Applied statistics moments

In recent years, non-compartmental or model-independent approaches to pharmacokinetic data analysis have been increasingly utilized since this approach permits the analysis of data without the use of a specific compartment model. Consequently, sophisticated, and often complex, computational methods are not required. The statistical or non-compartmental concept was first reported by Yamaoka in a general manner and by Cutler with specific application to mean absorption time. Riegelman and Collier reviewed and clarified these concepts and applied statistical moment theory to the evaluation of in vivo absorption time. This concept has many additional significant applications in pharmacokinetic calculations. 

The alternative to compartmental analysis is statistical moment analysis. We have already indicated that the results of this approach strongly depend on the accuracy of the measurement process, especially for the estimation of the higher order moments. In view of the limitations of both methods, compartmental and statistical, it is recommended that both approaches be applied in parallel, whenever possible. Each method may contribute information that is not provided by the other. The result of compartmental analysis may fit closely to the data using a model that is inadequate [12]. Statistical moment theory may provide a model which is closer to reality, although being less accurate. The latter point has been made in paradigmatic form by Thom [13] and is represented in Fig. 39.16. 

Applying a chromatographic method it is sometimes possible to separate copolymer molecules according to their size Z and composition [5]. The SCD found in such a way can be compared with that calculated within the framework of the chosen kinetic model. The first- and second-order statistical moments of SCD are of special importance. 

The aim of the paper was to describe the process of grinding of raw materials used in the industrial-scale production of ceramic tiles, by applying the theory of statistical moments. Grinding was performed in industrial ball mills in ceramic tile factories Ceramika Paradyz Ltd. and Opoczno S.A. The ball mills operated in a batch mode. A mixture of feldspars and clay was comminuted. Its composition and fractions depended on the conditions that should be satisfied by raw materials for the production of wall tiles (monoporosis and stoneware) and terracotta. The ground material was subjected to a particle size analysis. Results of the analysis were used in the calculation of relationships applied in the theory of statistical moments. The main parameters, i.e. zero moment of the first order and central moments of the third and fourth order were determined. The values of central moments were used in the calculation of skewness and flatness coefficients. Additionally, changes of mean particle size in time were determined. 

In the absence of external fields the suspension under consideration is macroscopically isotropic (W = const). The applied field h (we denote it in the same way as above but imply the electric field and dipoles as well as the magnetic ones), orienting, statically or dynamically, the particles, thus induces a uniaxial anisotropy, which is conventionally characterized by the orientational order parameter tensor (Piin h)) defined by Eq. (4.358). (We remind the reader that for rigid dipolar particles there is no difference between the unit vectors e and .) As in the case of the internal order parameter S2, [see Eq. (4.81)], one may define the set of quantities (Pi(n h)) for an arbitrary l. Of those, the first statistical moment (Pi) is proportional to the polarization (magnetization) of the medium, and the moments with / > 2, although not having meanings of directly observable quantities, determine those via the chain-linked set [see Eq. (4.369)]. 

For the quantitative description of the sequence distribution in the multicomponent copolymers the statistical characteristics similar to the ones applied for the description of the binary copolymerization products were used [45, 104-110]. In their well-known paper [45] Alfrey and Goldfinger stated an exponential character of the run distribution f (n) for length n with copolymerization of any number m of monomer types. Besides this distribution and its statistical moments [104-107] other parameters as alternation degree were introduced [108,107], which equals the overall fraction of all heterotriads P(M Mj) (i 4= j), and also the parameter with the similar meaning called alternating order [109]. Tosi [110] suggested to use informational entropy as a quantitative measure of the randomness of the multicomponent copolymers ... 

We first review the fundamentals of small particle statistics as these apply to synthetic polymers. This is mainly concerned with the use of statistical moments to characterize molecular weight distributions. One of the characteristics of such a distribution is its central tendency, or average, and the following main topic shows how it is possible to determine various of these averages from measure-... 

An important limitation of compartment analysis is that it cannot be applied universally to any drug. A simpler approach that is useful in the case of bioequivalency testing is the model independent method. It is based on statistical-moment theory. This approach uses the mean residence time (MRT) as a measure of a statistical half-life of the drug in the body. The MRT can be calculated by dividing the area under the first-moment curve (AUMC) by the area under the plasma curve (AUC). ... 

Now, on applying the classical trajectory method, one should perform averaging over all solutions (49) and (52) to calculate the required statistical moments. Here, we calculate the first and second-order field intensity moments necessary for determination of the Fano factors. The mean intensities of the fundamental and harmonic modes are simply given by h = N2r2 and % = r2, respectively. The second-order moments of field intensity are found to be... 

Meaningful chromatographic data can be extracted from asymmetric peaks by digital integration or curve fitting routines applied to the chromatographic peak profile. The statistical moments of a chromatographic peak in units of time are defined by Eq. (1.37) to (1.39) [153,184,187,188]... 

Once the random field involved in the stochastic boundary value problem has been discretized, a solution method has to be adopted in order to solve the boundary value problem numerically. The choice of the solution method depends on the required statistical information of the solution. If only the first two statistical moments of the solution are of interest second moment analysis), the perturbafion method can be applied. However, if a. full probabilistic analysis is necessary, Galerkin schemes can be utilized or one has to resort to Monte Carlo simulations eventually in combination with a von Neumann series expansion. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

A recent breakthrough in molecular theory of hydrophobic effects was achieved by modeling the distribution of occupancy probabilities, the pn depicted in Figure 4, rather than applying a more difficult, direct theory of po for cavity statistics for liquid water (Pohorille and Pratt, 1990). This information theory (IT) approach (Hummer et al., 1996) focuses on the set of probabilities pn of finding n water centers inside the observation volume, with po being just one of the probabilities. Accurate estimates of the pn, and po in particular, are obtained using experimentally available information as constraints on the pn. The moments of the fluctuations in the number of water centers within the observation volume provide such constraints. 

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