Adaptive importance sampling for extreme quantile estimation with stochastic black box computer models

Quantile is an important quantity in reliability analysis, as it is related to the resistance level for defining failure events. This study develops a computationally efficient sampling method for estimating extreme quantiles using stochastic black box computer models. Importance sampling has been widely employed as a powerful variance reduction technique to reduce estimation uncertainty and improve computational efficiency in many reliability studies. However, when applied to quantile estimation, importance sampling faces challenges, because a good choice of the importance sampling density relies on information about the unknown quantile. We propose an adaptive method that refines the importance sampling density parameter toward the unknown target quantile value along the iterations. The proposed adaptive scheme allows us to use the simulation outcomes obtained in previous iterations for steering the simulation process to focus on important input areas. We prove some convergence properties of the proposed method and show that our approach can achieve variance reduction over crude Monte Carlo sampling. We demonstrate its estimation efficiency through numerical examples and wind turbine case study.

[1]  Paul Glasserman,et al.  1 Importance Sampling and Stratification for Value-at-Risk , 1999 .

[2]  Jan C. Neddermeyer.  Computationally Efficient Nonparametric Importance Sampling , 2008, 0805.3591.

[3]  Xi Chen,et al.  Efficient budget allocation strategies for elementary effects method in stochastic simulation , 2018 .

[4]  Petar M. Djuric,et al.  Adaptive Importance Sampling: The past, the present, and the future , 2017, IEEE Signal Processing Magazine.

[5]  Eunshin Byon,et al.  Uncertainty Quantification for Extreme Quantile Estimation With Stochastic Computer Models , 2021, IEEE Transactions on Reliability.

[6]  J. Oakley.  Estimating percentiles of uncertain computer code outputs , 2004 .

[7]  T. Hesterberg,et al.  Weighted Average Importance Sampling and Defensive Mixture Distributions , 1995 .

[8]  F BugalloMónica,et al.  Improving population Monte Carlo , 2017 .

[9]  Anthony O'Hagan,et al.  Diagnostics for Gaussian Process Emulators , 2009, Technometrics.

[10]  Eunshin Byon,et al.  Adaptive Learning in Time-Variant Processes With Application to Wind Power Systems , 2016, IEEE Transactions on Automation Science and Engineering.

[11]  B. Iooss,et al.  Controlled stratification for quantile estimation , 2008, 0802.2426.

[12]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[13]  Gerardo Rubino,et al.  Introduction to Rare Event Simulation , 2009, Rare Event Simulation using Monte Carlo Methods.

[14]  Eunshin Byon,et al.  BAYESIAN SPLINE METHOD FOR ASSESSING EXTREME LOADS ON WIND TURBINES , 2013, 1401.2760.

[15]  L. Jeff Hong,et al.  Kernel Smoothing for Nested Estimation with Application to Portfolio Risk Measurement , 2017, Oper. Res..

[16]  Mark Broadie,et al.  Efficient Risk Estimation via Nested Sequential Simulation , 2011, Manag. Sci..

[17]  K NakayamaMarvin,et al.  Confidence intervals for quantiles when applying variance-reduction techniques , 2012 .

[18]  Xi Chen,et al.  Stochastic kriging for conditional value-at-risk and its sensitivities , 2012, Proceedings Title: Proceedings of the 2012 Winter Simulation Conference (WSC).

[19]  D. Ruppert,et al.  Efficient Estimations from a Slowly Convergent Robbins-Monro Process , 1988 .

[20]  Marvin K. Nakayama,et al.  Confidence intervals for quantiles when applying variance-reduction techniques , 2012, TOMC.

[21]  Jean-Marie Cornuet,et al.  Adaptive Multiple Importance Sampling , 2009, 0907.1254.

[22]  Daniel W. Apley,et al.  Efficient Nested Simulation for Estimating the Variance of a Conditional Expectation , 2011, Oper. Res..

[23]  Nan Chen,et al.  Importance Sampling for Reliability Evaluation With Stochastic Simulation Models , 2015, Technometrics.

