On the quantification and efficient propagation of imprecise probabilities with copula dependence

Zhang, Jiaxin; Shields, Michael

doi:10.1016/j.ijar.2020.04.002

On the quantification and efficient propagation of imprecise probabilities with copula dependence

Journal Article · Thu Apr 30 00:00:00 EDT 2020 · International Journal of Approximate Reasoning

DOI:https://doi.org/10.1016/j.ijar.2020.04.002· OSTI ID:1651315

^[1]; Shields, Michael ^[2]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Johns Hopkins Univ., Baltimore, MD (United States)

This paper addresses the problem of quantification and propagation of uncertainties associated with dependence modeling when data for characterizing probability models are limited. Practically, the system inputs are often assumed to be mutually independent or correlated by a multivariate Gaussian distribution. However, this subjective assumption may introduce bias in the response estimate if the real dependence structure deviates from this assumption. In this work, we overcome this limitation by introducing a flexible copula dependence model to capture complex dependencies. Here, a hierarchical Bayesian multimodel approach is proposed to quantify uncertainty in dependence model-form and model parameters that result from small data sets. This approach begins by identifying, through Bayesian multimodel inference, a set of candidate marginal models and their corresponding model probabilities, and then estimating the uncertainty in the copula-based dependence structure, which is conditional on the marginals and their parameters. The overall uncertainties integrating marginals and copulas are probabilistically represented by an ensemble of multivariate candidate densities. A novel importance sampling reweighting approach is proposed to efficiently propagate the overall uncertainties through a computational model. Through an example studying the influence of constituent properties on the out-of-plane properties of transversely isotropic E-glass fiber composites, we show that the composite property with copula-based dependence model converges to the true estimate as data set size increases, while an independence or arbitrary Gaussian correlation assumption leads to a biased estimate.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: US Department of the Navy, Office of Naval Research (ONR); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1651315

Alternate ID(s):: OSTI ID: 1693549

Journal Information:: International Journal of Approximate Reasoning, Journal Name: International Journal of Approximate Reasoning Journal Issue: NA Vol. 122; ISSN 0888-613X

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (55)

