a meaningful tool for analyzing the dependence structure of a random vector from given samples. Usually kernel estimators or penalized maximum likelihood estimators are considered. We propose solving the Volterra integral equationof the given copula C. In the statistical framework, the copula C is not available and we replace it by the empirical copula of the pseudo samples, which converges to the unobservable copula C for large samples. Hence, we can treat the copula density estimation from given samples as an inverse problem and consider the instability of the inverse operator, which has an important impact if the input data of the operator equation are noisy. The wellknown curse of high dimensions usually results in huge nonsparse linear equations after discretizing the operator equation. We present a Petrov-Galerkin projection for the numerical computation of the linear integral equation. A special choice of test and ansatz functions leads to a very special structure of the linear equations, such that we are able to estimate the copula density also in higher dimensions.

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Copula density estimation by total variation penalized likelihood with linear equality constraints

Cited by 18 publications

References 35 publications

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

Nonparametric Copula Density Estimation Using a Petrov–Galerkin Projection

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