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Monte Carlo simulation

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Contents
  1. 1Principle
  2. 2Random numbers and sampling
  3. 3Number of runs and accuracy
    1. 3.1Worked example
  4. 4Use in uncertainty and sensitivity analysis
  5. 5Applications in fire risk analysis
  6. 6Limits and common misconceptions
  7. 7See also
  8. 8Notes
  9. 9References

Monte Carlo simulation is a technique for problems with a random or probabilistic character, in which a calculation is repeated many times with input values drawn at random from probability distributions.[1] Each run gives one possible outcome. Together, the runs approximate the probability distribution of the result, and the approximation improves as more runs are made.[2] In fire safety engineering, Monte Carlo simulation is a common tool of uncertainty analysis. It is used to propagate uncertain inputs through fire models, to estimate the risk to occupants, and to quantify the uncertainty in fire risk assessments for nuclear power plants.[3][4]

The method turns a single "best estimate" into a range of possible outcomes with their probabilities.[5] A design can then be judged by statements such as "a probability of 0.9 of having 4 min or more before the corridor reaches 65 °C".[6] In a comparison of methods for uncertainty and sensitivity analysis of large computer models, Monte Carlo sampling performed best overall.[7] Its main limits are the computing time for complex models and the quality of the input distributions.[2][8]

Principle

Monte Carlo analysis is used for two different kinds of problem. In the first, the process itself is random. The rate of flame spread, the growth of a fire and the response of people to an alarm are examples of quantities that may be treated as random. A random component then enters the model during each run.[1] In the second kind, the process is treated as deterministic, but its starting conditions and parameters are drawn at random from probability distributions.[1]

The second kind corresponds to the propagation of uncertain inputs through a model. The exact value of an input parameter may be unknown. It may still be possible to estimate its mean, its variance and the form of its distribution, and so to describe it as a probability density.[1] The model is then run many times, each time with a new set of sampled inputs. The collected outputs, for example fire conditions or numbers of casualties, form an estimate of the probability distribution of the result.[1]

The results are usually presented as a cumulative distribution. For n runs, the n values of an outcome are sorted from largest to smallest. The largest value is plotted against 1/n, the next against 2/n, and so on, until the smallest value is plotted against 1.[9] From such a curve, a designer can read the probability of having at least a certain time before a critical condition is reached.[9]

Outputs can be skewed even when the inputs are symmetrical. Certain combinations of input values can reinforce each other, which is typical of processes in which factors multiply. Distributions of fire damage are often skewed, with a long tail towards large losses, and can often be described by a log-normal distribution.[10] Such tails are where the worst-case scenarios are found.[6]

Setting up a study starts with choosing which inputs to treat as uncertain. Ideally, only parameters whose uncertainty is large enough to change decisions about the design are sampled.[6] Because they are not always known in advance, judgement and the results of earlier analyses are combined. All available knowledge about each input is built into its distribution, including measured values, known variations and statistics. Examples are the share of fires that start in each type of room and regional weather data.[5]

Random numbers and sampling

A Monte Carlo study needs a source of random numbers. Computers supply pseudo-random numbers between 0 and 1. These are not truly random, because a deterministic algorithm generates them and the sequence eventually repeats, but the algorithm makes this cycle as long as possible.[11] The samples are drawn from an assumed theoretical distribution. If no standard distribution describes a process, an empirical distribution based on data can be used instead.[11]

Methods exist for generating values from most of the well-known probability distributions, as well as from any empirical distribution.[1] The inverse transformation method converts a uniform random number r into a value x with the desired distribution. It uses the cumulative distribution function F(x), which runs from 0 to 1, and requires that its inverse is known:[12]

F(x)=r⇒x=F−1(r)

For example, a variable with the probability density f(x)=2x between 0 and 1 has F(x)=x2, so its values are generated by taking the square root of uniform random numbers.[12] When the inverse function cannot be found, the rejection method can be used. Random pairs of numbers are generated, and only the pairs that fall under the scaled density curve are accepted. The same idea can be used to evaluate definite integrals.[13]