Bayesian model selection for D-vine pair-copula constructions Min, Aleksey; Czado, Claudia Canadian Journal of Statistics, Vol. 39, Issue 2 https://doi.org/10.1002/cjs.10098	journal	May 2011
Truncated regular vines in high dimensions with application to financial data Brechmann, E. C.; Czado, C.; Aas, K. Canadian Journal of Statistics, Vol. 40, Issue 1 https://doi.org/10.1002/cjs.10141	journal	February 2012
An overview of robust Bayesian analysis Berger, James O.; Moreno, Elías; Pericchi, Luis Raul Test, Vol. 3, Issue 1 https://doi.org/10.1007/BF02562676	journal	June 1994
Fuzzy sets Zadeh, L. A. Information and Control, Vol. 8, Issue 3 https://doi.org/10.1016/S0019-9958(65)90241-X	journal	June 1965
Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration Li, R.; Ghanem, R. Probabilistic Engineering Mechanics, Vol. 13, Issue 2 https://doi.org/10.1016/S0266-8920(97)00020-9	journal	April 1998
Towards a unified theory of imprecise probability Walley, Peter International Journal of Approximate Reasoning, Vol. 24, Issue 2-3 https://doi.org/10.1016/S0888-613X(00)00031-1	journal	May 2000
Different methods are needed to propagate ignorance and variability Ferson, Scott; Ginzburg, Lev R. Reliability Engineering & System Safety, Vol. 54, Issue 2-3 https://doi.org/10.1016/S0951-8320(96)00071-3	journal	November 1996
A D-vine copula quantile regression approach for the prediction of residential heating energy consumption based on historical data Niemierko, Rochus; Töppel, Jannick; Tränkler, Timm Applied Energy, Vol. 233-234 https://doi.org/10.1016/j.apenergy.2018.10.025	journal	January 2019
The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets Zhang, Jiaxin; Shields, Michael D. Computer Methods in Applied Mechanics and Engineering, Vol. 334 https://doi.org/10.1016/j.cma.2018.01.045	journal	June 2018
Coastal flooding: A copula based approach for estimating the joint probability of water levels and waves Masina, Marinella; Lamberti, Alberto; Archetti, Renata Coastal Engineering, Vol. 97 https://doi.org/10.1016/j.coastaleng.2014.12.010	journal	March 2015
Long-term performance assessment and design of offshore structures Zhang, Yi; Beer, Michael; Quek, Ser Tong Computers & Structures, Vol. 154 https://doi.org/10.1016/j.compstruc.2015.02.029	journal	July 2015
Roles of load temporal correlation and deterioration-load dependency in structural time-dependent reliability Wang, Cao; Zhang, Hao Computers & Structures, Vol. 194 https://doi.org/10.1016/j.compstruc.2017.09.001	journal	January 2018
An efficient D-vine copula-based coupling uncertainty analysis for variable-stiffness composites Li, Qidi; Cai, Yong; Wang, Hu Composite Structures, Vol. 219 https://doi.org/10.1016/j.compstruct.2019.03.067	journal	July 2019
Selecting and estimating regular vine copulae and application to financial returns Dißmann, J.; Brechmann, E. C.; Czado, C. Computational Statistics & Data Analysis, Vol. 59 https://doi.org/10.1016/j.csda.2012.08.010	journal	March 2013
Dependence modelling in ultra high dimensions with vine copulas and the Graphical Lasso Müller, Dominik; Czado, Claudia Computational Statistics & Data Analysis, Vol. 137 https://doi.org/10.1016/j.csda.2019.02.007	journal	September 2019
Sklar's theorem in an imprecise setting Montes, Ignacio; Miranda, Enrique; Pelessoni, Renato Fuzzy Sets and Systems, Vol. 278 https://doi.org/10.1016/j.fss.2014.10.007	journal	November 2015
Imprecise random variables, random sets, and Monte Carlo simulation Fetz, Thomas; Oberguggenberger, Michael International Journal of Approximate Reasoning, Vol. 78 https://doi.org/10.1016/j.ijar.2016.06.012	journal	November 2016
Bivariate quadratic copula constructions Wisadwongsa, Suttisak; Tasena, Santi International Journal of Approximate Reasoning, Vol. 92 https://doi.org/10.1016/j.ijar.2017.10.001	journal	January 2018
A scenario generation method based on the mixture vine copula and its application in the power system with wind/hydrogen production Qiu, Yibin; Li, Qi; Pan, Yuru International Journal of Hydrogen Energy, Vol. 44, Issue 11 https://doi.org/10.1016/j.ijhydene.2018.09.179	journal	February 2019
Pair-copula constructions of multiple dependence Aas, Kjersti; Czado, Claudia; Frigessi, Arnoldo Insurance: Mathematics and Economics, Vol. 44, Issue 2 https://doi.org/10.1016/j.insmatheco.2007.02.001	journal	April 2009
