Estimation of high-order moment-independent importance measures for Shapley value analysis

Moment-independent importance measures are increasingly used by practitioners to understand how output uncertainty may be shared between a set of stochastic inputs. Computing Borgonovo's sensitivity indices for a large group of inputs is still a challenging problem due to the curse of dimensionality and it is addressed in this article. An estimation scheme taking the most of recent developments in copula theory is developed. Furthermore, the concept of Shapley value is used to derive new sensitivity indices, which makes the interpretation of Borgonovo's indices much easier. The resulting importance measure offers a double advantage compared with other existing methods since it allows to quantify the impact exerted by one input variable on the whole output distribution after taking into account all possible dependencies and interactions with other variables. The validity of the proposed methodology is established on several analytical examples and the benefits in terms of computational efficiency are illustrated with real-life test cases such as the study of the water flow through a borehole. In addition, a detailed case study dealing with the atmospheric re-entry of a launcher first stage is completed.

[1]  E. Borgonovo.  Measuring Uncertainty Importance: Investigation and Comparison of Alternative Approaches , 2006, Risk analysis : an official publication of the Society for Risk Analysis.

[2]  F. Durante.  Construction of non-exchangeable bivariate distribution functions , 2009 .

[3]  Nils-Bastian Heidenreich,et al.  Bandwidth Selection Methods for Kernel Density Estimation - A Review of Performance , 2010 .

[4]  Zhenzhou Lu,et al.  Monte Carlo simulation for moment-independent sensitivity analysis , 2013, Reliab. Eng. Syst. Saf..

[5]  Alexander J. McNeil,et al.  Quantitative Risk Management: Concepts, Techniques and Tools Revised edition , 2015 .

[6]  J. Mielniczuk,et al.  Estimating the density of a copula function , 1990 .

[7]  Art B. Owen,et al.  Better estimation of small sobol' sensitivity indices , 2012, TOMC.

[8]  Stéphane Girard,et al.  A class of multivariate copulas based on products of bivariate copulas , 2015, J. Multivar. Anal..

[9]  Jérôme Morio,et al.  A nonparametric importance sampling estimator for moment independent importance measures , 2019, Reliab. Eng. Syst. Saf..

[10]  H. Joe.  Families of $m$-variate distributions with given margins and $m(m-1)/2$ bivariate dependence parameters , 1996 .

[11]  A. Frigessi,et al.  Pair-copula constructions of multiple dependence , 2009 .

[12]  G. Venter.  TAILS OF COPULAS , 2013 .

[13]  Art B. Owen,et al.  Sobol' Indices and Shapley Value , 2014, SIAM/ASA J. Uncertain. Quantification.

[14]  Gerhart I. Schuëller,et al.  Time Variant Reliability Analysis Utilizing Response Surface Approach , 1989 .

[15]  Qiao Liu,et al.  A new computational method of a moment-independent uncertainty importance measure , 2009, Reliab. Eng. Syst. Saf..

[16]  Claudia Czado,et al.  Selecting and estimating regular vine copulae and application to financial returns , 2012, Comput. Stat. Data Anal..

[17]  Emanuele Borgonovo,et al.  A new uncertainty importance measure , 2007, Reliab. Eng. Syst. Saf..

[18]  Jean-David Fermanian,et al.  Goodness-of-fit tests for copulas , 2005 .

[19]  F. Gamboa,et al.  Statistical inference for Sobol pick-freeze Monte Carlo method , 2013, 1303.6447.

[20]  Claudia Czado.  Recent Developments in Vine Copula Based Modeling , 2019 .

[21]  Nicolas Gayton,et al.  AK-MCSi: A Kriging-based method to deal with small failure probabilities and time-consuming models , 2018, Structural Safety.

[22]  Barry L. Nelson,et al.  Shapley Effects for Global Sensitivity Analysis: Theory and Computation , 2016, SIAM/ASA J. Uncertain. Quantification.

[23]  E. E. Myshetskaya,et al.  Monte Carlo estimators for small sensitivity indices , 2008, Monte Carlo Methods Appl..

[24]  L. Shapley.  A Value for n-person Games , 1988 .

[25]  Nils-Bastian Heidenreich,et al.  Bandwidth selection for kernel density estimation: a review of fully automatic selectors , 2013, AStA Advances in Statistical Analysis.

[26]  Zhenzhou Lu,et al.  A fast computational method for moment-independent uncertainty importance measure , 2014, Comput. Phys. Commun..

[27]  Emanuele Borgonovo,et al.  Copula theory and probabilistic sensitivity analysis: Is there a connection? , 2019, Eur. J. Oper. Res..

[28]  Nicolas Gayton,et al.  Nonparametric Importance Sampling Techniques for Sensitivity Analysis and Reliability Assessment of a Launcher Stage Fallout , 2019, Springer Optimization and Its Applications.

[29]  Mathieu Ribatet,et al.  Extreme value copulas and max-stable processes , 2013 .

[30]  B. Rémillard,et al.  Goodness-of-fit tests for copulas: A review and a power study , 2006 .

[31]  Fabian Spanhel,et al.  Simplified vine copula models: Approximations based on the simplifying assumption , 2015, Electronic Journal of Statistics.

[32]  L. Shampine.  Vectorized adaptive quadrature in MATLAB , 2008 .

[33]  T. J. Page.  Multivariate Statistics: A Vector Space Approach , 1984 .

[34]  Alexander J. McNeil,et al.  Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions , 2009, 0908.3750.

[35]  S. Kotz,et al.  The Meta-elliptical Distributions with Given Marginals , 2002 .

