Probabilistic models

Pit, F. (1993). Modelisation du melange pour la simulation d ecoulements reactifs turbulents Essais de modeles probabilistes euleriens-lagrangiens. Ph. D. thesis, Universite de Rouen, France. 

In order to illnstrate the effect of a probabilistic exposnre estimate versus a point estimate, a case study has been put together in which a German model point-estimate is compared to the German model probabilistic estimate when nsing a probability distribution generator (DistGen). 

Improving the detection limit in many cases is one of the most efficient ways to demonstrate lower exposure. Lor example the exposure to BADGE in canned foodstuffs (food and beverages) was estimated (Oldring et al. 2006) using a stochastic model (probabilistic - Monte-Carlo approach) with two different LODs of 0.3 t,g/dm and 0.5 t,g/dm and the exposure was effectively halved, primarily because many of the foodstuffs consumed were acidic, aqueous or alcoholic where the concentrations of BADGE and its regulated derivatives were non-detectable. 

The objective of the Risk Analysis is to identify risks that may endanger the safety of the plant and the health of the persotmel, or to impact the environment these risks should be controlled either by preventing them to occur in the first place or mitigating their consequences. During this process the analyst may have to model probabilistically in an integrative manner, human errors, hardware failures, environmental impacts, and the managerial effects (Kafka 1996). 

Currently, the landslide hazard spatial prediction methods can be divided into qualitative methods and quantitative methods. As we all know qualitative forecasting method mainly depends on the subjective experience and the predicted accuracy of qualitative methods is lower than it of quantitative methods. So the qualitative methods have been gradually replaced by the quantitative methods. Quantitative models can be divided into statistic analysis models, deterministic models, probabilistic model, fuzzy information optimization processing and neurd network models. 

The basis for the protective action distances given in Table 1 is based on analysis using state-of-the-art source term and vapor cloud dispersion modeling, probabilistic application of acmal atmospheric data, and information on toxicological exposure guidelines for each chemical. 

Dennis, C. Somaiya, K. 2004. Development and Use of the UK Railway Network s Safety Risk Model, Probabilistic Safety Assessment and Management PSAM 7—ESREL 04... 

Wei, Z., Yang, F., Lin, B., Luo, L., Konson, D. Nikbin, K. 2013. Deterministic and probabilistic creep-fatigue-oxidation crack growth modeling. Probabilistic Engineering Mechanics 33(0) 126-134. 

Modeling probabilistic and quantitative aspects, such as probabilistic faults and performability measures. 

Our component-oriented approach for analysing failure behaviour focuses on the transitions between modes of a component. Since we introduced a probability property into the failure model, we need a formal mechanism to verify the probabilistic model. Probabilistic model checking [7] is a suitable mechanism to use for verification in this situation. Probabilistic model checkers encode system models using Markov chains in this sense, they encode the probability of making a transition between states instead of simply the existence of a transition. The probabilistic model checking process is an automatic procedure for establishing if a desired property holds in a probabilistic system model. We exploit probabilistic model checking - and, in particular, PRISM - to accomplish three purposes. 

Modeling probabilistic aspects, such as random faults, repairs, and stochastic timing. 

Barbato M, Conte JP (2008) Spectral characteristics of non-stationary random processes theory and applications to linear structural models. Probabilist Eng Mech 23(4) 416 26... 

The unimolecular rate law can be justified by a probabilistic argument. The number (A Vdc x dc) of particles which react in a time dt is proportional both to this same time interval dt and to the number of particles present (A Vc x c). However, this probabilistic argument need not always be valid, as illustrated in figure A3.4.2 for a sunple model [20] ... 

Nature In monitoring a moving threadhne, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadhne by a broken-filament detector mounted adjacent to the threadhne. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include ... 

Model Validity Probabilistic failure models cannot be verified. Physical phenomena are observed in experiments and used in model correlations, but models are, at best, approximations of specific accident conditions. 

AL Delcher, S Kasif, HR Goldberg, WH Hsu. Protein secondary structure modelling with probabilistic networks. Intelligent Systems m Molecular Biology 1 109-117, 1993. 

Statistical methods for probabilistic design 4.2.1 Modelling data using statistical distributions... 

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. 

The use of computers is essential in probabilistic design (Siddal, 1983). However, research has shown that even the most complete computer supported analytical methods do not enable the designer to predict reliability with sufficiently low statistical risk (Fajdiga et al., 1996). Far more than try to decrease the statistical risk, which is probably impossible, it is hoped that the approach will make it possible to model a particular situation more completely, and from this provide the necessary redesign information which will generate a reliable design solution. 

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 ... 

All pictorial representations of molecules are simplified versions of our current model of real molecules, which are quantum mechanical, probabilistic collections of atoms as both particles and waves. These are difficult to illustrate. Therefore we use different types of simplified representations, including space-filling models ball-and-stick models, where atoms are spheres and bonds are sticks and models that illustrate surface properties. The most detailed representation is the ball-and-stick model. However, a model of a protein structure where all atoms are displayed is confusing because of the sheer amount of information present (Figure 2.9a). 

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,... 

Probabilistic CA. Probabilistic CA are cellular automata in which the deterministic state-transitions are replaced with specifications of the probabilities of the cell-value assignments. Since such systems have much in common with certain statistical mechanical models, analysis tools from physics are often borrowed for their study. Probabilistic CA are introduced in chapter 8. 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

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