Tail dependence functions and vine copulas Joe, Harry; Li, Haijun; Nikoloulopoulos, Aristidis K. Journal of Multivariate Analysis, Vol. 101, Issue 1 https://doi.org/10.1016/j.jmva.2009.08.002	journal	January 2010
Model selection in sparse high-dimensional vine copula models with an application to portfolio risk Nagler, T.; Bumann, C.; Czado, C. Journal of Multivariate Analysis, Vol. 172 https://doi.org/10.1016/j.jmva.2019.03.004	journal	July 2019
An innovating analysis of the Nataf transformation from the copula viewpoint Lebrun, Régis; Dutfoy, Anne Probabilistic Engineering Mechanics, Vol. 24, Issue 3 https://doi.org/10.1016/j.probengmech.2008.08.001	journal	July 2009
A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas Torre, Emiliano; Marelli, Stefano; Embrechts, Paul Probabilistic Engineering Mechanics, Vol. 55 https://doi.org/10.1016/j.probengmech.2018.08.001	journal	January 2019
Efficient Monte Carlo resampling for probability measure changes from Bayesian updating Zhang, Jiaxin; Shields, Michael D. Probabilistic Engineering Mechanics, Vol. 55 https://doi.org/10.1016/j.probengmech.2018.10.002	journal	January 2019
Propagation of uncertainty through multivariate functions in the framework of sets of probability measures Fetz, Th.; Oberguggenberger, M. Reliability Engineering & System Safety, Vol. 85, Issue 1-3 https://doi.org/10.1016/j.ress.2004.03.004	journal	July 2004
Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data Sankararaman, Shankar; Mahadevan, Sankaran Reliability Engineering & System Safety, Vol. 96, Issue 7 https://doi.org/10.1016/j.ress.2011.02.003	journal	July 2011
A copula-based sampling method for data-driven prognostics Xi, Zhimin; Jing, Rong; Wang, Pingfeng Reliability Engineering & System Safety, Vol. 132 https://doi.org/10.1016/j.ress.2014.06.014	journal	December 2014
Copula-based decomposition approach for the derivative-based sensitivity of variance contributions with dependent variables Wang, Pan; Lu, Zhenzhou; Zhang, Kaichao Reliability Engineering & System Safety, Vol. 169 https://doi.org/10.1016/j.ress.2017.09.012	journal	January 2018
Probability models for data-Driven global sensitivity analysis Hu, Zhen; Mahadevan, Sankaran Reliability Engineering & System Safety, Vol. 187 https://doi.org/10.1016/j.ress.2018.12.003	journal	July 2019
An enhanced copula-based method for data-driven prognostics considering insufficient training units Xi, Zhimin; Zhao, Xiangxue Reliability Engineering & System Safety, Vol. 188 https://doi.org/10.1016/j.ress.2019.03.015	journal	August 2019
Modeling risks in dependent systems: A Copula-Bayesian approach Pan, Yue; Ou, Shenwei; Zhang, Limao Reliability Engineering & System Safety, Vol. 188 https://doi.org/10.1016/j.ress.2019.03.048	journal	August 2019
Aleatory or epistemic? Does it matter? Kiureghian, Armen Der; Ditlevsen, Ove Structural Safety, Vol. 31, Issue 2 https://doi.org/10.1016/j.strusafe.2008.06.020	journal	March 2009
Multi-variate seismic demand modelling using copulas: Application to non-ductile reinforced concrete frame in Victoria, Canada Goda, Katsuichiro; Tesfamariam, Solomon Structural Safety, Vol. 56 https://doi.org/10.1016/j.strusafe.2015.05.004	journal	September 2015
Synthesis of wind tunnel and climatological data for estimating design wind effects: A copula based approach Warsido, Workamaw P.; Bitsuamlak, Girma T. Structural Safety, Vol. 57 https://doi.org/10.1016/j.strusafe.2015.07.004	journal	November 2015
The effect of copulas on time-variant reliability involving time-continuous stochastic processes Rózsás, Árpád; Mogyorósi, Zsuzsa Structural Safety, Vol. 66 https://doi.org/10.1016/j.strusafe.2017.02.004	journal	May 2017
The role of copulas in random fields: Characterization and application Wang, Fan; Li, Heng Structural Safety, Vol. 75 https://doi.org/10.1016/j.strusafe.2018.05.006	journal	November 2018
Imprecise probabilities in engineering analyses Beer, Michael; Ferson, Scott; Kreinovich, Vladik Mechanical Systems and Signal Processing, Vol. 37, Issue 1-2 https://doi.org/10.1016/j.ymssp.2013.01.024	journal	May 2013
On the quantification and efficient propagation of imprecise probabilities resulting from small datasets Zhang, Jiaxin; Shields, Michael D. Mechanical Systems and Signal Processing, Vol. 98 https://doi.org/10.1016/j.ymssp.2017.04.042	journal	January 2018
