Lognormal Approximations of Fault Tree Uncertainty Distributions

Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed‐form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling‐based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed‐form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks’s method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models.     

Description and Propagation of Uncertainty in Input Parameters in Environmental Fate Models

Today, chemical risk and safety assessments rely heavily on the estimation of environmental fate by models. The key compound‐related properties in such models describe partitioning and reactivity. Uncertainty in determining these properties can be separated into random and systematic (incompleteness) components, requiring different types of representation. Here, we evaluate two approaches that are suitable to treat also systematic errors, fuzzy arithmetic, and probability bounds analysis. When a best estimate (mode) and a range can be computed for an input parameter, then it is possible to characterize the uncertainty with a triangular fuzzy number (possibility distribution) or a corresponding probability box bound by two uniform distributions. We use a five‐compartment Level I fugacity model and reported empirical data from the literature for three well‐known environmental pollutants (benzene, pyrene, and DDT) as illustrative cases for this evaluation. Propagation of uncertainty by discrete probability calculus or interval arithmetic can be done at a low computational cost and gives maximum flexibility in applying different approaches. Our evaluation suggests that the difference between fuzzy arithmetic and probability bounds analysis is small, at least for this specific case. The fuzzy arithmetic approach can, however, be regarded as less conservative than probability bounds analysis if the assumption of independence is removed. Both approaches are sensitive to repeated parameters that may inflate the uncertainty estimate. Uncertainty described by probability boxes was therefore also propagated through the model by Monte Carlo simulation to show how this problem can be avoided.     

Conundrums with Uncertainty Factors

The practice of uncertainty factors as applied to noncancer endpoints in the IRIS database harkens back to traditional safety factors. In the era before risk quantification, these were used to build in a “margin of safety.” As risk quantification takes hold, the safety factor methods yield to quantitative risk calculations to guarantee safety. Many authors believe that uncertainty factors can be given a probabilistic interpretation as ratios of response rates, and that the reference values computed according to the IRIS methodology can thus be converted to random variables whose distributions can be computed with Monte Carlo methods, based on the distributions of the uncertainty factors. Recent proposals from the National Research Council echo this view. Based on probabilistic arguments, several authors claim that the current practice of uncertainty factors is overprotective. When interpreted probabilistically, uncertainty factors entail very strong assumptions on the underlying response rates. For example, the factor for extrapolating from animal to human is the same whether the dosage is chronic or subchronic. Together with independence assumptions, these assumptions entail that the covariance matrix of the logged response rates is singular. In other words, the accumulated assumptions entail a log‐linear dependence between the response rates. This in turn means that any uncertainty analysis based on these assumptions is ill‐conditioned; it effectively computes uncertainty conditional on a set of zero probability. The practice of uncertainty factors is due for a thorough review. Two directions are briefly sketched, one based on standard regression models, and one based on nonparametric continuous Bayesian belief nets.     

First-Order Reliability Analysis of Public Health Risk Assessment

This paper demonstrates a new methodology for probabilistic public health risk assessment using the first-order reliability method. The method provides the probability that incremental lifetime cancer risk exceeds a threshold level, and the probabilistic sensitivity quantifying the relative impact of considering the uncertainty of each random variable on the exceedance probability. The approach is applied to a case study given by Thompson et al. ⁽¹⁾ on cancer risk caused by ingestion of benzene-contaminated soil, and the results are compared to that of the Monte Carlo method. Parametric sensitivity analyses are conducted to assess the sensitivity of the probabilistic event with respect to the distribution parameters of the basic random variables, such as the mean and standard deviation. The technique is a novel approach to probabilistic risk assessment, and can be used in situations when Monte Carlo analysis is computationally expensive, such as when the simulated risk is at the tail of the risk probability distribution.     

