FIGURE 4.1 Monte Carlo analysis

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 

As practiced, Monte Carlo is not only a statistical method, but also a process that involves numerous cascading decisions involving statisticians, toxicologists, and risk assessors. The degree of belief inherent in the Monte Carlo outputs is as much a function of the numerous decisions the investigator makes during the course of the analysis as it is the correct selection of the sampling distributions. 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

Monte Carlo sampling is discussed extensively in Hammersley and Morton (1956), Hammersley and Handscomb (1964), Kloek and Van Dijk (1978), and Wilson (1984). For Monte Carlo results to be believable, the convergence properties of the Monte Carlo estimators must be met. Several statistical and practical limitations exist in this regard. The most important practical limitations of Monte Carlo are the following  

2) Use of Monte Carlo sampling with a large number of assumed independent parameters, particularly when the parameters are highly correlated 

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FIGURE 4.3 Exceedence curve for annual maximum atrazine concentrations in Tennessee pond water, based on exposure simulation using Monte Carlo analysis. 

Table 6.3 lists the summary statistical measures yielded by 3 analyses of this hypothetical calculation. The 2nd column gives the results that might be obtained by a standard Monte Carlo analysis under an independence assumption (the dotted lines in Figure 6.7). The 3rd and 4th columns give results from probability bounding analyses, either with or without an assumption of independence. 

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). 

The influence of the D2 partial pressure is similar for all catalysts as can be seen in Figure 4B. The Monte-Carlo analysis of this selectivity is given in Table 6. These analyses reveal that increasing the D2 partial pressure resulted in an increased contribution of the G-T)1 intermediate, whereas the amount of the di-o-r)2 intermediate was unaffected. The contribution of the 7t-r 2 and di-G-p1 intermediates was decreased with increasing D2 partial pressure, but the average number of rotations of the n-r intermediate was increased with higher D2 partial pressures. 

In Figure 3 (experiments) and Table 4 (results of Monte-Carlo analysis) give the selectivities in the H/D exchange of CP with the same two support materials. On the Si02 support, the... 

As is shown in Figure 6 (experiments) and Table 4 (Monte-Carlo analysis), a general trend is that Pt catalysts with supports of higher acidity lead to a higher contribution of the a-T)1 (Dl) and di-o-T)2 (D2) intermediates. As the ASA and LTL supports have similar metal particle sizes, this cannot be explained by particle size effects. Apparently, acidic supports enhance... 

Figure A2.8 Examples of the curves that can be plotted from this type of nested Monte Carlo analysis...

An added benefit of Monte Carlo analysis is that a common by-product of this computerized examination is a sensitivity analysis that shows how much each predictor variable contributed to the uncertainty or variability of the predictions. This, in turn, tells both the risk assessor and risk manager which portion of the variability is from natural fluctuation versus how much is caused by lack of knowledge. Given this information, decisions can be made as to where the most cost-effective allocation of resources may occur to refine the estimate of exposure and risk. In the example, the sensitivity analysis shown in Figure 4 presents the apportionment of variance for the model. 

The spreadsheets, however, may give an oversimplified result. In the case of a 100,000 accident that has a 5 percent probability of occurrence in each year, for example, the spreadsheets will show an annual cost of 5000. The actual cost will never be 5000 - it will be either 0 or 100,000 in any given year. A software program called TCAce was developed to properly accommodate uncertainties. Users enter data into TCAce, similarly to the way it is entered into the manual spreadsheets, but with more flexibility. TCAce can then perform an analysis similar to the simple analysis provided by the spreadsheets. It can also perform a Monte Carlo analysis, resulting in more realistic probable costs. Figure 6.11 shows the inconclusive result of an analysis between installing a water recovery system and... 

Figure 6.11. Using Monte Carlo analysis, the software program TCAce calculates the probability that an option will cost less than the baseline.

Figure 22.11 shows the conversion calculated by a Monte Carlo analysis, taking the deviations for the measured values in Table 22.8 into account Different conversion rates were calculated varying from 90 to 99%. The standard deviation is 0.79-0.89%. A systematic error is considered for the residual steam content between 1.02 and 1.67%. Cases (a)-(c) consider 1.02% and case (d) 1.67%. Finally, the calculated average values differ from the postulated conversion by 1-1.7%. 

Figure 10.16 A Comparison of the Profitability of Two Projects Showing the NPV with Respect to the Estimated Cumulative Probability from a Monte-Carlo Analysis...

As in previous examples, executing Listing 9.10 also produces a Monte Carlo analysis of the distribution of parameter values. With 8 parameters, there are a large number of joint parameter distributions that could be explored and studied. In many such types of data fitting what is most desired is an accurate estimate of the location of the peak heights (C2 and C5 in the model equation). Joint probability distributions for the parameters associated with one of the peaks, Ci, C2 and C3 are shown in Figures 9.38, 9.39 and 9.40. While there is some correlation among the three parameters, the three values can be seen to be relatively independent of each other. 

Figure 9.44 shows the joint probabilities between the Ci and C2 model parameters as determined in the Monte Carlo analysis of the model and data. While the analysis accurately determines the best fit values of the two model parameters, the joint probability figure shows that the two parameters are very closely correlated. In fact file data points collapse to essentially a straight line with negative slope in-... 

In viewing the joint probabilities from the Monte Carlo analysis, it is found that all the parameters are relatively independent of each other. The strongest correlation is between the C3 and C4 parameters, so only this joint probability is shown here in Figure 9.46. To calculate the correlation between the C3 and C4 parameters requires additional modifications to the code of Listing 9.6 from the changes... 

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%.

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

Figure 3. Histogram of Monte Carlo simulation for a synthetic alpha-spectrometric analysis (Table 1) of a sample with near-secular equilibrium No age can be calculated for the measured ...

A total of 10,000 iterations or calculations of dose were performed as part of this simulation, and Figure 4 shows the resulting distribution of average daily doses of chlorpyrifos as determined by the Monte Carlo simulation. Common practice in exposure and risk assessment is to characterize the 50th percentile as a "typical" exposure and the 95th percentile as the "reasonable maximum" exposure.4 The distributional analysis for these calculated doses... 

Figure 2. Multiscale modeling hierarchy. AIMD ab initio molecular dynamics, MD molecular dynamics, KMC kinetic Monte Carlo modeling, and FEA finite element analysis.

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