Failure behavior modeling and reliability estimation of product based on vine-copula and accelerated degradation data Xu, Dan; Xing, Mengli; Wei, Qidong Mechanical Systems and Signal Processing, Vol. 113 https://doi.org/10.1016/j.ymssp.2017.07.035	journal	December 2018
The Kolmogorov-Smirnov Test for Goodness of Fit Massey, Frank J. Journal of the American Statistical Association, Vol. 46, Issue 253 https://doi.org/10.1080/01621459.1951.10500769	journal	March 1951
Predictive inference for bivariate data: Combining nonparametric predictive inference for marginals with an estimated copula Coolen-Maturi, Tahani; Coolen, Frank P. A.; Muhammad, Noryanti Journal of Statistical Theory and Practice, Vol. 10, Issue 3 https://doi.org/10.1080/15598608.2016.1184112	journal	April 2016
The t Copula and Related Copulas Demarta, Stefano; McNeil, Alexander J. International Statistical Review, Vol. 73, Issue 1 https://doi.org/10.1111/j.1751-5823.2005.tb00254.x	journal	April 2005
Analysis of Composite Materials—A Survey Hashin, Z. Journal of Applied Mechanics, Vol. 50, Issue 3 https://doi.org/10.1115/1.3167081	journal	September 1983
Micromechanics Prediction of the Shear Strength of Carbon Fiber/Epoxy Matrix Composites: The Influence of the Matrix and Interface Strengths King, T. R.; Blackketter, D. M.; Walrath, D. E. Journal of Composite Materials, Vol. 26, Issue 4 https://doi.org/10.1177/002199839202600406	journal	April 1992
Measurement of the Out-of-Plane Shear Response of Thick Section Composite Materials Using the V-Notched Beam Specimen Gipple, K. L.; Hoyns, D. Journal of Composite Materials, Vol. 28, Issue 6 https://doi.org/10.1177/002199839402800604	journal	April 1994
Multimodel Inference: Understanding AIC and BIC in Model Selection Burnham, Kenneth P.; Anderson, David R. Sociological Methods & Research, Vol. 33, Issue 2 https://doi.org/10.1177/0049124104268644	journal	November 2004
Maximum likelihood estimation of mixed C-vines with application to exchange rates Czado, Claudia; Schepsmeier, Ulf; Min, Aleksey Statistical Modelling: An International Journal, Vol. 12, Issue 3 https://doi.org/10.1177/1471082X1101200302	journal	May 2012
Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling Schefzik, Roman; Thorarinsdottir, Thordis L.; Gneiting, Tilmann Statistical Science, Vol. 28, Issue 4 https://doi.org/10.1214/13-STS443	journal	November 2013
Sequential Bayesian Model Selection of Regular Vine Copulas Gruber, Lutz; Czado, Claudia Bayesian Analysis, Vol. 10, Issue 4 https://doi.org/10.1214/14-BA930	journal	December 2015
Upper and Lower Probabilities Induced by a Multivalued Mapping Dempster, A. P. The Annals of Mathematical Statistics, Vol. 38, Issue 2 https://doi.org/10.1214/aoms/1177698950	journal	April 1967
Remarks on a Multivariate Transformation Rosenblatt, Murray The Annals of Mathematical Statistics, Vol. 23, Issue 3 https://doi.org/10.1214/aoms/1177729394	journal	September 1952
Modeling Dependence with C- and D-Vine Copulas: The R Package CDVine Brechmann, Eike Christian; Schepsmeier, Ulf Journal of Statistical Software, Vol. 52, Issue 3 https://doi.org/10.18637/jss.v052.i03	journal	January 2013
A New Measure of Rank Correlation Kendall, M. G. Biometrika, Vol. 30, Issue 1/2 https://doi.org/10.2307/2332226	journal	June 1938
Multivariate extreme-value distributions with applications to environmental data Joe, Harry Canadian Journal of Statistics, Vol. 22, Issue 1 https://doi.org/10.2307/3315822	journal	March 1994

Similar Records

Copula cosmology: Constructing a likelihood function

Journal Article · Fri Jan 14 23:00:00 EST 2011 · Physical Review. D, Particles Fields · OSTI ID:21503927

Imprecise global sensitivity analysis using bayesian multimodel inference and importance sampling

Journal Article · Sat Aug 08 20:00:00 EDT 2020 · Mechanical Systems and Signal Processing · OSTI ID:1651313

A Copula-Based Conditional Probabilistic Forecast Model for Wind Power Ramps

Journal Article · Mon May 28 20:00:00 EDT 2018 · IEEE Transactions on Smart Grid · OSTI ID:1457667

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
Bayesian Multimodel Inference
Composites
Copula Theory
Dependence Modeling
Imprecise Probability
Small Data
Uncertainty Quantification

On the quantification and efficient propagation of imprecise probabilities with copula dependence

Citation Formats

References (55)

Similar Records

Related Subjects