Statistical foundations

H. J. Kme2er, Nonequilibrium Thermodynamics and its Statistical Foundations, Clarendon Press, Oxford, 1981. 

S. Blenk, H. Ehrentraut, W. Muschik. Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation. Physica A 77 119-138, 1991. 

MSN.77. 1. Prigogine, Microscopic aspects of entropy and the statistical foundations of nonequilibrium thermodynamics. Proceedings, International Symposium on Foundations of Continuum Thermodynamics, Bussaco, Delgado Domingo, M. N. R. Nina and J. H. Whitelaw, eds., Lisboa, 1974,... 

H. J. Kreuzer (1981) Nonequilibrium Thermodynamics and Its Statistical Foundations, Oxford University Press, London. 

It is a well-understood method with a solid statistical foundation. 

Even neglecting (i) and (ii) above, the description s statistical foundations... 

The required statistical foundations have been outlined in chapter 1.3. Basically, two approaches are open, the canonical and the grand canonical. In the former, the adsorbate is considered as a closed system, characterized by the number of adsorbate molecules N. the area A and the temperature T. By model considerations, the canonical partition function Q(N.A.T) is formulated from... 

Let us. before giving illustrations, discuss some aspects of the statistical foundations. Model assumptions regarding the mode of adsorption (patchwise or random mobile or localized mono- or multilayer with or without lateral interaction ) are reflected in the natures of the local partition functions and in the way they combine to Q[N,N, T). Again, no general solution can be given models of different degrees of sophistication can be developed. 

The estimates in Item 1 are then computed along with any of Items 2, 3. and 4 that the user requests. Thus, NPHIV(JMOD).GT.O is used in the call of GREGPLUS to request auxiliary function computations for model JMOD KVAR(0).NE.0 to request calculations of goodness of fit and posterior probability share for each model considered and JNEXT, NMOD. and IDSIGN control selection of the next event condition. These calculations are described below, and their statistical foundations are presented in Chapters 6 and 7. 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

The statistical foundation of the cell model has been studied by J. Kirkwood.il Some further improvements have been recently introduced by J. E. Mayer and G. Careri. j... 

DOE is the fastest route to a profitable, reliable, robust, validated process. DOE s requirement of a rigorous design methodology that passes peer review with the scientists most knowledgeable about the process ensures scientific soundness. The depth of DOE s statistical foundation that enables the measurement of multiple effects and interactions in a single set of experiments proves DOE s statistical validity. DOE is also a resource conservator, since it requires less time and... 

Yoon, C. McGraw, R. 2004a Representation of generally mixed multivariate aerosols by the quadrature method of moments I. Statistical foundation. Journal of Aerosol Science 35, 561-576. 

Three popnlar early methods are dne to Chon and Fasman, Ganuer et al., and Lim." ° The Chon-Fasman method is based on statistical propensities of residues to form an a-helix or a P-strand, combined with a series of rules. Many methods, like Chon-Fasman, are based on statistical analyses of structural databases. The GOR method adopts an information theory approach. Although the theory is a little daunting, this is transparent to the user and the method has a firm statistical foundation that permitted extension to more subtle analyses of a larger structural database." Methods may also be based on the physico-chemical principles underlying protein structure. For example, a-helices often have a distinct pattern of hydrophobic residues, leading to a hydrophobic face that can facilitate the packing of the helix in the protein. A series of such rules forms the basis of the Lim method. ... 

The transfer to a quasi-steady adsorption allows to transfer Eq. (7.75) into Eq. (7.5). Thus, the attempt of Mikhailovskij is the non-equilibrium statistical foundation of the macro-kinetic approach used throughout this chapter. 

S. Golden, Quantum Statistical Foundations of Chemical Kinetics, Qarendon Press, Oxford, 1969. 

I. Statistical Foundation Probability theories (Chapter 2), statistical quality... 

First, we briefly review statistical foundations of the Monte Carlo (MC) integration. The heart of the MC integration is the central limit theorem, which states the characterization for the average of random samples. Suppose d-dimensional space, the corresponding position vector X = (xi, X2,..., Xd), and a set of mutually independent M position vectors X, / = 1,2,..., M which are distributed according to the probability density P(X). The probability density P(X) should satisfy the following relations. 

Cheminformatics is a relatively new disciphne that encompasses a number of different fields. Many people come to cheminformatics from other fields, or use cheminformatic methods as an adjunct to experimental work. Standards for how cheminformatics professionals should be trained are still emerging. Hopefully as the field matures, practitioners will begin to appreciate the importance of a firm grounding in Statistics. Without an appreciation for the statistical foundations in our methods, it is difficult to see how our field will progress. This paper should not be considered a comprehensive guide to model building or evaluation. The objective here was just to point out a few commonly encountered pitfalls. The interested reader is urged to consult any of a number of excellent Biostatistics or data analysis texts [31,49]. Another excellent additional source of information on model evaluation and comparison is the recent work of Anthony Nicholls [50-52]. 

Golden, S. (1969). Quantum statistical foundation of chemical kinetics. Clarendon, Oxford. 

Considerable progress has been made in developing size scaling methods however, statistical foundations for the other areas are lacking. As a result, confidence intervals cannot be calculated for component strength and life predictions. 

The statistical foundation of rate theory is as follows the E value which characterizes the band spreading results from the contribution of some elementary processes. The concentration distribution throughout the band is described by a Gaussian relationship (eqn (2.14)) whose width is given by the standard deviation (eqn (2.16)). For random processes, according to the theory of random evolution, a, the standard deviation is given by ... 

Chapter 7 is concerned with the statistical foundation of population balance models. The chapter deals with master density formulations leading to mean field equations for the average behavior of the system and fluctuations about average behavior. This represents the subject of stochastic population dynamics applicable to small systems the relevance of which to engineering is discussed. Departures of the mean field equations from population balance equations are demonstrated. The mean field equations so obtained suffer from lack of closure. Closure approximations are presented suggesting more complex mean field equations than population balance along with applications. 

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