Inputs are often not independent of each other. Outdoor temperature, for example, is related to wind, humidity and the likelihood of open windows. If such correlations are not modelled, the generator may produce unrealistic scenarios, such as a fire on a freezing day in August in California with all windows open.[5] Besides simple random sampling, other methods such as Latin hypercube sampling and quasi-random sampling are available.[5] Fire risk assessments for nuclear power plants typically use Monte Carlo or Latin hypercube sampling.[4]

Number of runs and accuracy

More runs give a more accurate picture of the output distribution, but the gain is not linear and depends on the sampling method. A study with 2000 runs may give no more insight than one with 500.[5] The number of runs needed depends on the number of uncertain inputs, the computing time of each run and the statistical significance required.[5]

When the aim is to estimate a probability, each run can be seen as a trial that either exceeds a limit or does not. The number of exceedances k in N runs then follows a binomial distribution, with mean Np and variance Np(1−p).[14] Dividing by N gives the standard error of the estimated probability p^=k/N, which is the familiar standard error of a proportion:[15]

SE(p^)=p(1−p)N

To halve the error, four times as many runs are needed. The normal approximation behind a 95 % interval of about two standard errors works when Np and N(1−p) are both at least 5.[14][15] Rare events therefore need many runs. For a probability of 0.01 and 1000 runs, the standard error is about 0.003, almost a third of the estimate itself.[14]

When the aim is to find out which inputs matter, correlation coefficients between inputs and outputs are calculated. These coefficients lie between 0 and 1.[5] A formula attributed to Hald relates the number of runs n to the statistical significance of a correlation coefficient c. In it, t is the test statistic of a t-test, linked to the chosen confidence level, typically 95 %:[5]

t=c1−c2n−2


Symbol Description Unit
r uniform random number between 0 and 1 –
x sampled value of an input unit of the input
f(x),F(x) probability density and cumulative distribution of x 1/unit, –
N,n number of runs –
k number of runs that exceed a limit –
p,p^ true and estimated probability of exceeding the limit –
SE standard error of the estimate –
c correlation coefficient between an input and an output –
t test statistic of the t-test –

Worked example

A validation report of the NRC uses an example with electrical cables that fail at 330 °C. The fire model FDS predicts a maximum cable temperature of 300 °C in a room at 20 °C, and the question is how likely failure is.[16] Using the validation statistics of FDS for target temperatures, the true temperature rise can be treated as normally distributed. It has a mean of about 283 °C and a standard deviation of about 48 °C.[17] The critical rise is 330 − 20 = 310 °C. Evaluated exactly with the complementary error function, the probability of exceeding it is about 0.29.[18]

The same probability can be found by Monte Carlo simulation. Each run draws a uniform random number, converts it into a temperature rise with the inverse of the normal distribution, and checks whether the rise exceeds 310 °C.[12] A set of runs with a fixed seed gave the results in the table. The standard errors follow from the binomial formula with p = 0.286.[14]

Number of runs Runs above 310 °C Estimated probability Standard error
100 24 0.24 0.045
1,000 276 0.276 0.014
10,000 2,896 0.290 0.0045
100,000 28,445 0.284 0.0014

With 100 runs, the estimate is still about one standard error away from the exact value. With 10,000 or more runs, it agrees to within about 0.01. Each tenfold increase in runs reduces the standard error by a factor of about 3.2, the square root of 10.[14]

Hald's formula gives a similar warning for sensitivity studies. With t = 1.96, about 95 runs are needed before a correlation of 0.2 between an input and an output is statistically significant, and about 383 runs for a correlation of 0.1.[5] With 500 runs, correlations above about 0.09 can be distinguished from chance.

