Probabilistic distribution

Vermeire et al. (1999) have published a discussion paper with focus on assessment factors for human health risk assessment. The status quo with regard to assessment factors is reviewed and the paper discusses the development of a formal, harmonized set of assessment factors. Options are presented for a set of default values and probabilistic distributions for assessment factors based on the state of the art. Methods of combining default values or probabUistic distributions of assessment factors (Section 5.11) are also described. In relation to assessment factors, the authors recommended ... 

A more recent Dutch report (Vermeire et al. 2001) provides a practical guide for the application of probabilistic distributions of default assessment factors in human health risk assessments, and it is stated that the proposed distributions will be applied in risk assessments of new and existing substances and biocides prepared at RIVM (the National Institute of Public Health and the Environment) and TNO. The report concentrated on the quantification of default distributions of the assessment factors related to interspecies extrapolation (animal-to-human), intraspecies extrapolation (human-to-human), and exposure duration extrapolation. 

The use of default assessment factors is recommended in risk assessments, when justifiable, although the scientific background for such factors in general was considered unsatisfactory. The default assessment factors suggested are summarized in Table 5.2. It is recommended to use assessment factors derived from probabilistic distributions in favor of deterministic assessment factors, see Table 5.2. 

According to Vermeire et al. (1999, 2001), several theoretical probabilistic distributions have been proposed. Distributions proposed by Price et al. (1997), Swartout et al. (1998), and Slob and Pieters (1998) as cited in Vermeire et al. (1999, 2001), were considered to be consistent with the current use of the default factor of 10 and these authors found the traditional factor 10 to be conservative, see also Section 5.11. 

They suggested the effect parameter the Critical Effect Dose (CED, a benchmark dose. Section 4.2.5) derived from the dose-response data by regression analysis. This CED was defined as the dose at which the average animal shows the Critical Effect Size (CES) for a particular toxicological endpoint, below which there is no reason for concern. The distribution of the CED can probabilistically be combined with probabilistic distributions of assessment factors for deriving standards... 

Characterization of a databased distribution is only available at present for the interspecies factor (Section 5.3.3) and for the factor for duration of exposure (Section 5.6). A probabilistic distribution has also been proposed for the interindividual factor (Section 5.4.2). 

Like many data, emission and exposure data are presented as constant values, often a mean with standard deviation. In environmental risk assessment, however, awareness is growing that a stochastic or probabilistic approach is more suitable to obtain insight in the possible risk of chemicals. This also requires expressing exposure data as statistical, probabilistic distributions. Also in this case, the focus should be extended to mixtures. 

Improve methods for identification of the probabilistic distribution of short-and long-term exposure of possible chemical mixtures for ecosystems and humans. 

The SVM method, introduced by Vapnik (32) in 1995, is applicable for both classification and regression problems. In case of classification, SVM are used to determine a boundary, a hyperplane, which separates classes independently of the probabilistic distributions of samples in the data set and maximizes the distance between these classes. The decision boundary is determined calculating a function f(x) = y(x) (32-34). The technique is gaining popularity fast in... 

A thermodynamic system in thermal equilibrium at a temperature T has its energy probabilistically distributed among all the different energy states E. At any temperature (thermal energy kT) there is a finite probability P(E) of the system being in a high energy state... 

This section illustrates application of the mass balance technique for two exploration prospects in the East Texas Basin. Gas volumes were computed using both the conventional reservoir engineering and mass balance approaches. Furthermore, uncertainty was incorporated into the calculations with a Monte Carlo simulation technique that generates probabilistic distributions of gas volumes. 

Conditional and Joint Pmbability Distributions In addition to its graphical structure, a Bayesian network needs to be speerfied by the conditional probability distribution of eaeh node given its parents. Let A and D be variables of interest with a direct causal (parental) relationship in Example 11.6. This relationship can be represented by a conditional probabUity distribution P D A) which represents the probabilistic distribution of child node D given the information of parent node A. When both child and parent nodes arc discrete variables, a contingeney table can summarize the conditional probabdities for aU possible states given each of its parent node states. For continuous variables, a eonditional probability density function needs to be defined. For the combination of continuous and discrete nodes, a mixture distribution, for example, mixture normal distribution, will be required (Imoto et al., 2002). 

Figure 8.A.I. Orientation of a proximal water molecule relative to the solute surfaee normal. Two moleeules, each with four equivalent HB vectors, are shown sehematieally with elathrate-like and inverted orientations, respectively. Of the elathrate-like ease, three out of four of the angles 0 are tetrahedral (0,) and the remaining angle is 0°. Sueh orientation leads to probabilistic distribution of COS0 maximizing at -0.336 and 1. In contrast, inverted orientation would lead to a eos0 distribution mirroring that of the former and maximizing at -1 and 0.336. Adapted with permission from Biophys. J., 76 (1999), 1734-1743. Copyright (1999) Elsevier.

Kinetic theory was formulated to model the conversion degree of a material from one state to another. At each temperature, a FRP material can be considered as a mixture of materials in different states, with changing mechanical properties. The content of each state varies with temperature, thus the composite material shows temperature-dependent properties. If the quantity of material in each state is known and a probabilistic distribution function accounting the contribution from each material state to the effective properties of the mixture is available, the mechanical properties of the mixture can be estimated over the whole temperature range. 

Given the absence of statistics for occurred events, the only way to estimate the failure probability of nuclear vessels is by an analytical way on the basis of the probabilistic distribution of the involved parameters and of the available fracture mechanics models. The relevant parameters include toughness of the material, the number of cracks initially present in the component, the probability that they are detected during the pre-operational and in-service tests, the fatigue crack growth rate, etc. 

The climatic actions may often be described by the Extreme value distributions. The characteristic value of a climatic action is defined in Eurocodes as the upper fractile of the probabilistic distribution for the basic time period corresponding to the 2% probabihty of atmual exceeding. The design value of a climatic action is considered as 0,996% fractile of the probabilistic distribution for structures in the reUabUity class RC2. 

It is shown that for thermal actions on bridges the Weibull distribution is often fitting well as the skewness a of the probabilistic distribution based on evaluated statistical data is considerably less (0,1 to 0,6) than the skewness of Gumbel distribution. 

The partial factor yt specified for thermal actions versus the coefficient of variation Vf for considered Gumbel, Weibull and Lognormal distributions is illustrated in Figure 1. It is shown that for the same coefficient of variation the value of y factor significantly depends on apphed probabilistic distribution. 

When probabilistic distribution of// is close to normal one, according to Equation (39), the probability... 

Cooke 2001) including prior training and calibration steps in order to retrieve the expertise in the form of probabilistic distributions, quantiles etc. and possibly organize posterior consensus. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

At the end of nineteenth century, based on a concept of probabilistic distribution of energy states, Ludwig Boltzmann (1844-1906) presented an innovative interpretation that the entropy S is the most feasible number of microscopic energy states... 

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