Probabilistic Modeling

Final element assemblies are modeled like any combination of components using the system reliability engineering techniques. Failure mode classification is generally straightforward. The key variables to include in the evaluation of failure rates are  

Stress level service category (clean, fouling, severe, etc.), and 

Valves that require tight shutoff will have higher failure rates because certain stress events that damage the seat or the ball will be classified a failure. Such events would not be classified as a failure if a small leakage is allowed. 

The application of a valve will affect its failure rates and failure modes. One key variable is position of the valve in order for a safety instrumented function to achieve its safe state. The valve must be designed to be close to trip or open to trip. There are generally no tight shut-off requirements when the valve opens to perform the trip. 

Valves are exposed to highly variable process conditions that affect the failure rates and failure modes of all valves. Two primary categories of service have been used - clean service and severe service. 

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In the probabilistic design calculations, the value of Kt would be determined from the empirical models related to the nominal part dimensions, including the dimensional variation estimates from equations 4.19 or 4.20. Norton (1996) models Kt using power laws for many standard cases. Young (1989) uses fourth order polynomials. In either case, it is a relatively straightforward task to include Kt in the probabilistic model by determining the standard deviation through the variance equation. 

In experimental load studies, the measurable variables are often surface strain, acceleration, weight, pressure or temperature (Haugen, 1980). A discussion of the techniques on how to measure the different types of load parameters can be found in Figliola and Beasley (1995). The measurement of stress directly would be advantageous, you would assume, for use in subsequent calculations to predict reliability. However, no translation of the dimensional variability of the part could then be accounted for in the probabilistic model to give the stress distribution. A better test would be to output the load directly as shown and then use the appropriate probabilistic model to determine the stress distribution. 

Before a probabilistic model can be developed, the variables involved must be determined. It is assumed that the variables all follow the Normal distribution and that they are statistically independent, i.e. not correlated in anyway. The scatter of the pre-load, F, using an air tool with a clutch is approximately 30% of the mean, which gives the coefficient of variation, = 0.1, assuming 3cr covers this range, therefore ... 

Siu, N. and G. Apostolakis, 1982, Probabilistic Models for Cable Tray Fires, Rclinbility Engineering 3, p 213. 

Siu, N., 1980, Probabilistic Models for the Behavior of Compartment Fires, UCLA-ENG-8090,... 

A Hidden Markov Model (HMM) is a general probabilistic model for sequences of symbols. In a Markov chain, the probability of each symbol depends only on the preceding one. Hidden Markov models are widely used in bioinformatics, most notably to replace sequence profile in the calculation of sequence alignments. 

LAMBE j (2002) The use of food consumption data in assessments of exposure to food chemicals using the application of probabilistic modelling. Proc Nutrn Soc. 61 11-18. 

F. Rosenblatt, The perceptron a probabilistic model for information storage and organization in the brain. Psycholog. Rev., 65 (1958) 386-408. 

As probabilistic exposure and risk assessment methods are developed and become more frequently used for environmental fate and effects assessment, OPP increasingly needs distributions of environmental fate values rather than single point estimates, and quantitation of error and uncertainty in measurements. Probabilistic models currently being developed by the OPP require distributions of environmental fate and effects parameters either by measurement, extrapolation or a combination of the two. The models predictions will allow regulators to base decisions on the likelihood and magnitude of exposure and effects for a range of conditions which vary both spatially and temporally, rather than in a specific environment under static conditions. This increased need for basic data on environmental fate may increase data collection and drive development of less costly and more precise analytical methods. 

Stochastic or probabilistic techniques can be applied to either the moisture module, or the solution of equation (3) — or for example the models of Schwartz Crowe (13) and Tang et al. (16), or can lead to new conceptual model developments as for example the work of Jury (17). Stochastic or probabilistic modeling is mainly aimed at describing breakthrough times of overall concentration threshold levels, rather than individual processes or concentrations in individual soil compartments. Coefficients or response functions and these models have to be calibrated to field data since major processes are studied via a black-box or response function approach and not individually. Other modeling concepts may be related to soil models for solid waste sites and specialized pollutant leachate issues (18). 

Schwartz, F.W., A. Growe (1980). A deterministic probabilistic model for contaminant transport, US NRC, NUREG/CR-1609, Washington, DC. 

It does not contain a probabilistic modeling component that simulates variability therefore, it is not used to predict PbB probability distributions in exposed populations. Accordingly, the current version will not predict the probability that children exposed to lead in environmental media will have PbB concentrations exceeding a health-based level of concern (e.g., 10 pg/dL). Efforts are currently underway to explore applications of stochastic modeling methodologies to investigate variability in both exposure and biokinetic variables that will yield estimates of distributions of lead concentrations in blood, bone, and other tissues. 

Coating Mass Uniformity and Distribution. Basically, there have been two approaches to model the accumulation of mass (coating material) on the surface of bed particles (i) the use of population balances and (ii) the probabilistic modelling of the spray-particle interaction. We will look at each of these approaches and see how it may be possible to combine these methods to give a fuller picture of coating performance. 

Probabilistic modelling works by taking a random sample from the distribution of additive levels for a given food and combining it with a random sample from the distribution of food consumption for that food. This sampling is repeated several thousand times until a smooth intake distribution curve is produced. 

Probabilistic modelling has been widely applied for determining acute intakes of pesticide residues. The method works well because for any randomly selected individual, the level of pesticide residue in a given food item is a... 

Durbin, R., Eddy, S., Krogh, A., and Mitchison, G. (1998). Biological Sequence Analysis Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, Cambridge, Massachusetts. 

Ho CK (2004) Probabilistic modeling of peracutaneous absorption for risk-based exposure assessments and transdermal drug delivery. Statistical Methodology 1 47-69... 

Segal E, Taskar B, Gasch A, Friedman N, Koller D. 2001. Rich probabilistic models for gene expression. Bioinformatics 17Suppl 1 S243. 

More sophisticated probabilistic models are used by EPA to comply with the aggregate and cumulative risk provisions of the FQPA. These models consider rolling windows of exposure, toxicological equivalence factors for pesticides that have common toxicological mechanisms, and include methods to incorporate exposure from drinking water and residential pesticide use into the pesticide exposure estimates. 

A more general statistical (probabilistic) model of the system takes into account both an offset (Pq) and uncertainty (r,). 

Figure 4.5 Graphs of two fitted probabilistic models of the form y, = Po +...

Figure 5.4 Graph of the fitted probabilistic model >>, = PiXu + r...

Let us fit the probabilistic model, = Po + r, to the same data (see Figure 5.10). If the least squares approach to the fitting of this model is employed, the appropriate matrices and results are exactly those given in Section 5.2 where the same model was fit to the different factor levels j , = 3, yj, = 3, = 6, y, = 5. This identical... 

Before leaving this chapter we will consider one final model, the purely probabilistic model y = 0 + r, (see Section 4.2 and Equation 4.3). Whether obtained at different levels of the factor or at the same level (replicates), there is a possibility that the two observed responses, y, = 3 and y,2 = 5, belong to a population for which the mean (p) is zero. The fact that the two numbers we have... 

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