Integration of Probabilistic Exposure Assessment and Probabilistic Hazard Characterization

A method is proposed for integrated probabilistic risk assessment where exposure assessment and hazard characterization are both included in a probabilistic way. The aim is to specify the probability that a random individual from a defined (sub)population will have an exposure high enough to cause a particular health effect of a predefined magnitude, the critical effect size ( CES ). The exposure level that results in exactly that CES in a particular person is that person's individual critical effect dose ( ICED ). Individuals in a population typically show variation, both in their individual exposure ( IEXP ) and in their ICED . Both the variation in IEXP and the variation in ICED are quantified in the form of probability distributions. Assuming independence between both distributions, they are combined (by Monte Carlo) into a distribution of the individual margin of exposure ( IMoE ). The proportion of the IMoE distribution below unity is the probability of critical exposure ( PoCE ) in the particular (sub)population. Uncertainties involved in the overall risk assessment (i.e., both regarding exposure and effect assessment) are quantified using Monte Carlo and bootstrap methods. This results in an uncertainty distribution for any statistic of interest, such as the probability of critical exposure ( PoCE ). The method is illustrated based on data for the case of dietary exposure to the organophosphate acephate. We present plots that concisely summarize the probabilistic results, retaining the distinction between variability and uncertainty. We show how the relative contributions from the various sources of uncertainty involved may be quantified.     

A Combined Monte Carlo and Possibilistic Approach to Uncertainty Propagation in Event Tree Analysis

In risk analysis, the treatment of the epistemic uncertainty associated to the probability of occurrence of an event is fundamental. Traditionally, probabilistic distributions have been used to characterize the epistemic uncertainty due to imprecise knowledge of the parameters in risk models. On the other hand, it has been argued that in certain instances such uncertainty may be best accounted for by fuzzy or possibilistic distributions. This seems the case in particular for parameters for which the information available is scarce and of qualitative nature. In practice, it is to be expected that a risk model contains some parameters affected by uncertainties that may be best represented by probability distributions and some other parameters that may be more properly described in terms of fuzzy or possibilistic distributions. In this article, a hybrid method that jointly propagates probabilistic and possibilistic uncertainties is considered and compared with pure probabilistic and pure fuzzy methods for uncertainty propagation. The analyses are carried out on a case study concerning the uncertainties in the probabilities of occurrence of accident sequences in an event tree analysis of a nuclear power plant.     

Probabilistic Reliability Modeling for Oil Exploration & Production (E&P) Facilities in the Tallgrass Prairie Preserve

The aging domestic oil production infrastructure represents a high risk to the environment because of the type of fluids being handled (oil and brine) and the potential for accidental release of these fluids into sensitive ecosystems. Currently, there is not a quantitative risk model directly applicable to onshore oil exploration and production (E&P) facilities. We report on a probabilistic reliability model created for onshore exploration and production (E&P) facilities. Reliability theory, failure modes and effects analysis (FMEA), and event trees were used to develop the model estimates of the failure probability of typical oil production equipment. Monte Carlo simulation was used to translate uncertainty in input parameter values to uncertainty in the model output. The predicted failure rates were calibrated to available failure rate information by adjusting probability density function parameters used as random variates in the Monte Carlo simulations. The mean and standard deviation of normal variate distributions from which the Weibull distribution characteristic life was chosen were used as adjustable parameters in the model calibration. The model was applied to oil production leases in the Tallgrass Prairie Preserve, Oklahoma. We present the estimated failure probability due to the combination of the most significant failure modes associated with each type of equipment (pumps, tanks, and pipes). The results show that the estimated probability of failure for tanks is about the same as that for pipes, but that pumps have much lower failure probability. The model can provide necessary equipment reliability information for proactive risk management at the lease level by providing quantitative information to base allocation of maintenance resources to high-risk equipment that will minimize both lost production and ecosystem damage.     

Uncertain Numbers and Uncertainty in the Selection of Input Distributions—Consequences for a Probabilistic Risk Assessment of Contaminated Land

Risks from exposure to contaminated land are often assessed with the aid of mathematical models. The current probabilistic approach is a considerable improvement on previous deterministic risk assessment practices, in that it attempts to characterize uncertainty and variability. However, some inputs continue to be assigned as precise numbers, while others are characterized as precise probability distributions. Such precision is hard to justify, and we show in this article how rounding errors and distribution assumptions can affect an exposure assessment. The outcome of traditional deterministic point estimates and Monte Carlo simulations were compared to probability bounds analyses. Assigning all scalars as imprecise numbers (intervals prescribed by significant digits) added uncertainty to the deterministic point estimate of about one order of magnitude. Similarly, representing probability distributions as probability boxes added several orders of magnitude to the uncertainty of the probabilistic estimate. This indicates that the size of the uncertainty in such assessments is actually much greater than currently reported. The article suggests that full disclosure of the uncertainty may facilitate decision making in opening up a negotiation window. In the risk analysis process, it is also an ethical obligation to clarify the boundary between the scientific and social domains.     