[24]  Patrick Moriarty,et al.  Database for validation of design load extrapolation techniques , 2008 .

[25]  A. Krzyżak,et al.  Nonparametric recursive quantile estimation , 2014 .

[26]  Eunshin Byon,et al.  Uncertainty Quantification of Stochastic Simulation for Black-box Computer Experiments , 2018 .

[27]  Jérôme Morio,et al.  Extreme quantile estimation with nonparametric adaptive importance sampling , 2012, Simul. Model. Pract. Theory.

[28]  B. Jonkman.  Turbsim User's Guide: Version 1.50 , 2009 .

[29]  M. Muresan.  A concrete approach to classical analysis , 2009 .

[30]  Eunshin Byon,et al.  Condition Monitoring of Wind Power System With Nonparametric Regression Analysis , 2014, IEEE Transactions on Energy Conversion.

[31]  David Luengo,et al.  Generalized Multiple Importance Sampling , 2015, Statistical Science.

[32]  P. Glynn.  IMPORTANCE SAMPLING FOR MONTE CARLO ESTIMATION OF QUANTILES , 2011 .

[33]  R. Rigby,et al.  Generalized additive models for location, scale and shape , 2005 .

[34]  George Michailidis,et al.  Sequential Experiment Design for Contour Estimation From Complex Computer Codes , 2008, Technometrics.

[35]  Bing Wang,et al.  On the monotonic performance of stochastic kriging predictors , 2015, 2015 Winter Simulation Conference (WSC).

[36]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[37]  Jason Jonkman,et al.  FAST User's Guide , 2005 .

[38]  Barry L. Nelson,et al.  Efficient generation of cycle time‐throughput curves through simulation and metamodeling , 2005, WSC '05.

[39]  Sandeep Juneja,et al.  Nested Simulation in Portfolio Risk Measurement , 2008, Manag. Sci..

[40]  Ping Zhang.  Nonparametric Importance Sampling , 1996 .

[41]  Daniel Egloff,et al.  QUANTILE ESTIMATION WITH ADAPTIVE IMPORTANCE SAMPLING , 2010, 1002.4946.

[42]  Junho Song,et al.  Cross-Entropy-Based Adaptive Importance Sampling Using Gaussian Mixture , 2013 .

[43]  Barry L. Nelson,et al.  Stochastic kriging for simulation metamodeling , 2008, 2008 Winter Simulation Conference.

[44]  Gilles Pagès,et al.  Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling , 2008, Monte Carlo Methods Appl..

[45]  Stéphane Bulteau,et al.  A new importance sampling Monte Carlo method for a flow network reliability problem , 2002 .

[46]  Lance Manuel,et al.  On the use of a large database of simulated wind turbine loads to aid in assessing design standard provisions , 2013 .

[47]  A. Owen,et al.  Safe and Effective Importance Sampling , 2000 .

[48]  Luca Martino,et al.  Improving population Monte Carlo: Alternative weighting and resampling schemes , 2016, Signal Process..

[49]  Lance Manuel,et al.  Statistical Extrapolation Methods for Estimating Wind Turbine Extreme Loads , 2007 .

[50]  Eunshin Byon,et al.  Computationally Efficient Uncertainty Minimization in Wind Turbine Extreme Load Assessments , 2016 .

[51]  M. Balesdent,et al.  Kriging-based adaptive Importance Sampling algorithms for rare event estimation , 2013 .

[52]  M. Borel.  Les probabilités dénombrables et leurs applications arithmétiques , 1909 .

[53]  J. Beck,et al.  A new adaptive importance sampling scheme for reliability calculations , 1999 .

[54]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

[55]  P. Glasserman,et al.  Variance Reduction Techniques for Estimating Value-at-Risk , 2000 .

[56]  V. Roshan Joseph,et al.  Composite Gaussian process models for emulating expensive functions , 2012, 1301.2503.

[57]  Mike Ludkovski,et al.  Replication or Exploration? Sequential Design for Stochastic Simulation Experiments , 2017, Technometrics.

[58]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.