Rare Event Estimation Research Articles

Support vector machine-based importance sampling for rare event estimation

Structural reliability analysis aims at computing failure probability with respect to prescribed performance function. To efficiently estimate the structural failure probability, a novel two-stage meta-model importance sampling based on the support vector machine (SVM) is proposed. Firstly, a quasi-optimal importance sampling density function is approximated by SVM. To construct the SVM model, a multi-point enrichment algorithm allowing adding several training points in each iteration is employed. Then, the augmented failure probability and quasi-optimal importance sampling samples can be obtained by the trained SVM model. Secondly, the current SVM model is further polished by selecting informative training points from the quasi-optimal importance sampling samples until it can accurately recognize the states of samples, and the correction factor is estimated by the well-trained SVM model. Finally, the failure probability is obtained by the product of augmented failure probability and correction factor. The proposed method provides an algorithm to efficiently deal with multiple failure regions and rare events. Several examples are performed to illustrate the feasibility of the proposed method.

Mathematical Formulation and Analytic Solutions for Uncertainty Analysis in Probabilistic Safety Assessment of Nuclear Power Plants

Monte Carlo simulations are widely used for uncertainty analysis in the probabilistic safety assessment of nuclear power plants. Despite many advantages, such as its general applicability, a Monte Carlo simulation has inherent limitations as a simulation-based approach. This study provides a mathematical formulation and analytic solutions for the uncertainty analysis in a probabilistic safety assessment (PSA). Starting from the definitions of variables, mathematical equations are derived for synthesizing probability density functions for logical AND, logical OR, and logical OR with rare event approximation of two independent events. The equations can be applied consecutively when there exist more than two events. For fail-to-run failures, the probability density function for the unavailability has the same probability distribution as the probability density function (PDF) for the failure rate under specified conditions. The effectiveness of the analytic solutions is demonstrated by applying them to an example system. The resultant probability density functions are in good agreement with the Monte Carlo simulation results, which are in fact approximations for those from the analytic solutions, with errors less than 12.6%. Important theoretical aspects are examined with the analytic solutions such as the validity of the use of a right-unbounded distribution to describe the uncertainty in the unavailability/probability. The analytic solutions for uncertainty analysis can serve as a basis for all other methods, providing deeper insights into uncertainty analyses in probabilistic safety assessment.

Multiple Importance Sampling for Symbol Error Rate Estimation of Maximum-Likelihood Detectors in MIMO Channels

In this paper we propose a multiple importance sampling (MIS) method for the efficient symbol error rate (SER) estimation of maximum likelihood (ML) multiple-input multipleoutput (MIMO) detectors. Given a transmitted symbol from the input lattice, obtaining the SER requires the computation of an integral outside its Voronoi region in a high-dimensional space, for which a closed-form solution does not exist. Hence, the SER must be approximated through crude or naive Monte Carlo (MC) simulations. This practice is widely used in the literature despite its inefficiency, particularly severe at high signal-to-noise-ratio (SNR) or for systems with stringent SER requirements. It is well-known that more sophisticated MC-based techniques such as MIS, when carefully designed, can reduce the variance of the estimators in several orders of magnitude with respect to naive Monte Carlo in rare-event estimation, or equivalently, they need significantly less samples for attaining a desired performance. The proposed MIS method provides unbiased SER estimates by sampling from a mixture of components that are carefully chosen and parametrized. The number of components, the parameters of the components, and their weights in the mixture, are automatically chosen by the proposed method. As a result, the proposed method is flexible, easy-to-use, theoretically sound, and presents a high performance in a variety of scenarios. We show in our simulations that SERs lower than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup> can be accurately estimated with just 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> random samples.

Pricing discretely-monitored double barrier options with small probabilities of execution

In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.

A Universal Splitting Estimator for the Performance Evaluation of Wireless Communications Systems

We propose a unified rare-event estimator for the performance evaluation of wireless communication systems. The estimator is derived from the well-known multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to the simulation and estimation of time-independent problems, because splitting requires an underlying continuous-time Markov process whose trajectories can be split. We tackle this problem by embedding the static problem of interest within a continuous-time Markov process, so that the target time-independent distribution becomes the distribution of the Markov process at a given time instant. The main feature of the proposed multilevel splitting algorithm is its large scope of applicability. For illustration, we show how the same algorithm can be applied to the problem of estimating the cumulative distribution function (CDF) of sums of random variables (RVs), the CDF of partial sums of ordered RVs, the CDF of ratios of RVs, and the CDF of weighted sums of Poisson RVs. We investigate the computational efficiency of the proposed estimator via a number of simulation studies and find that it compares favorably with existing estimators.