Methods for Uncertainty Analysis: A Comparative Survey

This paper presents a survey and comparative evaluation of methods which have been developed for the determination of uncertainties in accident consequences and probabilities, for use in probabilistic risk assessment. The methods considered are: analytic techniques, Monte Carlo simulation, response surface approaches, differential sensitivity techniques, and evaluation of classical statistical confidence bounds. It is concluded that only the response surface and differential sensitivity approaches are sufficiently general and flexible for use as overall methods of uncertainty analysis in probabilistic risk assessment. The other methods considered, however, are very useful in particular problems.     

Discrete Probability Distributions for Probabilistic Fracture Mechanics

Recently, discrete probability distributions (DPDs) have been suggested for use in risk analysis calculations to simplify the numerical computations which must be performed to determine failure probabilities. Specifically, DPDs have been developed to investigate probabilistic functions, that is, functions whose exact form is uncertain. The analysis of defect growth in materials by probabilistic fracture mechanics (PFM) models provides an example in which probabilistic functions play an important role. This paper compares and contrasts Monte Carlo simulation and DPDs as tools for calculating material failure due to fatigue crack growth. For the problem studied, the DPD method takes approximately one third the computation time of the Monte Carlo approach for comparable accuracy. It is concluded that the DPD method has considerable promise in low-failure-probability calculations of importance in risk assessment. In contrast to Monte Carlo, the computation time for the DPD approach is relatively insensitive to the magnitude of the probability being estimated.     

Sensitivity Analysis of a Two‐Dimensional Quantitative Microbiological Risk Assessment: Keeping Variability and Uncertainty Separated

The aim of quantitative microbiological risk assessment is to estimate the risk of illness caused by the presence of a pathogen in a food type, and to study the impact of interventions. Because of inherent variability and uncertainty, risk assessments are generally conducted stochastically, and if possible it is advised to characterize variability separately from uncertainty. Sensitivity analysis allows to indicate to which of the input variables the outcome of a quantitative microbiological risk assessment is most sensitive. Although a number of methods exist to apply sensitivity analysis to a risk assessment with probabilistic input variables (such as contamination, storage temperature, storage duration, etc.), it is challenging to perform sensitivity analysis in the case where a risk assessment includes a separate characterization of variability and uncertainty of input variables. A procedure is proposed that focuses on the relation between risk estimates obtained by Monte Carlo simulation and the location of pseudo‐randomly sampled input variables within the uncertainty and variability distributions. Within this procedure, two methods are used—that is, an ANOVA‐like model and Sobol sensitivity indices—to obtain and compare the impact of variability and of uncertainty of all input variables, and of model uncertainty and scenario uncertainty. As a case study, this methodology is applied to a risk assessment to estimate the risk of contracting listeriosis due to consumption of deli meats.     

An Improved Framework for Uncertainty Analysis: Accounting for Unsuspected Errors

I use an analogy with the history of physical measurements, population and energy projections, and analyze the trends in several data sets to quantify the overconfidence of the experts in the reliability of their uncertainty estimates. Data sets include (i) time trends in the sequential measurements of the same physical quantity; (ii) national population projections; and (iii) projections for the U.S., energy sector. Probabilities of large deviations for the true values are parametrized by an exponential distribution with the slope determined by the data. Statistics of past errors can be used in probabilistic risk assessment to hedge against unsuspected uncertainties and to include the possibility of human error into the framework of uncertainty analysis. By means of a sample Monte Carlo simulation of cancer risk caused by ingestion of benzene in soil, I demonstrate how the upper 95th percentiles of risk are changed when unsuspected uncertainties are included. I recommend to inflate the estimated uncertainties by default safety factors determined from the relevant historical data sets.     