Use in uncertainty and sensitivity analysis

Monte Carlo sampling is one of three main ways to examine the uncertainty and sensitivity of complex computer models. The others are the direct calculation of derivatives and the replacement of the model by a fitted response surface.[2] A comparison by Iman and Helton found that the direct method gives good local information but does not extend well to global conclusions. Response surfaces were not recommended, because fire models are often too complex to be represented by a simple function.[7] Monte Carlo is a well-accepted method, although it is not necessarily the best choice in all cases.[2]

The same runs also support a sensitivity analysis. Regressing the outputs on the inputs creates a linear model of the simulation model, and the regression coefficients correspond to the partial derivatives of the outputs with respect to the inputs.[10] An importance analysis ranks the inputs by their correlation with the output. A correlation of 0 means that the uncertainty of an input has no effect on the uncertainty of the output.[19] If an input has little effect, there is no need to spend effort on refining its value.[1] One handbook chapter suggests varying each assumed input by 10 % and treating the output as insensitive to that input if it changes by less than 10 %.[20]

Monte Carlo methods also serve fire modelling directly. A methodology for performance-based design uses Monte Carlo sampling to generate a whole distribution of design fire scenarios. The set contains typical fires as well as the worst cases in the tails of the distribution.[6] Each candidate design is then evaluated against every scenario, and designs can be compared through the cumulative distributions of their results.[5][21]

Applications in fire risk analysis

Monte Carlo simulation is widely used in quantitative risk assessment. The CRISP II model of the Fire Research Station in the United Kingdom is a Monte Carlo simulation of an entire fire scenario.[10] It combines a two-zone smoke model for multiple rooms with a detailed model of human behaviour and movement. An occupant whose fractional effective dose reaches 100 % is counted as a fatality, and the risk is expressed as the number of deaths.[10] Earlier stochastic models of fire growth used Monte Carlo simulation to generate random chains of critical events, such as a fire passing through a given heat release rate.[22]

Fire risk assessments for nuclear power plants use the method to treat uncertainty in the inputs. The random course of events after a fire, such as equipment failures and operator actions, is represented by the structure of the model, including event tree analysis.[23] The fire frequencies and failure probabilities in these trees are themselves uncertain because knowledge is incomplete. These epistemic uncertainties are described by probability distributions and propagated with Monte Carlo or Latin hypercube sampling. The result is a probability distribution for the core damage frequency rather than a single value.[4] One handbook chapter describes direct Monte Carlo simulation as perhaps the easiest way to do this with current software.[24]

Monte Carlo methods are also used in decisions about fire safety measures. A cost-benefit model of mandatory residential sprinklers in the United States propagated the uncertainty in fire statistics and costs by Monte Carlo simulation.[25] The net cost per premature death averted varied widely. It ranged from $1.4 million for new mobile homes in small communities in the South to $35.1 million for retrofitting dwellings in the West, a factor of about 25.[26] The median for the national average case was $7.3 million.[25]

Limits and common misconceptions

A common misconception is that more runs always make a result more reliable. More runs only reduce the sampling error. The result depends on the input distributions supplied to the model, and data to define such distributions are often scarce.[1][8] Where hard data do not exist, expert elicitation is needed to quantify the uncertainty, and correlations between inputs must be represented.[5]

Computing time limits the use of Monte Carlo with detailed models. Large fire models are often time-consuming to run, have many inputs and outputs, and behave in nonlinear ways.[2] A single run of the CFD model FDS can take hours or days, whereas zone models give answers in seconds to minutes.[27] Because the number of runs depends on the computing time of each run, this difference strongly limits how many CFD runs a Monte Carlo study can afford.[5]

A second misconception is that every uncertainty should be sampled in the same way. In risk assessments, the random variability of events and the lack of knowledge about parameters should be kept apart where possible. The first is represented in the event trees, and only the second is propagated by sampling.[23][24] Finally, a strong correlation between an input and an output found in a Monte Carlo study does not prove a causal link. Counterintuitive correlations should be checked with engineering judgement.[19]