PNS146 BEYOND DEFAULTS - ADAPTING PRINCIPLED PRIOR SELECTION IN BAYESIAN NETWORK META-ANALYSIS

Bayesian methods provide a logical and consistent decision making framework to guide evidence synthesis and commonly used for Network Meta-Analysis (NMA). Bayesian analyses allow for estimation of rare events and can protect against issues of multiple comparisons through incorporation of genuine prior information. We adapt recent guidance documents for principled prior selection to NMA. We provide a summary of recent publications emphasizing the importance of prior selection for leveraging the full benefits of a Bayesian decision making framework. We then adapt and provide a worked example of methods based on simulation from the prior predictive distribution to facilitate understanding of the implication of priors on decision making. Standard default priors recommended in NMA methods guidance documents for health technology assessment imply unrealistic bounds for ratio-based measures and impossible values for continuous summaries like the mean difference. Prior predictive distributions for these statistics imply that prior belief places more weight on implausibly large values than are commonly seen across a variety of domains. A worked example in neonatology shows how the use visualizations of the prior predictive distribution can be incorporated in the protocol development stage in order to set priors for analysis that respect genuine expert domain knowledge. We show that conclusions of NMAs where any node is sparsely informed can differ when reasonable priors are selected. Bayesian inference allows for the incorporation of prior information with data in order to support rational decision-making. The use of standard, default “vague” priors in an effort to “let the data speak for themselves” is likely to be inconsistent with genuine domain knowledge and could meaningfully change conclusions of NMAs. Priors chosen at the protocol development stage based on visualizations of the prior predictive distribution may lead to more consistent decisions.

Vaccine skepticism reflects basic cognitive differences in mortality-related event frequency estimation

Vaccines have prevented and nearly eliminated several deadly diseases, yet they face skepticism from the public. One potential driver of vaccine skepticism is how people process event frequencies such as rare adverse reactions to vaccines. Misestimations may distort the perceived risks of vaccinating. The current study examined how vaccine skepticism is related to accuracy in event frequency processing. In Experiment 1, participants (n = 158) estimated the frequencies of several vital statistics (e.g., ‘How many people die per year in the U.S. from emphysema?’). Higher levels of vaccine skepticism were associated with lower accuracy in frequency estimation and over-estimation of rare events. In Experiment 2 (n = 109), we again found that vaccine skepticism was negatively associated with vital statistic estimation accuracy but not for emotionally neutral or positive events. These results suggest that vaccine skepticism may arise from basic individual differences in processing events associated with mortality or negative affect.

Subset simulation method including fitness-based seed selection for reliability analysis

Probability estimation of rare events is a challenging task in the reliability theory. Subset simulation (SS) is a robust simulation technique that transforms a rare event into a sequence of multiple intermediate failure events with large probabilities and efficiently approximates the mentioned probability. Proper handling of a reliability problem by this method requires employing a suitable sampling approach to transmit samples toward the failure set. Markov Chain Monte Carlo (MCMC) is a suitable sampling approach that solves the SS transition phase using the failed sample of each simulation level as the seed of next samples. This paper is aimed to study the seed selection effect on the SS accuracy through several seed selection approaches inspired by the genetic algorithm and particle filter and using the main PDF of the variables to assign a mass function probability to each subset sample in the failure domain. Roulette wheel (I, II), tournament and proportional probability techniques are then employed to choose the weighed samples as seeds to be placed in the MCMC to transmit the samples. To examine the capability of each approach, reliabilities of some engineering problems were investigated and results showed that the proposed approaches could find proper failure sets better than the original SS method, especially in problems with several failure domains.