The Effect of Neglecting Correlations When Propagating Uncertainty and Estimating the Population Distribution of Risk

Interest in examining both the uncertainty and variability in environmental health risk assessments has led to increased use of methods for propagating uncertainty. While a variety of approaches have been described, the advent of both powerful personal computers and commercially available simulation software have led to increased use of Monte Carlo simulation. Although most analysts and regulators are encouraged by these developments, some are concerned that Monte Carlo analysis is being applied uncritically. The validity of any analysis is contingent on the validity of the inputs to the analysis. In the propagation of uncertainty or variability, it is essential that the statistical distribution of input variables are properly specified. Furthermore, any dependencies among the input variables must be considered in the analysis. In light of the potential difficulty in specifying dependencies among input variables, it is useful to consider whether there exist rules of thumb as to when correlations can be safely ignored (i.e., when little overall precision is gained by an additional effort to improve upon an estimation of correlation). We make use of well-known error propagation formulas to develop expressions intended to aid the analyst in situations wherein normally and lognormally distributed variables are linearly correlated.     

The Repeatability of Uncertainty and Sensitivity Analyses for Complex Probabilistic Risk Assessments

The performance of a probabilistic risk assessment (PRA) for a nuclear power plant is a complex undertaking, involving the assembly of an accident frequency analysis, an accident progression analysis, a source term analysis, and a consequence analysis. Each of these analyses is, in itself, quite complex. Uncertainties enter into a PRA from each of these analyses. An important focus in recent PRAs has been to incorporate these uncertainties at each stage of the analysis, propagate the subsequent uncertainties through the entire analysis, and include uncertainty in the final results. Monte Carlo procedures based on Latin hypercube sampling provide one way to perform propagations of this type. In this paper, the results of two complete and independent Monte Carlo calculations for a recently completed PRA for a nuclear power plant are compared as a means of providing empirical evidence on the repeatability of uncertainty and sensitivity analyses for large-scale PRA calculations. These calculations use the same variables and analysis structure with two independently generated Latin hypercube samples. The results of the two calculations show a high degree of repeatability for the analysis of a very complex system.     

A Probabilistic Analysis of the Passive–Restraint Question

The benefits and costs of automobile safety policies are compared using a methodology which explicitly quantifies the uncertainties. The policies addressed include both voluntary and compulsory manual belt usage, nondetachable passive (automatic) seat belts, and air bags. Estimates of the effectiveness and usage rates of these alternatives were obtained in the form of subjective probability distributions from eight experts. Their opinions were combined using equal weighting. The direct economic costs of the technologies were also estimated probabilistically. The number of lives saved and the net benefits of the policies were calculated probabilistically for a range of values of lifesaving. Probabilistic computations and sensitivity analysis were performed by the Demos modelling system using Monte Carlo simulation. The results are highly uncertain and quite sensitive to the value of lifesaving. Nevertheless, they imply that repeal of the passive-restraint standard is defensible according to the net-benefit criterion only if a relatively low value is assigned to lifesaving. The degree of uncertainty emphasizes the potential value of demonstration programs to obtain better information.     

Feasibility Study on the Use of Probabilistic Migration Modeling in Support of Exposure Assessment from Food Contact Materials

The use of probabilistic approaches in exposure assessments of contaminants migrating from food packages is of increasing interest but the lack of concentration or migration data is often referred as a limitation. Data accounting for the variability and uncertainty that can be expected in migration, for example, due to heterogeneity in the packaging system, variation of the temperature along the distribution chain, and different time of consumption of each individual package, are required for probabilistic analysis. The objective of this work was to characterize quantitatively the uncertainty and variability in estimates of migration. A Monte Carlo simulation was applied to a typical solution of the Fick's law with given variability in the input parameters. The analysis was performed based on experimental data of a model system (migration of Irgafos 168 from polyethylene into isooctane) and illustrates how important sources of variability and uncertainty can be identified in order to refine analyses. For long migration times and controlled conditions of temperature the affinity of the migrant to the food can be the major factor determining the variability in the migration values (more than 70% of variance). In situations where both the time of consumption and temperature can vary, these factors can be responsible, respectively, for more than 60% and 20% of the variance in the migration estimates. The approach presented can be used with databases from consumption surveys to yield a true probabilistic estimate of exposure.     

