Gamma distribution models

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... 

TABLE 41.6 Example NONMEM Code 2 Cell Transit Gamma Distribution Model... 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

An Excel spreadsheet (Example8-7.xls) was used to determine the various RTD functions and the computer program PROGS 1 was used to simulate the model response curve with the experimental data. The results show the equivalent number of ideally mixed stages (nCSTRs) for the RTD is 13.2. The Gamma distribution function from Equation 8-143 is ... 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

The exponential distribution with parameter X is the distribution of waiting times ( distance in time) between events which take place at a mean rate of X. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and X l, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution /(t), such as... 

Sun.Y.N. and W.J. Jusko. 1998. Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics. 

One possible model choice for p(k) that is of widespread use in statistical applications, because of its simplicity and flexibility, is the two-parameter gamma distribution 13... 

It is also worthy of mention that a gamma distribution function proposed by Djordjevic [118] for modeling in vitro dissolution profiles implies a relevant type of time dependency for the amount of drug dissolved. 

Figure 8.2 One-compartment model with gamma-distributed elimination flow rate k Gam(2, 2). The solid line represents the expected profile E[q(t)], and dashed lines, the confidence intervals E [q (t) JVar [<7 ( )].

We report the one-compartment probabilistic transfer model receiving the drug particles by an absorption process. In this model, the elimination rate h was fixed and the absorption constant hev was random. For the stochastic context, the difference hev — h = w is assumed to follow the gamma distribution, i.e., W Gam(A, //.) with density / (w, A, //.) and E [W] =... 

This approach is presented for the two-compartment model of Section 9.2.7. At the second level in (9.16), we assume that A is a gamma-distributed random variable, A Gam(A2,p2)- The Laplace transform of the state probability is... 

Djordjevic, A. and Mendas, I., Method for modelling in vitro dissolution profiles of drugs using gamma distribution, European Journal of Pharmaceutics and Biopharmaceutics, Vol. 44, No. 2, 1997, pp. 215-217. 

Depending on the distribution chosen, as few as three fitting parameters may be required to define a distribution of diffusion rates. In some cases, a single distribution was used to describe both fast and slow rates of sorption and desorption, and in other cases fast and slow mass transfer were captured with separate distributions of diffusion rates. For example, Werth et al. [42] used the pore diffusion model with nonlinear sorption to predict fast desorption, and a gamma distribution of diffusion rate constants to describe slow desorption. 

If we let MLk and MGk denote the model assumptions that the expression data of gene k follow either a lognormal or a Gamma distribution, the posterior probability... 

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