Importance Sampling for Slow-Fast Diffusions Based on Moderate Deviations

We consider systems of slow-fast diffusions with small noise in the slow component. We construct provably logarithmic asymptotically optimal importance schemes for the estimation of rare events based on the moderate deviations principle. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Moderate deviations--based importance sampling offers a viable alternative to large deviations importance sampling when the events are not too rare. In particular, in many cases of interest one can indeed construct the required change of measure in closed form, a task which is more complicated using the large deviations--based importance sampling, especially when it comes to multiscale dynamically evolving processes. The presence of multiple scales and the fact that we do not make any periodicity assumptions for the coefficients driving the processes complicate the design and the analysis of efficient importance sampling schemes. Simulation studies illustrate the theory.

Construction and quantification of a fire probabilistic safety assessment model using support system initiating event fault trees

This study investigated the effect of the fire-induced initiating event fault trees (IE FTs) related to the support systems of a reference plant, Hanul 3,4, on the risk quantification results by comparing the results of two (2) different kinds of fire probabilistic safety assessment (PSA) models: one based on the FT approach (with IE FTs) and the other based on the conventional approach (without IE FTs). The results show that the FT modeling of fire-induced IEs decreases fire-induced core damage frequencies (CDFs) if the IE FTs involve relatively many AND logic gates, because the fire-induced component failure events, the probabilities of which are less than '1′(TRUE), constitute the fire-induced IE FTs. The results also show that the CDFs increase if the fire-induced IE FTs involve relatively many OR logic gates. This is because a rare event approximation leads to overestimation of the real fire-induced risks.

Simplified complexity: Analytical strategies for conflict event research

Scholars who have sought to identify the triggers of rare political events have met with limited success. With respect to civil war, studies teach us to expect conflict where it is feasible. However, although we understand where civil conflict occurs, we do not quite understand when it occurs. Focusing on civil conflict, I argue that time-variant and time-invariant explanations relate to the outcome by means of two distinct causal processes, which has implications for the identification of triggers of rare events. I provide an easily implementable approach to improve rare event estimation that uses matching to leverage constant attributes to estimate the effects of rare predictors. I demonstrate the utility of this procedure by providing an aggregate and disaggregate example of civil conflict onset estimation.

Adaptive Kriging Stochastic Sampling and Density Approximation and Its Application to Rare-Event Estimation

AbstractThis paper examines the task of approximating or generating samples according to a target probability distribution when this distribution is expressed as a function of the response of an en...

Efficient Pricing of Barrier Options on High Volatility Assets Using Subset Simulation

In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.

Bayesian inference with Subset Simulation: Strategies and improvements

Bayesian Updating with Structural reliability methods (BUS) reinterprets the Bayesian updating problem as a structural reliability problem; i.e. a rare event estimation. The BUS approach can be considered an extension of rejection sampling, where a standard uniform random variable is added to the space of random variables. Each generated sample from this extended random variable space is accepted if the realization of the uniform random variable is smaller than the likelihood function scaled by a constant c. The constant c has to be selected such that 1∕c is not smaller than the maximum of the likelihood function, which, however, is typically unknown a-priori. A c chosen too small will have negative impact on the efficiency of the BUS approach when combined with sampling-based reliability methods. For the combination of BUS with Subset Simulation, we propose an approach, termed aBUS, for adaptive BUS, that does not require c as input. The proposed algorithm requires only minimal modifications of standard BUS with Subset Simulation. We discuss why aBUS produces samples that follow the posterior distribution - even if 1∕c is selected smaller than the maximum of the likelihood function. The performance of aBUS in terms of the computed evidence required for Bayesian model class selection and in terms of the produced posterior samples is assessed numerically for different example problems. The combination of BUS with Subset Simulation (and aBUS in particular) is well suited for problems with many uncertain parameters and for Bayesian updating of models where it is computationally demanding to evaluate the likelihood function.

Probability estimation of rare extreme events in the case of small samples: Technique and examples of analysis of earthquake catalogs

The most general approach to studying the recurrence law in the area of the rare largest events is associated with the use of limit law theorems of the theory of extreme values. In this paper, we use the Generalized Pareto Distribution (GPD). The unknown GPD parameters are typically determined by the method of maximal likelihood (ML). However, the ML estimation is only optimal for the case of fairly large samples (>200–300), whereas in many practical important cases, there are only dozens of large events. It is shown that in the case of a small number of events, the highest accuracy in the case of using the GPD is provided by the method of quantiles (MQs). In order to illustrate the obtained methodical results, we have formed the compiled data sets characterizing the tails of the distributions for typical subduction zones, regions of intracontinental seismicity, and for the zones of midoceanic (MO) ridges. This approach paves the way for designing a new method for seismic risk assessment. Here, instead of the unstable characteristics—the uppermost possible magnitude Mmax—it is recommended to use the quantiles of the distribution of random maxima for a future time interval. The results of calculating such quantiles are presented.

