Distributed model examples

THE SIMPLEST DISTRIBUTED MODEL Example 1 The Tubular Reactor... 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

Confidence levels on the reliability estimates from the SSI model can be determined and are useful when the PDFs for stress and strength are based on only small amounts of data or where critical reliability projects are undertaken. However, approaches to determine these confidence levels only strictly apply when stress and strength are characterized by the Normal distribution. Detailed examples can be found in Kececioglu (1972) and Sundararajan and Witt (1995). 

Budgets and cycles can be considered on very different spatial scales. In this book we concentrate on global, hemispheric and regional scales. The choice of a suitable scale (i.e. the size of the reservoirs), is determined by the goals of the analysis as well as by the homogeneity of the spatial distribution. For example, in carbon cycle models it is reasonable to consider the atmosphere as one reservoir (the concentration of CO2 in the atmosphere is fairly uniform). On the other hand, oceanic carbon content and carbon exchange processes exhibit large spatial variations and it is reasonable to separate the... 

For a limited number of exposure pathways (primarily inhalation of air in the vicinity of sources), pollutant fate and distribution models have been adapted to estimate population exposure. Examples of such models include the SAI and SRI methodologies developed for EPA s Office of Air Quality Planning and Standards (1,2), the NAAQS Exposure Model (3), and the GEMS approach developed for EPA s Office of Toxic Substances (4). In most cases, however, fate model output will serve as an independent input to an exposure estimate. 

Although it is possible to derive a PDF transport equation for stochastic model for the Fagrangian turbulence frequency a> (t) is developed along the lines of those discussed in Section 6.7. The goal of these models is to reproduce as many of the relevant one-point, two-time statistics of the Fagrangian fluid-particle turbulence frequency, o>+(t), as possible. Examples of two such models (log-normal model (Jayesh and Pope 1995) and gamma-distribution model (Pope and Chen 1990 Pope 1991a Pope 1992)) can be found in Pope (2000). Here we will... 

The measurement of fluorescence lifetimes is an integral part of the anisotropy, energy transfer, and quenching experiment. Also, the fluorescence lifetime provides potentially useful information on the fluorophore environment and therefore provides useful information on membrane properties. An example is the investigation of lateral phase separations. Recently, interest in the fluorescence lifetime itself has increased due to the introduction of the lifetime distribution model as an alternative to the discrete multiexponential approach which has been prevalent in the past. 

In the following model example, we assume that each species involved in the binding process has a spherical shape and that the FGs on its surface are distributed in such a way that each pair of FGs on the surface (i.e., exposed to the solvent) is independently solvated. In other words, the conditional solvation Gibbs energy of the ith FG (given the hard core H) is independent of the presence or absence of any other FGs. Formally, this is equivalent to taking only the first sum over i in the expansion on the rhs of Eq. (9.4.2). 

The most simple model for the dielectric properties of foods is called the distributive model. Here, the dielectric properties of each constituent of the food are added together according to their fractional make-up of the total product. The model assumes that the various constituents of the food are distributed uniformly throughout the product. For example, Figure 3 shows the total dielectric loss factor for a 0.5 molar aqueous solution of water at two temperatures. Note that the total loss factor, e"t is the sum of the ionic and polar contributions, e"c and e" An example of loss factor properties of mustard, ketchup, mayonnaise and water is shown in Figure 8. A comparison of food constituents important in determining dielectric properties is shown in Table 2, (USDA1963). 

Single-sourcing restrictions can be employed between different echelons of a supply chain. Their objective often is to reduce the complexity of the supply network. For example, in a production-distribution model a restriction might be included to ensure that each customer is only served by one distribution center (e.g., Tsiakis et al. 2001, p. 3590 Geoffrion and Graves 1974, p. 823). 

Distribution and density Mean Variance Model Example... 

Recently NOESY MAS was used to study molecular motions in technically relevant materials such as rubbers [46, 47]. For the evaluation of these parameters, it is necessary to understand the cross-relaxation process in the presence of anisotropic motions and under sample spinning. Such a treatment is provided in [47] and the cross-relaxation rates were found to weakly depend on fast motions in the Larmor-frequency range and strongly on slow motions of the order of the spinning frequency vR. Explicit expressions for the vR dependent cross-relaxation rates were derived for different motional models. Examples explicitly discussed were based on a heterogeneous distribution of correlation times [1,8,48] or on a multi-step process in the most simple case assuming a bimodal distribution of correlation times [49-51]. 

In general, a covariate distribution model considers only the covariates influencing the PK and/or PD of the compound of interest. For example, if the covariates age, sex, and weight are identified as important covariates than correlated covariates like height, body mass index, and others might not be incorporated. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

To determine precision, we need to know something about the manner in which data is customarily distributed. For example, high precision (i.e., the data are very close together) produces a very narrow distribution, while low precision (i.e., the data are spread far apart) produces a wide distribution. Assuming that the data are normally distributed (which holds true for many cases and can be used as an approximation in many other cases) allows us to use the well understood mathematical distribution known as the normal or Gaussian error distribution. The advantage to using such a model is that we can compare the collected data with a well understood statistical model to determine the precision of the data. 

Recently, Gillespie (2001) introduced an approximate approach, termed the r-leap method, for solving stochastic models. The main idea is the same as in the WP-KMC method. One selects a time increment r that is larger than the microscopic KMC time increment, and multiple molecular bundles of fast events occur. However, one now samples how many times each reaction will be executed from a Poisson rather than a uniform random number distribution. Prototype examples indicate that the r-leap method provides comparable noise with the microscopic KMC when the leap condition is satisfied, i.e., the time increments are such that the populations do not change significantly between time steps. 

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