Accounting for Both Random Errors and Systematic Errors in Uncertainty Propagation Analysis of Computer Models Involving Experimental Measurements with Monte Carlo Methods

A Monte Carlo method is presented to study the effect of systematic and random errors on computer models mainly dealing with experimental data. It is a common assumption in this type of models (linear and nonlinear regression, and nonregression computer models) involving experimental measurements that the error sources are mainly random and independent with no constant background errors (systematic errors). However, from comparisons of different experimental data sources evidence is often found of significant bias or calibration errors. The uncertainty analysis approach presented in this work is based on the analysis of cumulative probability distributions for output variables of the models involved taking into account the effect of both types of errors. The probability distributions are obtained by performing Monte Carlo simulation coupled with appropriate definitions for the random and systematic errors. The main objectives are to detect the error source with stochastic dominance on the uncertainty propagation and the combined effect on output variables of the models. The results from the case studies analyzed show that the approach is able to distinguish which error type has a more significant effect on the performance of the model. Also, it was found that systematic or calibration errors, if present, cannot be neglected in uncertainty analysis of models dependent on experimental measurements such as chemical and physical properties. The approach can be used to facilitate decision making in fields related to safety factors selection, modeling, experimental data measurement, and experimental design.     

Influence of Distributional Shape of Substance Parameters on Exposure Model Output

Uncertainty of environmental concentrations is calculated with the regional multimedia exposure model of EUSES 1.0 by considering probability input distributions for aqueous solubility, vapor pressure, and octanol-water partition coefficient, K(ow). Only reliable experimentally determined data are selected from available literature for eight reference chemicals representing a wide substance property spectrum. Monte Carlo simulations are performed with uniform, triangular, and log-normal input distributions to assess the influence of the choice of input distribution type on the predicted concentration distributions. The impact of input distribution shapes on output variance exceeds the effect on the output mean by one order of magnitude. Both are affected by influence and uncertainty (i.e., variance) of the input variable as well. Distributional shape has no influence when the sensitivity function of the respective parameter is perfectly linear. For nonlinear relationships, overlap of probability mass of input distribution with influential ranges of the parameter space is important. Differences in computed output distribution are greatest when input distributions differ in the most influential parameter range.     

Quantifying Water Pathogen Risk in an Epidemiological Framework

Traditionally, microbial risk assessors have used point estimates to evaluate the probability that an individual will become infected. We developed a quantitative approach that shifts the risk characterization perspective from point estimate to distributional estimate, and from individual to population. To this end, we first designed and implemented a dynamic model that tracks traditional epidemiological variables such as the number of susceptible, infected, diseased, and immune, and environmental variables such as pathogen density. Second, we used a simulation methodology that explicitly acknowledges the uncertainty and variability associated with the data. Specifically, the approach consists of assigning probability distributions to each parameter, sampling from these distributions for Monte Carlo simulations, and using a binary classification to assess the output of each simulation. A case study is presented that explores the uncertainties in assessing the risk of giardiasis when swimming in a recreational impoundment using reclaimed water. Using literature-based information to assign parameters ranges, our analysis demonstrated that the parameter describing the shedding of pathogens by infected swimmers was the factor that contributed most to the uncertainty in risk. The importance of other parameters was dependent on reducing the a priori range of this shedding parameter. By constraining the shedding parameter to its lower subrange, treatment efficiency was the parameter most important in predicting whether a simulation resulted in prevalences above or below non outbreak levels. Whereas parameters associated with human exposure were important when the shedding parameter was constrained to a higher subrange. This Monte Carlo simulation technique identified conditions in which outbreaks and/or nonoutbreaks are likely and identified the parameters that most contributed to the uncertainty associated with a risk prediction.     

Bayesian Forecasting via Deterministic Model

Rational decision making requires that the total uncertainty about a variate of interest (a predictand) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Suppose the state-of-knowledge is embodied in a deterministic model, which is imperfect and outputs only an estimate of the predictand. Fundamentals are presented of two Bayesian methods for producing a probabilistic forecast via any deterministic model. The Bayesian Processor of Forecast (BPF) quantifies the total uncertainty in terms of a posterior distribution, conditional on model output. The Bayesian Forecasting System (BFS) decomposes the total uncertainty into input uncertainty and model uncertainty, which are characterized independently and then integrated into a predictive distribution. The BFS is compared with Monte Carlo simulation and ensemble forecasting technique, none of which can alone produce a probabilistic forecast that quantifies the total uncertainty, but each can serve as a component of the BFS.