An adaptive directional importance sampling method for structural reliability analysis

The importance sampling is merged with directional simulation in this paper. A sampling function is defined on the unit hyper sphere which samples random directions. The directions are sampled around a direction that aims to the design point. The sampling function uses spherical coordinates to generate random directions. The method is made adaptive by a closed form updating rule to renew the sampling parameters. To reduce the number of calls on the limit state function, a root finding procedure is put forward. The proposed method is tested with well-known test problems and its performance is compared with the conventional directional simulation. The results demonstrate the accuracy and efficiency of the proposed method for rare event estimation.

Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics

Many interesting rare events in molecular systems, like ligand association, protein folding or conformational changes, occur on timescales that often are not accessible by direct numerical simulation. Therefore, rare event approximation approaches like interface sampling, Markov state model building, or advanced reaction coordinate-based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches. How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so we also allow for understanding deep connections between the different approaches.

Rare event simulation via importance sampling for linear SPDE’s

The goal of this paper is to develop provably efficient importance sampling Monte Carlo methods for the estimation of rare events within the class of linear stochastic partial differential equations. We find that if a spectral gap of appropriate size exists, then one can identify a lower dimensional manifold where the rare event takes place. This allows one to build importance sampling changes of measures that perform provably well even pre-asymptotically (i.e. for small but non-zero size of the noise) without degrading in performance due to infinite dimensionality or due to long simulation time horizons. Simulation studies supplement and illustrate the theoretical results.

Estimating rare events in biochemical systems using conditional sampling.

The paper focuses on development of variance reduction strategies to estimate rare events in biochemical systems. Obtaining this probability using brute force Monte Carlo simulations in conjunction with the stochastic simulation algorithm (Gillespie's method) is computationally prohibitive. To circumvent this, important sampling tools such as the weighted stochastic simulation algorithm and the doubly weighted stochastic simulation algorithm have been proposed. However, these strategies require an additional step of determining the important region to sample from, which is not straightforward for most of the problems. In this paper, we apply the subset simulation method, developed as a variance reduction tool in the context of structural engineering, to the problem of rare event estimation in biochemical systems. The main idea is that the rare event probability is expressed as a product of more frequent conditional probabilities. These conditional probabilities are estimated with high accuracy using Monte Carlo simulations, specifically the Markov chain Monte Carlo method with the modified Metropolis-Hastings algorithm. Generating sample realizations of the state vector using the stochastic simulation algorithm is viewed as mapping the discrete-state continuous-time random process to the standard normal random variable vector. This viewpoint opens up the possibility of applying more sophisticated and efficient sampling schemes developed elsewhere to problems in stochastic chemical kinetics. The results obtained using the subset simulation method are compared with existing variance reduction strategies for a few benchmark problems, and a satisfactory improvement in computational time is demonstrated.

Risk assessment of rare events

Rare events often result in large impacts and are hard to predict. Risk analysis of such events is a challenging task because there are few directly relevant data to form a basis for probabilistic risk assessment. Due to the scarcity of data, the probability estimation of a rare event often uses precursor data. Precursor-based methods have been widely used in probability estimation of rare events. However, few attempts have been made to estimate consequences of rare events using their precursors. This paper proposes a holistic precursor-based risk assessment framework for rare events. The Hierarchical Bayesian Approach (HBA) using hyper-priors to represent prior parameters is applied to probability estimation in the proposed framework. Accident precursor data are utilized from an information theory perspective to seek the most informative precursor upon which the consequence of a rare event is estimated. Combining the estimated probability and consequence gives a reasonable assessment of risk. The assessed risk is updated as new information becomes available to produce a dynamic risk profile. The applicability of the methodology is tested through a case study of an offshore blowout accident. The proposed framework provides a rational way to develop the dynamic risk profile of a rare event for its prevention and control.