Monte Carlo uncertainty analysis

Greenland, S. (2001). Sensitivity analysis, Monte Carlo risk analysis, and Bayesian uncertainty assessment. Risk Anal 21, 579-583. 

Balsamo A, Di Ciommo M, Mugno R, Rebaglia BI, Ricci E, Grella R (1999) Evaluation of CMM uncertainty through Monte Carlo simulations. Ann CIRP -Manuf Technol 48(l) 425-428 Cox MG, Dainton MP, Harris PM (2001) Software specifications for uncertainty calculation and associated statistical analysis. National Physics Laboratory (NPL) report CMSC lOAll. National Physics Laboratory, Teddington... 

The Systems Module constructs and displays fault trees using EASYFLOW which aic read automatically to generate minimal cutsets that can be transferred, for solution, to SETS. CAFT A. or IRRAS and then transferred to RISKMAN for point estimates and uncertainty analysi,s using Monte Carlo simulations or Latin hypercube. Uncertainty analysis is performed on the systems lev el using a probability quantification model and using Monte Carlo simulations from unavailability distributions. 

MONTE - Performs a Monte Carlo uncertainty analysis using the uncertainties in the data to estimate the uncertainty in the calculation of the system and subsystem failure probabilities. 

Blau et al. [22] have applied probabilistic network models to model resource needs and success probabilities in pharmaceutical and agrochemical development, through Monte Carlo analysis. This requires solving the problem of scheduling a portfolio of projects under uncertainty about progression. This approach is tractable for drug development. However, the inherent complex-... 

Figure 22.4 Monte Carlo techniques were used to simulate different hypothetical individuals for different instances of the trial design, using variability and uncertainty distributions from the model analysis. The result is a collection of predicted outcomes, shown as a binned histogram (top figure). Success was defined as a difference in end point measurement of X or smaller between drug and comparator. Likelihood of success (shown in the bottom figure as a cumulative probability) for this example (low/medium drug dose and high comparator dose) is seen to be low, about 33%.

Cronin WJ, Oswald EJ, Shelley ML, et al. 1995. A trichloroethylene risk assessment using a Monte Carlo analysis of parameter uncertainty in conjunction with physiologically-based pharmacokinetic modeling. Risk Anal 15 555-565. 

RISK RISK performs risk analysis using Monte Carlo simulation to conduct sensitivity and uncertainty analyses... 

Local sensitivity analysis is of limited value when the chemical system is non-linear. In this case global methods, which vary the parameters over the range of their possible values, are preferable. Two global uncertainty methods have been used in this work, a screening method, the so-called Morris One-At-A-Time (MOAT) analysis and a Monte Carlo analysis with Latin Hypercube Sampling (Saltelli et al., 2000 Zador et al., submitted, 20041). The analyses were performed by varying rate parameters, branching ratios and constrained concentrations within their uncertainty interval,... 

Monte-Carlo analysis is typically applied to handle uncertainty if several data sources are available to identify most probable costs. 

Monte Carlo—A statistical technique commonly used to quantitatively characterize the uncertainty and variability in estimates of exposure or risk. The analysis uses statistical sampling techniques to obtain a probabilistic approximation to the solution of a mathematical equation or model. 

The estimates of an NCE s expected sales trajectory can be analyzed using a Monte Carlo technique, which produces a range of possible outcomes, given uncertainty about the actual magnitudes of various input assumptions. The results of a Monte Carlo analysis... 

Eurthermore, uncertainties in the exposure assessment should also be taken into account. However, no generally, internationally accepted principles for addressing these uncertainties have been developed. For predicted exposure estimates, an uncertainty analysis involving the determination of the uncertainty in the model output value, based on the collective uncertainty of the model input parameters, can be performed. The usual approach for assessing this uncertainty is the Monte Carlo simulation. This method starts with an analysis of the probability distribution of each of the variables in the uncertainty analysis. In the simulation, one random value from each distribution curve is drawn to produce an output value. This process is repeated many times to produce a complete distribution curve for the output parameter. 