See also

Notes

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Phillips (2002), p. 5-115.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Notarianni (2002), p. 5-51.
  3. ↑ Phillips (2002), p. 5-112.
  4. ↑ 4.0 4.1 4.2 Nowlen et al. (2005b), p. 15-2.
  5. ↑ 5.00 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10 5.11 Notarianni (2002), p. 5-55.
  6. ↑ 6.0 6.1 6.2 6.3 Notarianni (2002), pp. 5-53–5-54.
  7. ↑ 7.0 7.1 Notarianni (2002), pp. 5-51–5-52.
  8. ↑ 8.0 8.1 Notarianni (2002), p. 5-44.
  9. ↑ 9.0 9.1 Notarianni (2002), p. 5-56.
  10. ↑ 10.0 10.1 10.2 10.3 Phillips (2002), p. 5-116.
  11. ↑ 11.0 11.1 Phillips (2002), p. 5-120.
  12. ↑ 12.0 12.1 12.2 Phillips (2002), pp. 5-120–5-121.
  13. ↑ Phillips (2002), pp. 5-121–5-122.
  14. ↑ 14.0 14.1 14.2 14.3 14.4 Hall (2002a), p. 1-190.
  15. ↑ 15.0 15.1 Hall (2002c), p. 1-198.
  16. ↑ Stroup et al. (2016), pp. 6-3–6-4.
  17. ↑ Stroup et al. (2016), p. 6-57.
  18. ↑ Stroup et al. (2016), pp. 6-3–6-4. The report states a probability of about 0.40 for this example; evaluating its Equation 6-9 with the values it gives yields about 0.29.
  19. ↑ 19.0 19.1 Notarianni (2002), p. 5-59.
  20. ↑ Phillips (2002), pp. 5-119–5-120.
  21. ↑ Notarianni (2002), p. 5-58.
  22. ↑ Ramachandran (2002c), p. 3-389.
  23. ↑ 23.0 23.1 Nowlen et al. (2005b), pp. U-4–U-5.
  24. ↑ 24.0 24.1 Siu (2002), pp. 5-221–5-222.
  25. ↑ 25.0 25.1 Notarianni (2002), p. 5-62.
  26. ↑ Notarianni (2002), p. 5-62. The chapter describes this variation as "greater than a factor of 35"; the two values it gives differ by a factor of about 25.
  27. ↑ Stroup et al. (2016), p. 6-56.

References

  • Hall, J. R., Jr. (2002a). Probability concepts. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 1-182–1-192). National Fire Protection Association; Society of Fire Protection Engineers.
  • Hall, J. R., Jr. (2002c). Statistics. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 1-193–1-202). National Fire Protection Association; Society of Fire Protection Engineers.
  • Notarianni, K. A. (2002). Uncertainty. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 5-40–5-64). National Fire Protection Association; Society of Fire Protection Engineers.
  • Nowlen, S. P., Najafi, B., Wyant, F. J., Forester, J., Kazarians, M., Kolaczkowski, A., Joglar, F., Funk, D., Hannaman, G., & Anoba, R. (2005b). EPRI/NRC-RES fire PRA methodology for nuclear power facilities: Vol. 2. Detailed methodology (NUREG/CR-6850; EPRI 1011989). Electric Power Research Institute; U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research.
  • Phillips, W. G. B. (2002). Computer simulation for fire protection engineering (D. K. Beller & R. F. Fahy, Rev.). In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 5-112–5-124). National Fire Protection Association; Society of Fire Protection Engineers.
  • Ramachandran, G. (2002c). Stochastic models of fire growth. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-381–3-401). National Fire Protection Association; Society of Fire Protection Engineers.
  • Siu, N. (2002). Fire risk assessment for nuclear power plants. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 5-214–5-226). National Fire Protection Association; Society of Fire Protection Engineers.
  • Stroup, D., Lindeman, A., McGrattan, K., Peacock, R., Overholt, K., Joglar, F., LeStrange, S., & Montanez, S. (2016). Verification and validation of selected fire models for nuclear power plant applications: Supplement 1 (NUREG-1824, Supplement 1; EPRI 3002002182). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
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