[36]  K. S. Tan,et al.  Vine copula models with GLM and sparsity , 2017 .

[37]  W. V. Harper,et al.  Sensitivity/uncertainty analysis of a borehole scenario comparing Latin Hypercube Sampling and deterministic sensitivity approaches , 1983 .

[38]  A. Bowman,et al.  Applied smoothing techniques for data analysis : the kernel approach with S-plus illustrations , 1999 .

[39]  Yarema Okhrin,et al.  Properties of hierarchical Archimedean copulas , 2013 .

[40]  Eric Bouyé,et al.  Copulas for Finance - A Reading Guide and Some Applications , 2000 .

[41]  T. J. Mitchell,et al.  Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction , 1993 .

[42]  Yarema Okhrin,et al.  On the structure and estimation of hierarchical Archimedean copulas , 2013 .

[43]  Markus Junker,et al.  Estimating the tail-dependence coefficient: Properties and pitfalls , 2005 .

[44]  A. Kiureghian,et al.  STRUCTURAL RELIABILITY UNDER INCOMPLETE PROBABILITY INFORMATION , 1986 .

[45]  B. Iooss,et al.  A Review on Global Sensitivity Analysis Methods , 2014, 1404.2405.

[46]  Thibault Vatter,et al.  Generalized Additive Models for Pair-Copula Constructions , 2016, Journal of Computational and Graphical Statistics.

[47]  Emanuele Borgonovo,et al.  Copula-based sensitivity measures of computer experiments , 2015 .

[48]  Zhenzhou Lu,et al.  Moment‐Independent Sensitivity Analysis Using Copula , 2014, Risk analysis : an official publication of the Society for Risk Analysis.

[49]  U. Schepsmeier.  A goodness-of-fit test for regular vine copula models , 2013, 1306.0818.

[50]  Kjersti Aas,et al.  On the simplified pair-copula construction - Simply useful or too simplistic? , 2010, J. Multivar. Anal..

[51]  Jonathan A. Tawn,et al.  Bivariate extreme value theory: Models and estimation , 1988 .

[52]  Olivier Scaillet,et al.  The estimation of copulas : theory and practice , 2007 .

[53]  Art B. Owen,et al.  On Shapley Value for Measuring Importance of Dependent Inputs , 2016, SIAM/ASA J. Uncertain. Quantification.

[54]  B. A. Worley.  Deterministic uncertainty analysis , 1987 .

[55]  Emanuele Borgonovo,et al.  Global sensitivity measures from given data , 2013, Eur. J. Oper. Res..

[56]  S. Tarantola,et al.  Moment Independent Importance Measures: New Results and Analytical Test Cases , 2011, Risk analysis : an official publication of the Society for Risk Analysis.

[57]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[58]  Jérôme Morio,et al.  ESTIMATION OF MOMENT INDEPENDENT IMPORTANCE MEASURES USING A COPULA AND MAXIMUM ENTROPY FRAMEWORK , 2018, 2018 Winter Simulation Conference (WSC).

[59]  Saralees Nadarajah,et al.  A Compendium of Copulas , 2018 .

[60]  Claudia Czado,et al.  Simplified pair copula constructions - Limitations and extensions , 2013, J. Multivar. Anal..

[61]  Roger M. Cooke,et al.  Probability Density Decomposition for Conditionally Dependent Random Variables Modeled by Vines , 2001, Annals of Mathematics and Artificial Intelligence.

[62]  Christian Schellhase,et al.  Estimating non-simplified vine copulas using penalized splines , 2016, Statistics and Computing.

[63]  Zhenzhou Lu,et al.  A vine copula–based method for analyzing the moment-independent importance measure of the multivariate output , 2018, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability.

[64]  Song-xi Chen,et al.  Beta kernel estimators for density functions , 1999 .

[65]  T. Bedford,et al.  Vines: A new graphical model for dependent random variables , 2002 .

[66]  Claudia Czado,et al.  Nonparametric estimation of simplified vine copula models: comparison of methods , 2017, 1701.00845.

[67]  Claudia Czado,et al.  Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas , 2015, J. Multivar. Anal..

[68]  Emanuele Borgonovo,et al.  Sampling strategies in density-based sensitivity analysis , 2012, Environ. Model. Softw..

[69]  Zhenzhou Lu,et al.  A new method for evaluating Borgonovo moment-independent importance measure with its application in an aircraft structure , 2014, Reliab. Eng. Syst. Saf..

[70]  Roger M. Cooke,et al.  Uncertainty Analysis with High Dimensional Dependence Modelling , 2006 .

[71]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[72]  Fabrizio Durante,et al.  Copula Theory and Its Applications , 2010 .

[73]  Nicolas Gayton,et al.  AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation , 2011 .

[74]  C. Genest,et al.  Statistical Inference Procedures for Bivariate Archimedean Copulas , 1993 .

[75]  Nicolas Gayton,et al.  A combined Importance Sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models , 2013, Reliab. Eng. Syst. Saf..

[76]  Eckhard Liebscher,et al.  Construction of asymmetric multivariate copulas , 2008 .

[77]  C. Genest,et al.  Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask , 2007 .

[78]  S. Satchell,et al.  THE BERNSTEIN COPULA AND ITS APPLICATIONS TO MODELING AND APPROXIMATIONS OF MULTIVARIATE DISTRIBUTIONS , 2004, Econometric Theory.

[79]  Emanuele Borgonovo,et al.  Sensitivity analysis: A review of recent advances , 2016, Eur. J. Oper. Res..