Probabilistic calculations

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

A popular way of determining the standard deviation for use in the probabilistic calculations is to estimate it by equation 4.21 which is based on the bilateral tolerance, t, and various empirical factors as shown in Table 4.7 (Dieter, 1986 Haugen, 1980 Smith, 1995). The factors relate to the fact that the more parts produced, the more confidence there will be in producing capable tolerances ... 

Historically, in probabilistic calculations, the standard deviation, cr, is expressed as t/3 (Dieter, 1986 Haugen, 1980 Smith, 1995 Welling and Lynch, 1985), which... 

It is frequently stated that probabilistic methods require more data than deterministic methods. This is not literally true it is possible to perform probabilistic calculations with input distributions based on small datasets or expert judgement. It is true that distributions derived from small datasets or expert judgement are likely to be very uncertain. However, if these uncertainties can be adequately represented within the probabilistic assessment, or dealt with by making conservative assumptions for the affected inputs, then probabilistic methods should still provide a useful refinement. Even in those cases where the uncertainties are too great to provide reliable estimates of exposure, probabilistic analysis may still be useful as a form of sensitivity analysis to identify priorities for data collection. 

In the final step (step 7) an overall analysis of the outcome of steps 1 through 6 for the endpoint of interest is performed. The toxicity of the parent compound is assessed via the information obtained for all the query compounds (metabolites, reaction products, analogues). The overall assessment should make use of all the available information (testing and nontesting data). A formal logical approach, such as decision theory, that evaluates all the possible options and uses probabilistic calculations to identify the best decision could be used to support the overall assessment process. This implies that additional decision analysis tools will need to be integrated into the DSS. 

Estimation of Risk. The estimation of risk contains uncertainties, based on the lack of specific data (such as exposure information) and/or the lack of understanding of the mechanism of toxic action of a compound. Between the extremes of acturial risk, which is based on enough information that "time has removed the uncertainty," such as the probability of death as cited in an insurance table, and theoretical risk, which is based on probabilistic calculations of events which have never actually occurred (e.g., nuclear "winter" (7)) lies a wide continuum into which most estimates of human health effects fall. In real-life situations, many assumptions are made in evaluating risk in order to make a conclusion, and these assumptions lead to uncertainties in the final result. These uncertainties should be understood as limitations to the best guess science can presently make. Although one response to this uncertainty, in the face of an outcome as fearsome as cancer, is to deny that there is a lack of certainty, the more reasonable response is to try to estimate the uncertainty, making it clear that any estimate is bracketed by these possible errors. 

Safety Lifecycle analyses heavily involve probabilistic calculations to verify the integrity of the safety design. 

The standards allow any internationally accepted methodology for the purpose of performing the probabilistic calculations and the most common methods are described in this book -- simplified equation, fault trees and Markov models. 

Probabilistic calculations are done to determine if the design meets the safety integrity requirement after ... 

Figure 7-11. SILver " Screen Shot with Probabilistic Calculations...

In the opinion of committee members on functional safety standards, some of the above factors cannot be practically quantified, e.g., systematic faults like software bugs or procedural errors. Hence functional safety standards provide requirements for protection against systematic faults as well as requirements to do probabilistic calculations to protect against random failures. For the typical SIF solutions being reviewed in this chapter the results of probabilistic SIL verification calculations, including architecture limitations per lEC 61508 (Ref. 1), will be used to demonstrate whether the design satisfies the SIL requirements. 

To verify that the design meets the requirements, a probabilistic calculation must be done to determine PFDavg, MTTFS and Architecture... 

In the case of probabilistic calculation and the ultimate limit state of the structure the load combination we take following ... 

ABSTRACT DDFPM has been developed as an alternative for Monte Carlo in the assessment of structural reliability in probabilistic calculations (Marek et al. 1995). Input random quantities (such as the load, geometry, material properties, or imperfections) are expressed as histograms in the calculations. In the probabilistic calculations, all input random variables are combined with each other. The munber of possible combinations is equal to the product of classes (intervals) of all input variables. With rather many input random variables, the number of combination is very high. Only a small portion of possible combinations results, typically, in failures. When DDFPM is used, the calculation takes too much time, because combinations are taken into account that does not contribute to the failure. Efforts to reduce the number of calculation operations have resulted into the development of algorithms that provide the munerical solution of the integral that defines formally the failure probability with rather many random variables ... 

Ad 3) definitions of those parts/zones of the histograms with variables that might, but not necessarily, participate in the failure probability and performance only of those probabilistic calculations mentioned... 

Janas, P., Krejsa, M., Krejsa, V 2006. Stmctural Reliability Assessment Using Direct Determined Fully Probabilistic Calculation (DDFPM). In International Asranet colloquium. Glasgow, UK ISBN 0-9553550-0-1/978-0-9553550-0, pp.8 and text on CD, ISBN 0-9553550-0-1/978-0-9553550-0-4 (in EngUsh). 

Goble, William M., Safety Instrumented Systems Verification Practical Probabilistic Calculation, ISA The Instrumentation, Systems, and Automation Society, 2006. 

H. Cheddie, Safety Instrumented Systems Verification Practical Probabilistic Calculations, IHS. 

Probabilistic design. For probabilistic calculations, Eq. (15.2) should be used together with a normal distribution and variation coefficient of CoV = 0.07. For prediction or comparison of measurements, the same Eq. (15.2) is used, but now for instance with the 5% lower and upper exceedance lines. 

The reliability of Eq. (15.8) is described by taking the coefficients 4.75 and 2.6 as normally distributed stochastic parameters with means of 4.75 and 2.6 and standard deviations a = 0.5 and 0.35, respectively. For probabilistic calculations, Eq. (15.8) should be taken together with these stochastic coefficients. For predictions of measurements or comparison with measurements also Eq. (15.8) should be taken with, for instance, 5% upper and lower exceedance curves. 

The mean prediction for impulsive conditions at a composite vertical structure is given by Eq. (16.8), and Fig. 16.15. The reliability of this equation is described by considering the scatter in the logarithm of the data about the mean prediction l°Sio(9measured) logiQ( predicted) IS taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation a = 0.28 (i.e., 68% of predictions lie within the range of x/ 1.9). For probabilistic calculations, Eq. (16.8) should be taken together with these stochastic coefficients ... 

A separated reliability analysis of structural and technical systems into dynamic by Eq. (12) and static by Eq. (14) or (15) parts helps designers to simplify probabilistic calculations. The presented unsophisticated probability-based approaches, design models and calculation equations may be convenient for many practitioners and can stimulate designers to use the full probabilistic methods in their practice more actively and effectively. 

The safety factors S represent estimates and empirical values and thus reflect to a certain extent the uncertainties regarding the history of both material and load [8]. Table 1.35 offers an insight. (The probabilistic or semi-probabilistic calculation method with boundary conditions enables a reduction to system-dependent safety factors [115]. However, considerable statistical effort is required to secure each individual dimensioning factor.)... 

This margin, serving to compensate the various uncertainties encountered (reliability of the data used, analysis of incompleteness, inaccuracy of the models, approximate calculations, etc.) is not huge, and therefore, it is necessary to perform analysis and probabilistic calculations with the greatest possible rigor, the best available data, and models close to actual physical systems. 

Perform probabilistic calculations to compute the probability distribution of ground motion at the site and select the design value at the appropriate probability level. 

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