Uncertainty analysis is increasingly used in ecological risk assessment and was the subject of an earlier Pellston workshop (Warren-Hicks and Moore 1998). The US Environmental Protection Agency (USEPA) has developed general principles for the use of Monte Carlo methods (USEPA 1997), which provide one of several approaches to incorporating variability and uncertainty in risk assessment. 

Uncertainty analysis for multiparameter models may require assigning sampling distributions to many random parameters. In which case, a single value is drawn from each of the respective sampling distributions during each Monte Carlo iteration. After each random draw, the generated values of the random parameters... 

The popularity of Monte Carlo for risk-based uncertainty analysis is somewhat driven by the fact that Monte Carlo is fundamentally easy to implement, particularly with the advent of the personal computer, and graphically based software like Crystal Ball (www.decisioneering.com) and Risk (www.palisade. com/risk.html). The availability of such software systems generally promotes the use of uncertainty analysis in ecological risk assessments, reducing the amount of mathematical and statistical knowledge required of the user to implement the... 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

If you can perform an uncertainty analysis with a calculator, do not use Monte Carlo analysis or at least compare the calculator and simulation method for consistency. 

Dilks DW, Canale RP, Meier PG. 1989. Analysis of model uncertainty using Bayesian Monte Carlo. In Proceedings of ASCE Specialty Conference on Environmental Engineering. New York American Society of Civil Engineers, p 571-577. 

The following section describes the most commonly used technique for propagating variability and parameter uncertainty separately, 2nd-order Monte Carlo analysis. A brief case study illustrating the technique is included in Section 7.3. 

Second-order Monte Carlo analysis consists of 2 loops, the inner loop representing variability and the outer loop representing parameter uncertainty. To conduct an analysis, the following steps are required (also see Figure 7.1) ... 

Specify the number of inner and outer loop simulations for the 2nd-order Monte Carlo analysis. In the 1st outer loop simulation, values for the parameters with uncertainty (either constants or random variables) are randomly selected from the outer loop distributions. These values are then used to specify the inner loop constants and random variable distributions. The analysis then proceeds for the number of simulations specified by the analyst for the inner loop. This is analogous to a Ist-order Monte Carlo analysis. The analysis then proceeds to the 2nd outer loop simulation and the process is repeated. When the number of outer loop simulations reaches the value specified by the analyst, the analysis is complete. The result is a distribution of distributions, a meta-distribution that expresses uncertainty both from uncertainty and from variability (Figure 7.1). 

There are some issues associated with 2nd-order Monte Carlo analysis. Computational time can be a problem because the necessary number of replicates is squared with 2nd-order Monte Carlo analyses (i.e., number of inner loop simulations times number of outer loop simulations). In practice, specifying variability and uncertainty with random variables is a difficult exercise because the analyst is essentially trying to quantify what he or she does not know or only partially understands. 

The major benefit of 2nd-order Monte Carlo analysis is that it allows analysts to propagate their uncertainty about distribution parameters in a probabilistic analysis. An analyst need not specify a precise estimate for an uncertain parameter value simply because one is needed to conduct the simulation. The relative importance of our inability to precisely specify values for constants or distributions for random variables can be determined by examining the spread of distributions in the output. If the spread is too wide to promote effective decision making, then additional research is required. 

Inputs for a 2nd-order Monte Carlo analysis to estimate exposure of Carolina wrens to a hypothetical pesticide (random variables are included in the inner loop of the Monte Carlo analysis, while random variables with uncertainty are included in both the inner and outer loops of the Monte Carlo analysis)... 

Burmaster and Anderson (1994) have compiled a list of principles of good practice, which were originally aimed at Monte Carlo simulations, but are valuable also for other techniques in uncertainty analysis. These recommendations later appeared in modified and supplemented form in various handbooks and other publications, e.g., the USEPA Guiding Principles for Monte Carlo Analysis (USEPA 1997). 

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